Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe ›...

7
Research Article Numerical Solution of Dispersive Optical Solitons with Schrödinger-Hirota Equation by Improved Adomian Decomposition Method H. O. Bakodah , 1 M. A. Banaja , 1 A. A. Alshaery, 1 and A. A. Al Qarni 2 1 Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia 2 Department of Mathematics, Faculty of Science, University of Bisha, Blqarn Campus, Saudi Arabia Correspondence should be addressed to A. A. Al Qarni; [email protected] Received 14 February 2019; Accepted 21 April 2019; Published 27 May 2019 Academic Editor: Mariano Torrisi Copyright © 2019 H. O. Bakodah et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we present new numerical results for the dispersive optical soliton solutions of the nonlinear Schr¨ odinger-Hirota equation. e spatio-temporal dispersion term is included, in addition to group velocity dispersion Kerr law of nonlinearity are studied. A general recursive numerical scheme for the equation is devised via the Improved Adomian Decomposition Method (IADM) and further sought for some analytical results for validation. e scheme is shown to be efficient and possessed high level of accuracy as demonstrated. 1. Introduction Nonlinear Schrodinger equations (NLSE) play a significant role in the fiber optics, optical soliton, prorogation of pulses in metamaterials and network engineering applications among others [1, 2]. us, different forms of NLSE exist and serve for a variety of purposes mostly in communication and networking engineering including, for example, the trans- continental and trans-oceanic data transfer requiring phase modulation [3] and beyond; see [4–16] for different math- ematical studies on these equations. Furthermore, due to their immense application, many analytical methods and few numerical methods have been proposed in the past decades. Such methods include the Backlund transformation method [17], Hirota's direct method [18, 19], tanh-sech method [20, 21], extended tanh method [22, 23], sine-cosine method [24], and homogeneous balance method [25]. However, we will propose in this paper a numerical method for solving a class of NLSE called the nonlin- ear Schrodinger-Hirota equation. e method will heavily depend on the Improved Adomian Decomposition Method (IADM) [26–31]. e IADM is an efficient numerical method for functional and integral solutions based on the Adomian decomposition method [32]. Further, a recent analytical study on the nonlinear Schrodinger-Hirota equation will be sought for to validate the proposed scheme. is form of solution is being reported for the first time in this paper. 2. Governing Equation e nonlinear Schrodinger equation used in modeling prop- agation of solitons through optical fibers with ird-Order Dispersion (TOD) is given by + 1 2 + || 2 = − , (1) where = (, ) is a complex-valued function of (space) and (time); is the coefficient of TOD. e first term on the leſt hand side is the linear temporal evolution, the second term is the group velocity dispersion term, and the third term accounts for Kerr law nonlinearity. Also when the group velocity is low, the TOD is justified. We now introduced the Lie symmetry concept to be able to study (1). With = (, ), let = − 3 [ + 2 ∫ −∞ () 2 , (2) Hindawi Mathematical Problems in Engineering Volume 2019, Article ID 2960912, 6 pages https://doi.org/10.1155/2019/2960912

Transcript of Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe ›...

Page 1: Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe › 2019 › 2960912.pdfMathematicalProblemsinEngineering [] M. Younis, H. U. Rehman,

Research ArticleNumerical Solution of Dispersive OpticalSolitons with Schroumldinger-Hirota Equation byImproved Adomian Decomposition Method

H O Bakodah 1 M A Banaja 1 A A Alshaery1 and A A Al Qarni 2

1Department of Mathematics Faculty of Science University of Jeddah Jeddah Saudi Arabia2Department of Mathematics Faculty of Science University of Bisha Blqarn Campus Saudi Arabia

Correspondence should be addressed to A A Al Qarni alyaa14092009hotmailcom

Received 14 February 2019 Accepted 21 April 2019 Published 27 May 2019

Academic Editor Mariano Torrisi

Copyright copy 2019 H O Bakodah et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we present new numerical results for the dispersive optical soliton solutions of the nonlinear Schrodinger-Hirotaequation The spatio-temporal dispersion term is included in addition to group velocity dispersion Kerr law of nonlinearity arestudied A general recursive numerical scheme for the equation is devised via the Improved Adomian Decomposition Method(IADM) and further sought for some analytical results for validation The scheme is shown to be efficient and possessed high levelof accuracy as demonstrated

1 Introduction

Nonlinear Schrodinger equations (NLSE) play a significantrole in the fiber optics optical soliton prorogation of pulses inmetamaterials and network engineering applications amongothers [1 2] Thus different forms of NLSE exist and servefor a variety of purposes mostly in communication andnetworking engineering including for example the trans-continental and trans-oceanic data transfer requiring phasemodulation [3] and beyond see [4ndash16] for different math-ematical studies on these equations Furthermore due totheir immense application many analytical methods and fewnumerical methods have been proposed in the past decadesSuch methods include the Backlund transformation method[17] Hirotas direct method [18 19] tanh-sech method [2021] extended tanh method [22 23] sine-cosine method [24]and homogeneous balance method [25]

However we will propose in this paper a numericalmethod for solving a class of NLSE called the nonlin-ear Schrodinger-Hirota equation The method will heavilydepend on the Improved Adomian Decomposition Method(IADM) [26ndash31]The IADM is an efficient numerical methodfor functional and integral solutions based on the Adomian

decompositionmethod [32] Further a recent analytical studyon the nonlinear Schrodinger-Hirota equation will be soughtfor to validate the proposed scheme This form of solution isbeing reported for the first time in this paper

2 Governing Equation

The nonlinear Schrodinger equation used in modeling prop-agation of solitons through optical fibers with Third-OrderDispersion (TOD) is given by

119894119906119905 + 12119906119909119909 + |119906|2 119906 = minus119894120582119906119909119909119909 (1)

where 119906 = 119906(119909 119905) is a complex-valued function of 119909 (space)and 119905 (time) 120582 is the coefficient of TOD The first term onthe left hand side is the linear temporal evolution the secondterm is the group velocity dispersion term and the thirdterm accounts for Kerr law nonlinearity Also when the groupvelocity is low the TOD is justified

We now introduced the Lie symmetry concept to be ableto study (1) With 119902 = 119902(119909 119905) let

119902 = 119906 minus 3119894120582 [119906119909 + 2119906int119909minusinfin

1003816100381610038161003816119906 (120585)10038161003816100381610038162 119889120585 (2)

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2960912 6 pageshttpsdoiorg10115520192960912

2 Mathematical Problems in Engineering

which transforms (1) to

119894119902119905 + 12119902119909119909 + 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894120582 (119902119909119909119909 + 6 100381610038161003816100381611990210038161003816100381610038162 119902119909) = 0 (3)

In (3) higher order terms are neglected [11 33 34] withKerr law nonlinearity that models transmission of dispersiveoptical solitons through nonlinear fibres We express (3) withgeneral coefficients as

119894119902119905 + 119886119902119909119909 + 119888 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894 (120574119902119909119909119909 + 120590 100381610038161003816100381611990210038161003816100381610038162 119902119909) = 0 (4)

where physically 120590 represents nonlinear dispersion Further(4) happens to be ill-posed due to the group velocity disper-sion term However with the addition of the spatiotemporaldispersion (STD) term (4) possesses a well-posedness [3536] given by

119894119902119905 + 119886119902119909119909 + 119887119902119909119905 + 119888 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894 (120574119902119909119909119909 + 120590 100381610038161003816100381611990210038161003816100381610038162 119902119909) = 0 (5)

where the coefficient 119887 represents STD Finally in presence ofperturbation terms (5) becomes [37ndash39]

119894119902119905 + 119886119902119909119909 + 119887119902119909119905 + 119888 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894 (120574119902119909119909119909+120590 100381610038161003816100381611990210038161003816100381610038162119902119909)= 119894120572119902119909 + 119894120582 (100381610038161003816100381611990210038161003816100381610038162 119902)119909 + 119894V (100381610038161003816100381611990210038161003816100381610038162)119909 119902 (6)

However we consider an analytical 1-soliton solution of (6) asspecial case presented byAlQarni et al [40] for the numericalsimulation sake in Section 4 given by

119902 (119909 119905) = 119860 sech [119861 (119909 minus V119905)] 119890119894(minus119896119897119909+120596119897119905+120579) (7)

where 119896 is the frequencies of the solitons 120596 is the wavenumber 120579 is the phase velocity and119860 and119861 are the amplitudeand width 119861 given respectively by

119860 = radic 2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)119886 minus 119887V120596 minus 3120574119896 (8)

119861 = radic2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)119888 + 120590119896 minus 120582119896 (9)

The constraint conditions for the existence of bright solitonsfrom (7)-(9) are as follows

(119886 minus 119887V120596 minus 3120574119896) (2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)) gt 0 (10)

(119888 + 120590119896 minus 120582119896) (2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)) gt 0 (11)

3 Numerical Method

In [26ndash31] authors introduced the IADM to convert a specialcase of the complex-valued system into a real-valued systemby splitting 119902(119909 119905) as

119902 (119909 119905) = 1199061 + 1198941199062 (12)

where (119906119896 119896 = 1 2) are real functions By substitutingequation (12) into (6) we obtain the following system

1199061119905 + 1198861199062119909119909 + 1198871199062119909119905 + 119888 (11990621 + 11990622) 1199062 + 1205741199061119909119909119909+ 120590 (11990621 + 11990622) 1199061119909 = 1205721199061119909 + 120582 ((11990621 + 11990622) 1199061)119909minus V (11990621119909 + 11990622119909) 1199061

(13)

1199062119905 minus 1198861199061119909119909 minus 1198871199061119909119905 minus 119888 (11990621 + 11990622) 1199061 + 1205741199062119909119909119909+ 120590 (11990621 + 11990622) 1199062119909 = 1205721199062119909 + 120582 ((11990621 + 11990622) 1199062)119909+ V (11990621119909 + 11990622119909) 1199062

(14)

where

1199061 (119909 0) = [119902 (119909 0)]119877 1199062 (119909 0) = [119902 (119909 0)]119868 (15)

The decomposition method [32] decomposes the solutioninto infinite sums of components defined by

1199061 (119909 119905) = infinsum119899=0

1199061119899 (119909 119905) (16)

1199062 (119909 119905) = infinsum119899=0

1199062119899 (119909 119905) (17)

where the components 1199061119899 1199062119899 119899 ge 0 will be determinedrecursively In an operator form with 119871 119905 = 120597120597119905 (14) and (15)become

119871 119905 (1199061 + 1198871199062119909) + 1198861199062119909119909 + 119888 (11990621 + 11990622) 1199062 + 1205741199061119909119909119909+ 120590 (11990621 + 11990622) 1199061119909 = 1205721199061119909 + 120582 ((11990621 + 11990622) 1199061)119909minus V (11990621119909 + 11990622119909) 1199061

(18)

119871 119905 (1199062 minus 11988711199061119909) minus 1198861199061119909119909 minus 119888 (11990621 + 11990622) 1199061 + 1205741199062119909119909119909+ 120590 (11990621 + 11990622) 1199062119909 = 1205721199062119909 + 120582 ((11990621 + 11990622) 1199062)119909+ V (11990621119909 + 11990622119909) 1199062

(19)

Taking the inverse operator 119871minus1119905 to both sides of (18) and (19)gives

1199061 (119909 119905) = 1199061 (119909 0) minus 119887 (1199062119909 (119909 119905) minus 1199062119909 (119909 0))minus 119871minus1119905 1198861199062119909119909 minus 119871minus1119905 1205741199061119909119909119909 + 119871minus1119905 1205721199061119909+ 119871minus1119905 1198601

(20)

1199062 (119909 119905) = 1199062 (119909 0) + 119887 (1199061119909 (119909 119905) minus 1199061119909 (119909 0))+ 119871minus1119905 1198861199061119909119909 minus 119871minus1119905 1205741199062119909119909119909 + 119871minus1119905 1205721199062119909+ 119871minus1119905 1198602

(21)

Mathematical Problems in Engineering 3

Assuming that the nonlinear terms in (20) and (21) arerepresented by the series

1198601 = minus119888 (11990621 + 11990622) 1199062 minus 120590 (11990621 + 11990622) 1199061119909+ 120582 ((11990621 + 11990622) 1199061)119909 + V (11990621119909 + 11990622119909) 1199061

(22)

1198602 = 119888 (11990621 + 11990622) 1199061 minus 120590 (11990621 + 11990622) 1199062119909+ 120582 ((11990621 + 11990622) 1199062)119909 + V (11990621119909 + 11990622119909) 1199062

(23)

1198601119899 1198602119899 are the so-calledAdomian polynomials thatcan be constructed for all forms of nonlinearity accordingto specific algorithms set by Adomian [32] Substituting thenonlinear terms into (22) and (23) and the solution from (16)and (17) into (20) and (21) gives

infinsum119899=0

1199061119899 (119909 119905)= 1199061 (119909 0) minus 119887infinsum

119899=0

(1199062119899 (119909 119905))119909 + 119887 (1199062119899 (119909 0))119909minus 119871minus1119905 119886infinsum

119899=0

(1199062119899 (119909 119905))119909119909minus 119871minus1119905 120574infinsum

119899=0

(1199061119899 (119909 119905))119909119909119909 + 119871minus1119905 120572infinsum119899=0

(1199061119899 (119909 119905))119909+ 119871minus1119905 infinsum119899=0

1198601119899

(24)

infinsum119899=0

1199062119899 (119909 119905)= 1199062 (119909 0) + 119887infinsum

119899=0

(1199061119899 (119909 119905))119909 minus 119887 (1199061119899 (119909 0))119909+ 119871minus1119905 119886infinsum

119899=0

(1199061119899 (119909 119905))119909119909minus 119871minus1119905 120574infinsum

119899=0

(1199062119899 (119909 119905))119909119909119909 + 119871minus1119905 120572infinsum119899=0

(1199062119899 (119909 119905))119909+ 119871minus1119905 infinsum119899=0

1198602119899

(25)

Following the decomposition analysis the following recur-sive relations are introduced

11990610 (119909 119905) = 1199061 (119909 0) + 119887 (1199062 (119909 0))119909 (26)

11990620 (119909 119905) = 1199062 (119909 0) minus 119887 (1199061 (119909 0))119909 (27)

1199061119896+1 (119909 119905) = minus119887 (1199062119896 (119909 119905))119909 minus 119871minus1119905 119886 (1199062119896 (119909 119905))119909119909minus 119871minus1119905 120574 (1199061119896 (119909 119905))119909119909119909+ 119871minus1119905 120572 (1199061119896 (119909 119905))119909 + 119871minus1119905 1198601119898

(28)

1199062119896+1 (119909 119905) = 119887 (1199061119896 (119909 119905))119909 + 119871minus1119905 119886 (1199061119896 (119909 119905))119909119909+ 119871minus1119905 120574 (1199062119899 (119909 119905))119909119909119909+ 119871minus1119905 120572 (1199062119899 (119909 119905))119909 + 119871minus1119905 1198602119898

(29)

Thus we determine 1199061 and 1199062 as follows1199061 = 11990610 + 11990611 + 11990612 + sdot sdot sdot1199062 = 11990620 + 11990621 + 11990622 + sdot sdot sdot (30)

and the overall approximate solution for (6) is obtained bysubstituting the above into (12) coupled to (26)-(29) to get

119902 (119909 119905) = 11990610 + 11990611 + 11990612 + sdot sdot sdot+ 119894 (11990620 + 11990621 + 11990622 + sdot sdot sdot) (31)

4 Numerical Results

In this section we consider three different cases for theSchrodinger-Hirota equation with spatiotemporal dispersiongiven in (6) to illustrate the application of the IADM schemewe presented in the above sectionWe also consider the brightsoliton solution given in (7)-(11) for numerical simulationwith the following fixed parameters 120596 = 1 1198961 = 119896 = 01120579 = 0 119888 = 1 119886 = 05We also represent the graphs for absolutevalue for each of exact solution 119902119890 and approximate solution119902119886Example 1 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 00 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The results and the profiles of this example are presentedin Table 1 and Figure 1

Example 2 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 01 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 2 and Figure 2

4 Mathematical Problems in Engineering

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 1 Comparison of the exact and approximate solution for Example 1 for minus20 le 119909 le 20

Table 1 The absolute error for Example 1 when 119909 = 20119905 119887 = 0120574 = 0 120574 = 06 120574 = 100 00 00 0001 0000010261373 000003209069 00000637830002 0000021764973 000006266170 00001259991303 0000034488427 000009173715 00001866700204 0000048407918 000011934200 00002458179205 0000063498224 000014550173 000030346539

Table 2 The absolute error for Example 2 when 119909 = 20119905 119887 = 01120574 = 0 120574 = 06 120574 = 100 547003times10minus7 75968times10minus7 87637times10minus701 0000013623790 000003154594 00000649859302 0000028096531 000006216823 00001291215103 0000043940472 000009113126 00001915515404 0000061129371 000011845995 00002522976405 0000079635465 000014417984 000031138194

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 2 Comparison of the exact and approximate solution for Example 2 for minus20 le 119909 le 20

Mathematical Problems in Engineering 5

Table 3 The absolute error for Example 3 when 119909 = 20119905 119887 = 06120574 = 0 120574 = 06 120574 = 100 000014373490 000017576127 00001956321901 000014417980 000011583001 00000927555102 000014701256 000005817085 00000075892103 000015221500 000000366844 00001054147304 000015976708 000004859351 00002007345805 000016964674 000009833252 000029356299

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 3 Comparison of the exact and approximate solution for Example 3 for minus20 le 119909 le 20

Example 3 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 06 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 3 and Figure 3

5 Conclusion

In this work an Improved Adomian DecompositionMethod (IADM) has been proposed to solve the nonlinearSchrodinger-Hirota equation in presence of severalHamiltonian type perturbation terms The obtainedresults possess high precision and converged to the exactsolution with less computational efforts It was noticedthat a high accuracy of results is obtained when is thecoefficient of 3OD 120574 = 0 and within the range 0 gt 120574 119887 lt 1(119887 is the coefficient of spatio-temporal dispersion term) ascommented also in [40] It is also clear from Figures 1ndash3 thatthe complete correspondence of the two solutions is indeedremarkable Thus the IADM is an efficient method fornonlinear Schrodinger equations since it shows a high levelof accuracy with lesser computational efforts as compared toother numerical methods This topic is still open for further

researches for different values of parameters and other typesof soliton solutions

Data Availability

All the data used for the numerical simulations and compar-ison purpose have been reported in the tables included andvisualized in the graphical illustrations and nothing is left

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y Kodama and A Hasegawa ldquoIV theoretical foundation ofoptical-soliton concept in fibersrdquo Progress in Optics vol 30 pp205ndash259 1992

[2] V E Zakharov and S Wabnitz Optical Solitons TheoreticalChallenges and Industrial Perspectives Springer New York NYUSA 1999

[3] M J Ablowitz and H Segur Solitons and the Inverse ScatteringTransform SIAM Philadelphia Pa USA 1981

[4] I Bernstein E Zerrad Q Zhou A Biswas and N MelikechildquoDispersive optical solitons with Schrodinger-Hirota equationby traveling wave hypothesisrdquo Optoelectronics and AdvancedMaterials ndash Rapid Communications vol 9 no 5-6 pp 792ndash7972015

[5] G P Agarwal Nonlinear Fiber Optics Academic Press 2001

6 Mathematical Problems in Engineering

[6] M Younis H U Rehman and F Tahir ldquoOptical Gaussonsand dark solitons in directional couplers with spatiotemporaldispersionrdquo Optical and Quantum Electronics vol 49 no 122018

[7] A Biswas ldquoStochastic perturbation of optical solitons inSchrodingerndashHirota equationrdquo Optics Communications vol239 no 4-6 pp 461ndash466 2004

[8] R K Dowluru and P R Bhima ldquoInfluences of third-orderdispersion on linear birefringent optical soliton transmissionsystemsrdquo Journal of Optics (India) vol 40 no 3 pp 132ndash1422011

[9] P DGreen andA Biswas ldquoBright and dark optical solitonswithtime-dependent coefficients in a non-Kerr lawmediardquoCommu-nications inNonlinear Science andNumerical Simulation vol 15no 12 pp 3865ndash3873 2010

[10] A J Mohamad Jawad M D Petkovic and A Biswas ldquoSolitonsolutions of BURgers equations and perturbed BURgers equa-tionrdquoAppliedMathematics and Computation vol 216 no 11 pp3370ndash3377 2010

[11] Y Kodama and A Hasegawa ldquoNonlinear pulse propagationin a monomode dielectric guiderdquo IEEE Journal of QuantumElectronics vol 23 no 5 pp 510ndash524 1987

[12] Y Kodama M Romagnoli S Wabnitz andMMidrio ldquoRole ofthird-order dispersion on soliton instabilities and interactionsin optical fibersrdquoOptics Expresss vol 19 no 3 pp 165ndash167 1994

[13] G-D Lin Y-T Gao X-L Gai and D-X Meng ldquoExtendeddouble Wronskian solutions to the WHItham-Broer-Kaupequations in shallow waterrdquo Nonlinear Dynamics vol 64 no1-2 pp 197ndash206 2011

[14] Q Cao T Zhang K Djidjeli G Price and E H Twizell ldquoSoli-ton solutions of a class of generalized nonlinear Schrodingerequationsrdquo Applied Mathematics vol 12 no 4 pp 389ndash3981997

[15] A K Sarma ldquoA comparative study of soliton switching in a two-and three-core coupler with TOD and IMDrdquoOptik vol 120 no8 pp 390ndash394 2009

[16] S Tinggen ldquoPropagation characteristics of dark soliton studyin optical fibers with slowly decreasing dispersionrdquo ChineseJournal of Computational Physics vol 13 pp 115ndash118 1996

[17] M RMiura Backlund Transformation Springer-Verlag BerlinGermany 1978

[18] RHirota ldquoExact solution of theKorteweg-de-Vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[19] R Hirota The Direct Method in Soliton Theory CambridgeUniversity Press Cambridge UK 2004

[20] W Malfliet and W Hereman ldquoThe tanh method I Exactsolutions of nonlinear evolution and wave equationsrdquo PhysicaScripta vol 54 no 6 pp 563ndash568 1996

[21] A M Wazwaz ldquoThe tanh method for traveling wave solutionsof nonlinear equationsrdquoAppliedMathematics andComputationvol 154 no 3 pp 713ndash723 2004

[22] S A El-Wakil and M A Abdou ldquoNew exact travelling wavesolutions using modified extended tanh-function methodrdquoChaos Solitons amp Fractals vol 31 no 4 pp 840ndash852 2007

[23] R I Nuruddeen ldquoElzaki decomposition method and its appli-cations in solving linear and nonlinear schrodinger equationsrdquoSohag Journal of Mathematics vol 4 no 2 pp 31ndash35 2017

[24] A M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[25] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[26] H O Bakodah M A Banaja A A Alqarni et al ldquoOpticalsolitons in birefringent fibers with adomian decompositionmethodrdquo Journal of Computational andTheoretical Nanosciencevol 12 no 12 pp 5846ndash5853 2015

[27] A A Qarni M A Banaja H O Bakodah M Mirzazadehand A Biswas ldquoOptical solitons with coupled nonlinearschrodingerrsquos equation in birefringent nano-fibers by adomiandecompositionmethodrdquo Journal of Computational andTheoret-ical Nanoscience vol 13 no 8 pp 5493ndash5498 2016

[28] A A Al Qarni M A Banaja H O Bakodah A A Alshaery FB Majid and A Biswas ldquoOptical solitons in birefringent fibersA numerical studyrdquo Journal of Computational and TheoreticalNanoscience vol 13 no 11 pp 9001ndash9013 2016

[29] A Al-Shareef A A Al Qarni S Al-Mohalbadi and HO Bakodah ldquoSoliton solutions and numerical treatment ofthe nonlinear schrodingerrsquos equation using modified adomiandecomposition methodrdquo Journal of Applied Mathematics andPhysics vol 04 no 12 pp 2215ndash2232 2016

[30] H O Bakodah A A Al Qarni M A Banaja Q Zhou S PMoshokoa andA Biswas ldquoBright anddark thirring optical soli-tons with improved adomian decomposition methodrdquo Optikvol 130 pp 1115ndash1123 2017

[31] M A Banaja A A Al Qarni H O Bakodah Q Zhou S PMoshokoa and A Biswas ldquoThe investigate of optical solitons incascaded systemby improved adomiandecomposition schemerdquoOptik vol 130 pp 1107ndash1114 2017

[32] G Adomian ldquoSolution of physical problems by decompositionrdquoComputers amp Mathematics with Applications vol 27 no 9-10pp 145ndash154 1994

[33] A Hasegawa and Y Kodama Solitons in Optical Communica-tions Oxford University Press Oxford UK 1995

[34] S Wabnitz Y Kodama and A B Aceves ldquoControl of opticalsoliton interactionsrdquo Optical Fiber Technology vol 1 no 3 pp187ndash217 1995

[35] X Geng and Y Lv ldquoDarboux transformation for an integrablegeneralization of the nonlinear Schrodinger equationrdquo Nonlin-ear Dynamics vol 69 no 4 pp 1621ndash1630 2012

[36] S Kumar K Singh and R K Gupta ldquoCoupled Higgs fieldequation and Hamiltonian amplitude equation lie classicalapproach and (GrsquoG)-expansion methodrdquo Pramana vol 79 no1 pp 41ndash60 2012

[37] M Inc A I Aliyu A Yusuf andD Baleanu ldquoDispersive opticalsolitons and modulation instability analysis of Schrodinger-Hirota equation with spatio-temporal dispersion and Kerr lawnonlinearityrdquo Superlattices andMicrostructures vol 113 pp 319ndash327 2018

[38] Y Yildirim ldquoOptical solitons to SchrodingerHirota equation inDWDM system with trial equation integration architecturerdquoOptik vol 18 pp 275ndash281 2019

[39] A Biswas Y Yildirim E Yasar et al ldquoDispersive optical solitonswith SchrodingerndashHirota model by trial equation methodrdquoOptik - International Journal for Light and Electron Optics vol162 pp 35ndash41 2018

[40] A A Al Qarni M A Banaja and H O Bakodah ldquoNumericalanalyses optical solitons in dual core couplers with kerr lawnonlinearityrdquo Applied Mathematics vol 06 no 12 pp 1957ndash1967 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 2: Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe › 2019 › 2960912.pdfMathematicalProblemsinEngineering [] M. Younis, H. U. Rehman,

2 Mathematical Problems in Engineering

which transforms (1) to

119894119902119905 + 12119902119909119909 + 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894120582 (119902119909119909119909 + 6 100381610038161003816100381611990210038161003816100381610038162 119902119909) = 0 (3)

In (3) higher order terms are neglected [11 33 34] withKerr law nonlinearity that models transmission of dispersiveoptical solitons through nonlinear fibres We express (3) withgeneral coefficients as

119894119902119905 + 119886119902119909119909 + 119888 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894 (120574119902119909119909119909 + 120590 100381610038161003816100381611990210038161003816100381610038162 119902119909) = 0 (4)

where physically 120590 represents nonlinear dispersion Further(4) happens to be ill-posed due to the group velocity disper-sion term However with the addition of the spatiotemporaldispersion (STD) term (4) possesses a well-posedness [3536] given by

119894119902119905 + 119886119902119909119909 + 119887119902119909119905 + 119888 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894 (120574119902119909119909119909 + 120590 100381610038161003816100381611990210038161003816100381610038162 119902119909) = 0 (5)

where the coefficient 119887 represents STD Finally in presence ofperturbation terms (5) becomes [37ndash39]

119894119902119905 + 119886119902119909119909 + 119887119902119909119905 + 119888 100381610038161003816100381611990210038161003816100381610038162 119902 + 119894 (120574119902119909119909119909+120590 100381610038161003816100381611990210038161003816100381610038162119902119909)= 119894120572119902119909 + 119894120582 (100381610038161003816100381611990210038161003816100381610038162 119902)119909 + 119894V (100381610038161003816100381611990210038161003816100381610038162)119909 119902 (6)

However we consider an analytical 1-soliton solution of (6) asspecial case presented byAlQarni et al [40] for the numericalsimulation sake in Section 4 given by

119902 (119909 119905) = 119860 sech [119861 (119909 minus V119905)] 119890119894(minus119896119897119909+120596119897119905+120579) (7)

where 119896 is the frequencies of the solitons 120596 is the wavenumber 120579 is the phase velocity and119860 and119861 are the amplitudeand width 119861 given respectively by

119860 = radic 2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)119886 minus 119887V120596 minus 3120574119896 (8)

119861 = radic2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)119888 + 120590119896 minus 120582119896 (9)

The constraint conditions for the existence of bright solitonsfrom (7)-(9) are as follows

(119886 minus 119887V120596 minus 3120574119896) (2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)) gt 0 (10)

(119888 + 120590119896 minus 120582119896) (2 (120596 + 119886119896 minus 120596119896119887 + 1198861198962 + 1205741198963)) gt 0 (11)

3 Numerical Method

In [26ndash31] authors introduced the IADM to convert a specialcase of the complex-valued system into a real-valued systemby splitting 119902(119909 119905) as

119902 (119909 119905) = 1199061 + 1198941199062 (12)

where (119906119896 119896 = 1 2) are real functions By substitutingequation (12) into (6) we obtain the following system

1199061119905 + 1198861199062119909119909 + 1198871199062119909119905 + 119888 (11990621 + 11990622) 1199062 + 1205741199061119909119909119909+ 120590 (11990621 + 11990622) 1199061119909 = 1205721199061119909 + 120582 ((11990621 + 11990622) 1199061)119909minus V (11990621119909 + 11990622119909) 1199061

(13)

1199062119905 minus 1198861199061119909119909 minus 1198871199061119909119905 minus 119888 (11990621 + 11990622) 1199061 + 1205741199062119909119909119909+ 120590 (11990621 + 11990622) 1199062119909 = 1205721199062119909 + 120582 ((11990621 + 11990622) 1199062)119909+ V (11990621119909 + 11990622119909) 1199062

(14)

where

1199061 (119909 0) = [119902 (119909 0)]119877 1199062 (119909 0) = [119902 (119909 0)]119868 (15)

The decomposition method [32] decomposes the solutioninto infinite sums of components defined by

1199061 (119909 119905) = infinsum119899=0

1199061119899 (119909 119905) (16)

1199062 (119909 119905) = infinsum119899=0

1199062119899 (119909 119905) (17)

where the components 1199061119899 1199062119899 119899 ge 0 will be determinedrecursively In an operator form with 119871 119905 = 120597120597119905 (14) and (15)become

119871 119905 (1199061 + 1198871199062119909) + 1198861199062119909119909 + 119888 (11990621 + 11990622) 1199062 + 1205741199061119909119909119909+ 120590 (11990621 + 11990622) 1199061119909 = 1205721199061119909 + 120582 ((11990621 + 11990622) 1199061)119909minus V (11990621119909 + 11990622119909) 1199061

(18)

119871 119905 (1199062 minus 11988711199061119909) minus 1198861199061119909119909 minus 119888 (11990621 + 11990622) 1199061 + 1205741199062119909119909119909+ 120590 (11990621 + 11990622) 1199062119909 = 1205721199062119909 + 120582 ((11990621 + 11990622) 1199062)119909+ V (11990621119909 + 11990622119909) 1199062

(19)

Taking the inverse operator 119871minus1119905 to both sides of (18) and (19)gives

1199061 (119909 119905) = 1199061 (119909 0) minus 119887 (1199062119909 (119909 119905) minus 1199062119909 (119909 0))minus 119871minus1119905 1198861199062119909119909 minus 119871minus1119905 1205741199061119909119909119909 + 119871minus1119905 1205721199061119909+ 119871minus1119905 1198601

(20)

1199062 (119909 119905) = 1199062 (119909 0) + 119887 (1199061119909 (119909 119905) minus 1199061119909 (119909 0))+ 119871minus1119905 1198861199061119909119909 minus 119871minus1119905 1205741199062119909119909119909 + 119871minus1119905 1205721199062119909+ 119871minus1119905 1198602

(21)

Mathematical Problems in Engineering 3

Assuming that the nonlinear terms in (20) and (21) arerepresented by the series

1198601 = minus119888 (11990621 + 11990622) 1199062 minus 120590 (11990621 + 11990622) 1199061119909+ 120582 ((11990621 + 11990622) 1199061)119909 + V (11990621119909 + 11990622119909) 1199061

(22)

1198602 = 119888 (11990621 + 11990622) 1199061 minus 120590 (11990621 + 11990622) 1199062119909+ 120582 ((11990621 + 11990622) 1199062)119909 + V (11990621119909 + 11990622119909) 1199062

(23)

1198601119899 1198602119899 are the so-calledAdomian polynomials thatcan be constructed for all forms of nonlinearity accordingto specific algorithms set by Adomian [32] Substituting thenonlinear terms into (22) and (23) and the solution from (16)and (17) into (20) and (21) gives

infinsum119899=0

1199061119899 (119909 119905)= 1199061 (119909 0) minus 119887infinsum

119899=0

(1199062119899 (119909 119905))119909 + 119887 (1199062119899 (119909 0))119909minus 119871minus1119905 119886infinsum

119899=0

(1199062119899 (119909 119905))119909119909minus 119871minus1119905 120574infinsum

119899=0

(1199061119899 (119909 119905))119909119909119909 + 119871minus1119905 120572infinsum119899=0

(1199061119899 (119909 119905))119909+ 119871minus1119905 infinsum119899=0

1198601119899

(24)

infinsum119899=0

1199062119899 (119909 119905)= 1199062 (119909 0) + 119887infinsum

119899=0

(1199061119899 (119909 119905))119909 minus 119887 (1199061119899 (119909 0))119909+ 119871minus1119905 119886infinsum

119899=0

(1199061119899 (119909 119905))119909119909minus 119871minus1119905 120574infinsum

119899=0

(1199062119899 (119909 119905))119909119909119909 + 119871minus1119905 120572infinsum119899=0

(1199062119899 (119909 119905))119909+ 119871minus1119905 infinsum119899=0

1198602119899

(25)

Following the decomposition analysis the following recur-sive relations are introduced

11990610 (119909 119905) = 1199061 (119909 0) + 119887 (1199062 (119909 0))119909 (26)

11990620 (119909 119905) = 1199062 (119909 0) minus 119887 (1199061 (119909 0))119909 (27)

1199061119896+1 (119909 119905) = minus119887 (1199062119896 (119909 119905))119909 minus 119871minus1119905 119886 (1199062119896 (119909 119905))119909119909minus 119871minus1119905 120574 (1199061119896 (119909 119905))119909119909119909+ 119871minus1119905 120572 (1199061119896 (119909 119905))119909 + 119871minus1119905 1198601119898

(28)

1199062119896+1 (119909 119905) = 119887 (1199061119896 (119909 119905))119909 + 119871minus1119905 119886 (1199061119896 (119909 119905))119909119909+ 119871minus1119905 120574 (1199062119899 (119909 119905))119909119909119909+ 119871minus1119905 120572 (1199062119899 (119909 119905))119909 + 119871minus1119905 1198602119898

(29)

Thus we determine 1199061 and 1199062 as follows1199061 = 11990610 + 11990611 + 11990612 + sdot sdot sdot1199062 = 11990620 + 11990621 + 11990622 + sdot sdot sdot (30)

and the overall approximate solution for (6) is obtained bysubstituting the above into (12) coupled to (26)-(29) to get

119902 (119909 119905) = 11990610 + 11990611 + 11990612 + sdot sdot sdot+ 119894 (11990620 + 11990621 + 11990622 + sdot sdot sdot) (31)

4 Numerical Results

In this section we consider three different cases for theSchrodinger-Hirota equation with spatiotemporal dispersiongiven in (6) to illustrate the application of the IADM schemewe presented in the above sectionWe also consider the brightsoliton solution given in (7)-(11) for numerical simulationwith the following fixed parameters 120596 = 1 1198961 = 119896 = 01120579 = 0 119888 = 1 119886 = 05We also represent the graphs for absolutevalue for each of exact solution 119902119890 and approximate solution119902119886Example 1 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 00 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The results and the profiles of this example are presentedin Table 1 and Figure 1

Example 2 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 01 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 2 and Figure 2

4 Mathematical Problems in Engineering

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 1 Comparison of the exact and approximate solution for Example 1 for minus20 le 119909 le 20

Table 1 The absolute error for Example 1 when 119909 = 20119905 119887 = 0120574 = 0 120574 = 06 120574 = 100 00 00 0001 0000010261373 000003209069 00000637830002 0000021764973 000006266170 00001259991303 0000034488427 000009173715 00001866700204 0000048407918 000011934200 00002458179205 0000063498224 000014550173 000030346539

Table 2 The absolute error for Example 2 when 119909 = 20119905 119887 = 01120574 = 0 120574 = 06 120574 = 100 547003times10minus7 75968times10minus7 87637times10minus701 0000013623790 000003154594 00000649859302 0000028096531 000006216823 00001291215103 0000043940472 000009113126 00001915515404 0000061129371 000011845995 00002522976405 0000079635465 000014417984 000031138194

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 2 Comparison of the exact and approximate solution for Example 2 for minus20 le 119909 le 20

Mathematical Problems in Engineering 5

Table 3 The absolute error for Example 3 when 119909 = 20119905 119887 = 06120574 = 0 120574 = 06 120574 = 100 000014373490 000017576127 00001956321901 000014417980 000011583001 00000927555102 000014701256 000005817085 00000075892103 000015221500 000000366844 00001054147304 000015976708 000004859351 00002007345805 000016964674 000009833252 000029356299

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 3 Comparison of the exact and approximate solution for Example 3 for minus20 le 119909 le 20

Example 3 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 06 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 3 and Figure 3

5 Conclusion

In this work an Improved Adomian DecompositionMethod (IADM) has been proposed to solve the nonlinearSchrodinger-Hirota equation in presence of severalHamiltonian type perturbation terms The obtainedresults possess high precision and converged to the exactsolution with less computational efforts It was noticedthat a high accuracy of results is obtained when is thecoefficient of 3OD 120574 = 0 and within the range 0 gt 120574 119887 lt 1(119887 is the coefficient of spatio-temporal dispersion term) ascommented also in [40] It is also clear from Figures 1ndash3 thatthe complete correspondence of the two solutions is indeedremarkable Thus the IADM is an efficient method fornonlinear Schrodinger equations since it shows a high levelof accuracy with lesser computational efforts as compared toother numerical methods This topic is still open for further

researches for different values of parameters and other typesof soliton solutions

Data Availability

All the data used for the numerical simulations and compar-ison purpose have been reported in the tables included andvisualized in the graphical illustrations and nothing is left

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y Kodama and A Hasegawa ldquoIV theoretical foundation ofoptical-soliton concept in fibersrdquo Progress in Optics vol 30 pp205ndash259 1992

[2] V E Zakharov and S Wabnitz Optical Solitons TheoreticalChallenges and Industrial Perspectives Springer New York NYUSA 1999

[3] M J Ablowitz and H Segur Solitons and the Inverse ScatteringTransform SIAM Philadelphia Pa USA 1981

[4] I Bernstein E Zerrad Q Zhou A Biswas and N MelikechildquoDispersive optical solitons with Schrodinger-Hirota equationby traveling wave hypothesisrdquo Optoelectronics and AdvancedMaterials ndash Rapid Communications vol 9 no 5-6 pp 792ndash7972015

[5] G P Agarwal Nonlinear Fiber Optics Academic Press 2001

6 Mathematical Problems in Engineering

[6] M Younis H U Rehman and F Tahir ldquoOptical Gaussonsand dark solitons in directional couplers with spatiotemporaldispersionrdquo Optical and Quantum Electronics vol 49 no 122018

[7] A Biswas ldquoStochastic perturbation of optical solitons inSchrodingerndashHirota equationrdquo Optics Communications vol239 no 4-6 pp 461ndash466 2004

[8] R K Dowluru and P R Bhima ldquoInfluences of third-orderdispersion on linear birefringent optical soliton transmissionsystemsrdquo Journal of Optics (India) vol 40 no 3 pp 132ndash1422011

[9] P DGreen andA Biswas ldquoBright and dark optical solitonswithtime-dependent coefficients in a non-Kerr lawmediardquoCommu-nications inNonlinear Science andNumerical Simulation vol 15no 12 pp 3865ndash3873 2010

[10] A J Mohamad Jawad M D Petkovic and A Biswas ldquoSolitonsolutions of BURgers equations and perturbed BURgers equa-tionrdquoAppliedMathematics and Computation vol 216 no 11 pp3370ndash3377 2010

[11] Y Kodama and A Hasegawa ldquoNonlinear pulse propagationin a monomode dielectric guiderdquo IEEE Journal of QuantumElectronics vol 23 no 5 pp 510ndash524 1987

[12] Y Kodama M Romagnoli S Wabnitz andMMidrio ldquoRole ofthird-order dispersion on soliton instabilities and interactionsin optical fibersrdquoOptics Expresss vol 19 no 3 pp 165ndash167 1994

[13] G-D Lin Y-T Gao X-L Gai and D-X Meng ldquoExtendeddouble Wronskian solutions to the WHItham-Broer-Kaupequations in shallow waterrdquo Nonlinear Dynamics vol 64 no1-2 pp 197ndash206 2011

[14] Q Cao T Zhang K Djidjeli G Price and E H Twizell ldquoSoli-ton solutions of a class of generalized nonlinear Schrodingerequationsrdquo Applied Mathematics vol 12 no 4 pp 389ndash3981997

[15] A K Sarma ldquoA comparative study of soliton switching in a two-and three-core coupler with TOD and IMDrdquoOptik vol 120 no8 pp 390ndash394 2009

[16] S Tinggen ldquoPropagation characteristics of dark soliton studyin optical fibers with slowly decreasing dispersionrdquo ChineseJournal of Computational Physics vol 13 pp 115ndash118 1996

[17] M RMiura Backlund Transformation Springer-Verlag BerlinGermany 1978

[18] RHirota ldquoExact solution of theKorteweg-de-Vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[19] R Hirota The Direct Method in Soliton Theory CambridgeUniversity Press Cambridge UK 2004

[20] W Malfliet and W Hereman ldquoThe tanh method I Exactsolutions of nonlinear evolution and wave equationsrdquo PhysicaScripta vol 54 no 6 pp 563ndash568 1996

[21] A M Wazwaz ldquoThe tanh method for traveling wave solutionsof nonlinear equationsrdquoAppliedMathematics andComputationvol 154 no 3 pp 713ndash723 2004

[22] S A El-Wakil and M A Abdou ldquoNew exact travelling wavesolutions using modified extended tanh-function methodrdquoChaos Solitons amp Fractals vol 31 no 4 pp 840ndash852 2007

[23] R I Nuruddeen ldquoElzaki decomposition method and its appli-cations in solving linear and nonlinear schrodinger equationsrdquoSohag Journal of Mathematics vol 4 no 2 pp 31ndash35 2017

[24] A M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[25] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[26] H O Bakodah M A Banaja A A Alqarni et al ldquoOpticalsolitons in birefringent fibers with adomian decompositionmethodrdquo Journal of Computational andTheoretical Nanosciencevol 12 no 12 pp 5846ndash5853 2015

[27] A A Qarni M A Banaja H O Bakodah M Mirzazadehand A Biswas ldquoOptical solitons with coupled nonlinearschrodingerrsquos equation in birefringent nano-fibers by adomiandecompositionmethodrdquo Journal of Computational andTheoret-ical Nanoscience vol 13 no 8 pp 5493ndash5498 2016

[28] A A Al Qarni M A Banaja H O Bakodah A A Alshaery FB Majid and A Biswas ldquoOptical solitons in birefringent fibersA numerical studyrdquo Journal of Computational and TheoreticalNanoscience vol 13 no 11 pp 9001ndash9013 2016

[29] A Al-Shareef A A Al Qarni S Al-Mohalbadi and HO Bakodah ldquoSoliton solutions and numerical treatment ofthe nonlinear schrodingerrsquos equation using modified adomiandecomposition methodrdquo Journal of Applied Mathematics andPhysics vol 04 no 12 pp 2215ndash2232 2016

[30] H O Bakodah A A Al Qarni M A Banaja Q Zhou S PMoshokoa andA Biswas ldquoBright anddark thirring optical soli-tons with improved adomian decomposition methodrdquo Optikvol 130 pp 1115ndash1123 2017

[31] M A Banaja A A Al Qarni H O Bakodah Q Zhou S PMoshokoa and A Biswas ldquoThe investigate of optical solitons incascaded systemby improved adomiandecomposition schemerdquoOptik vol 130 pp 1107ndash1114 2017

[32] G Adomian ldquoSolution of physical problems by decompositionrdquoComputers amp Mathematics with Applications vol 27 no 9-10pp 145ndash154 1994

[33] A Hasegawa and Y Kodama Solitons in Optical Communica-tions Oxford University Press Oxford UK 1995

[34] S Wabnitz Y Kodama and A B Aceves ldquoControl of opticalsoliton interactionsrdquo Optical Fiber Technology vol 1 no 3 pp187ndash217 1995

[35] X Geng and Y Lv ldquoDarboux transformation for an integrablegeneralization of the nonlinear Schrodinger equationrdquo Nonlin-ear Dynamics vol 69 no 4 pp 1621ndash1630 2012

[36] S Kumar K Singh and R K Gupta ldquoCoupled Higgs fieldequation and Hamiltonian amplitude equation lie classicalapproach and (GrsquoG)-expansion methodrdquo Pramana vol 79 no1 pp 41ndash60 2012

[37] M Inc A I Aliyu A Yusuf andD Baleanu ldquoDispersive opticalsolitons and modulation instability analysis of Schrodinger-Hirota equation with spatio-temporal dispersion and Kerr lawnonlinearityrdquo Superlattices andMicrostructures vol 113 pp 319ndash327 2018

[38] Y Yildirim ldquoOptical solitons to SchrodingerHirota equation inDWDM system with trial equation integration architecturerdquoOptik vol 18 pp 275ndash281 2019

[39] A Biswas Y Yildirim E Yasar et al ldquoDispersive optical solitonswith SchrodingerndashHirota model by trial equation methodrdquoOptik - International Journal for Light and Electron Optics vol162 pp 35ndash41 2018

[40] A A Al Qarni M A Banaja and H O Bakodah ldquoNumericalanalyses optical solitons in dual core couplers with kerr lawnonlinearityrdquo Applied Mathematics vol 06 no 12 pp 1957ndash1967 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 3: Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe › 2019 › 2960912.pdfMathematicalProblemsinEngineering [] M. Younis, H. U. Rehman,

Mathematical Problems in Engineering 3

Assuming that the nonlinear terms in (20) and (21) arerepresented by the series

1198601 = minus119888 (11990621 + 11990622) 1199062 minus 120590 (11990621 + 11990622) 1199061119909+ 120582 ((11990621 + 11990622) 1199061)119909 + V (11990621119909 + 11990622119909) 1199061

(22)

1198602 = 119888 (11990621 + 11990622) 1199061 minus 120590 (11990621 + 11990622) 1199062119909+ 120582 ((11990621 + 11990622) 1199062)119909 + V (11990621119909 + 11990622119909) 1199062

(23)

1198601119899 1198602119899 are the so-calledAdomian polynomials thatcan be constructed for all forms of nonlinearity accordingto specific algorithms set by Adomian [32] Substituting thenonlinear terms into (22) and (23) and the solution from (16)and (17) into (20) and (21) gives

infinsum119899=0

1199061119899 (119909 119905)= 1199061 (119909 0) minus 119887infinsum

119899=0

(1199062119899 (119909 119905))119909 + 119887 (1199062119899 (119909 0))119909minus 119871minus1119905 119886infinsum

119899=0

(1199062119899 (119909 119905))119909119909minus 119871minus1119905 120574infinsum

119899=0

(1199061119899 (119909 119905))119909119909119909 + 119871minus1119905 120572infinsum119899=0

(1199061119899 (119909 119905))119909+ 119871minus1119905 infinsum119899=0

1198601119899

(24)

infinsum119899=0

1199062119899 (119909 119905)= 1199062 (119909 0) + 119887infinsum

119899=0

(1199061119899 (119909 119905))119909 minus 119887 (1199061119899 (119909 0))119909+ 119871minus1119905 119886infinsum

119899=0

(1199061119899 (119909 119905))119909119909minus 119871minus1119905 120574infinsum

119899=0

(1199062119899 (119909 119905))119909119909119909 + 119871minus1119905 120572infinsum119899=0

(1199062119899 (119909 119905))119909+ 119871minus1119905 infinsum119899=0

1198602119899

(25)

Following the decomposition analysis the following recur-sive relations are introduced

11990610 (119909 119905) = 1199061 (119909 0) + 119887 (1199062 (119909 0))119909 (26)

11990620 (119909 119905) = 1199062 (119909 0) minus 119887 (1199061 (119909 0))119909 (27)

1199061119896+1 (119909 119905) = minus119887 (1199062119896 (119909 119905))119909 minus 119871minus1119905 119886 (1199062119896 (119909 119905))119909119909minus 119871minus1119905 120574 (1199061119896 (119909 119905))119909119909119909+ 119871minus1119905 120572 (1199061119896 (119909 119905))119909 + 119871minus1119905 1198601119898

(28)

1199062119896+1 (119909 119905) = 119887 (1199061119896 (119909 119905))119909 + 119871minus1119905 119886 (1199061119896 (119909 119905))119909119909+ 119871minus1119905 120574 (1199062119899 (119909 119905))119909119909119909+ 119871minus1119905 120572 (1199062119899 (119909 119905))119909 + 119871minus1119905 1198602119898

(29)

Thus we determine 1199061 and 1199062 as follows1199061 = 11990610 + 11990611 + 11990612 + sdot sdot sdot1199062 = 11990620 + 11990621 + 11990622 + sdot sdot sdot (30)

and the overall approximate solution for (6) is obtained bysubstituting the above into (12) coupled to (26)-(29) to get

119902 (119909 119905) = 11990610 + 11990611 + 11990612 + sdot sdot sdot+ 119894 (11990620 + 11990621 + 11990622 + sdot sdot sdot) (31)

4 Numerical Results

In this section we consider three different cases for theSchrodinger-Hirota equation with spatiotemporal dispersiongiven in (6) to illustrate the application of the IADM schemewe presented in the above sectionWe also consider the brightsoliton solution given in (7)-(11) for numerical simulationwith the following fixed parameters 120596 = 1 1198961 = 119896 = 01120579 = 0 119888 = 1 119886 = 05We also represent the graphs for absolutevalue for each of exact solution 119902119890 and approximate solution119902119886Example 1 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 00 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The results and the profiles of this example are presentedin Table 1 and Figure 1

Example 2 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 01 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 2 and Figure 2

4 Mathematical Problems in Engineering

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 1 Comparison of the exact and approximate solution for Example 1 for minus20 le 119909 le 20

Table 1 The absolute error for Example 1 when 119909 = 20119905 119887 = 0120574 = 0 120574 = 06 120574 = 100 00 00 0001 0000010261373 000003209069 00000637830002 0000021764973 000006266170 00001259991303 0000034488427 000009173715 00001866700204 0000048407918 000011934200 00002458179205 0000063498224 000014550173 000030346539

Table 2 The absolute error for Example 2 when 119909 = 20119905 119887 = 01120574 = 0 120574 = 06 120574 = 100 547003times10minus7 75968times10minus7 87637times10minus701 0000013623790 000003154594 00000649859302 0000028096531 000006216823 00001291215103 0000043940472 000009113126 00001915515404 0000061129371 000011845995 00002522976405 0000079635465 000014417984 000031138194

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 2 Comparison of the exact and approximate solution for Example 2 for minus20 le 119909 le 20

Mathematical Problems in Engineering 5

Table 3 The absolute error for Example 3 when 119909 = 20119905 119887 = 06120574 = 0 120574 = 06 120574 = 100 000014373490 000017576127 00001956321901 000014417980 000011583001 00000927555102 000014701256 000005817085 00000075892103 000015221500 000000366844 00001054147304 000015976708 000004859351 00002007345805 000016964674 000009833252 000029356299

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 3 Comparison of the exact and approximate solution for Example 3 for minus20 le 119909 le 20

Example 3 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 06 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 3 and Figure 3

5 Conclusion

In this work an Improved Adomian DecompositionMethod (IADM) has been proposed to solve the nonlinearSchrodinger-Hirota equation in presence of severalHamiltonian type perturbation terms The obtainedresults possess high precision and converged to the exactsolution with less computational efforts It was noticedthat a high accuracy of results is obtained when is thecoefficient of 3OD 120574 = 0 and within the range 0 gt 120574 119887 lt 1(119887 is the coefficient of spatio-temporal dispersion term) ascommented also in [40] It is also clear from Figures 1ndash3 thatthe complete correspondence of the two solutions is indeedremarkable Thus the IADM is an efficient method fornonlinear Schrodinger equations since it shows a high levelof accuracy with lesser computational efforts as compared toother numerical methods This topic is still open for further

researches for different values of parameters and other typesof soliton solutions

Data Availability

All the data used for the numerical simulations and compar-ison purpose have been reported in the tables included andvisualized in the graphical illustrations and nothing is left

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y Kodama and A Hasegawa ldquoIV theoretical foundation ofoptical-soliton concept in fibersrdquo Progress in Optics vol 30 pp205ndash259 1992

[2] V E Zakharov and S Wabnitz Optical Solitons TheoreticalChallenges and Industrial Perspectives Springer New York NYUSA 1999

[3] M J Ablowitz and H Segur Solitons and the Inverse ScatteringTransform SIAM Philadelphia Pa USA 1981

[4] I Bernstein E Zerrad Q Zhou A Biswas and N MelikechildquoDispersive optical solitons with Schrodinger-Hirota equationby traveling wave hypothesisrdquo Optoelectronics and AdvancedMaterials ndash Rapid Communications vol 9 no 5-6 pp 792ndash7972015

[5] G P Agarwal Nonlinear Fiber Optics Academic Press 2001

6 Mathematical Problems in Engineering

[6] M Younis H U Rehman and F Tahir ldquoOptical Gaussonsand dark solitons in directional couplers with spatiotemporaldispersionrdquo Optical and Quantum Electronics vol 49 no 122018

[7] A Biswas ldquoStochastic perturbation of optical solitons inSchrodingerndashHirota equationrdquo Optics Communications vol239 no 4-6 pp 461ndash466 2004

[8] R K Dowluru and P R Bhima ldquoInfluences of third-orderdispersion on linear birefringent optical soliton transmissionsystemsrdquo Journal of Optics (India) vol 40 no 3 pp 132ndash1422011

[9] P DGreen andA Biswas ldquoBright and dark optical solitonswithtime-dependent coefficients in a non-Kerr lawmediardquoCommu-nications inNonlinear Science andNumerical Simulation vol 15no 12 pp 3865ndash3873 2010

[10] A J Mohamad Jawad M D Petkovic and A Biswas ldquoSolitonsolutions of BURgers equations and perturbed BURgers equa-tionrdquoAppliedMathematics and Computation vol 216 no 11 pp3370ndash3377 2010

[11] Y Kodama and A Hasegawa ldquoNonlinear pulse propagationin a monomode dielectric guiderdquo IEEE Journal of QuantumElectronics vol 23 no 5 pp 510ndash524 1987

[12] Y Kodama M Romagnoli S Wabnitz andMMidrio ldquoRole ofthird-order dispersion on soliton instabilities and interactionsin optical fibersrdquoOptics Expresss vol 19 no 3 pp 165ndash167 1994

[13] G-D Lin Y-T Gao X-L Gai and D-X Meng ldquoExtendeddouble Wronskian solutions to the WHItham-Broer-Kaupequations in shallow waterrdquo Nonlinear Dynamics vol 64 no1-2 pp 197ndash206 2011

[14] Q Cao T Zhang K Djidjeli G Price and E H Twizell ldquoSoli-ton solutions of a class of generalized nonlinear Schrodingerequationsrdquo Applied Mathematics vol 12 no 4 pp 389ndash3981997

[15] A K Sarma ldquoA comparative study of soliton switching in a two-and three-core coupler with TOD and IMDrdquoOptik vol 120 no8 pp 390ndash394 2009

[16] S Tinggen ldquoPropagation characteristics of dark soliton studyin optical fibers with slowly decreasing dispersionrdquo ChineseJournal of Computational Physics vol 13 pp 115ndash118 1996

[17] M RMiura Backlund Transformation Springer-Verlag BerlinGermany 1978

[18] RHirota ldquoExact solution of theKorteweg-de-Vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[19] R Hirota The Direct Method in Soliton Theory CambridgeUniversity Press Cambridge UK 2004

[20] W Malfliet and W Hereman ldquoThe tanh method I Exactsolutions of nonlinear evolution and wave equationsrdquo PhysicaScripta vol 54 no 6 pp 563ndash568 1996

[21] A M Wazwaz ldquoThe tanh method for traveling wave solutionsof nonlinear equationsrdquoAppliedMathematics andComputationvol 154 no 3 pp 713ndash723 2004

[22] S A El-Wakil and M A Abdou ldquoNew exact travelling wavesolutions using modified extended tanh-function methodrdquoChaos Solitons amp Fractals vol 31 no 4 pp 840ndash852 2007

[23] R I Nuruddeen ldquoElzaki decomposition method and its appli-cations in solving linear and nonlinear schrodinger equationsrdquoSohag Journal of Mathematics vol 4 no 2 pp 31ndash35 2017

[24] A M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[25] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[26] H O Bakodah M A Banaja A A Alqarni et al ldquoOpticalsolitons in birefringent fibers with adomian decompositionmethodrdquo Journal of Computational andTheoretical Nanosciencevol 12 no 12 pp 5846ndash5853 2015

[27] A A Qarni M A Banaja H O Bakodah M Mirzazadehand A Biswas ldquoOptical solitons with coupled nonlinearschrodingerrsquos equation in birefringent nano-fibers by adomiandecompositionmethodrdquo Journal of Computational andTheoret-ical Nanoscience vol 13 no 8 pp 5493ndash5498 2016

[28] A A Al Qarni M A Banaja H O Bakodah A A Alshaery FB Majid and A Biswas ldquoOptical solitons in birefringent fibersA numerical studyrdquo Journal of Computational and TheoreticalNanoscience vol 13 no 11 pp 9001ndash9013 2016

[29] A Al-Shareef A A Al Qarni S Al-Mohalbadi and HO Bakodah ldquoSoliton solutions and numerical treatment ofthe nonlinear schrodingerrsquos equation using modified adomiandecomposition methodrdquo Journal of Applied Mathematics andPhysics vol 04 no 12 pp 2215ndash2232 2016

[30] H O Bakodah A A Al Qarni M A Banaja Q Zhou S PMoshokoa andA Biswas ldquoBright anddark thirring optical soli-tons with improved adomian decomposition methodrdquo Optikvol 130 pp 1115ndash1123 2017

[31] M A Banaja A A Al Qarni H O Bakodah Q Zhou S PMoshokoa and A Biswas ldquoThe investigate of optical solitons incascaded systemby improved adomiandecomposition schemerdquoOptik vol 130 pp 1107ndash1114 2017

[32] G Adomian ldquoSolution of physical problems by decompositionrdquoComputers amp Mathematics with Applications vol 27 no 9-10pp 145ndash154 1994

[33] A Hasegawa and Y Kodama Solitons in Optical Communica-tions Oxford University Press Oxford UK 1995

[34] S Wabnitz Y Kodama and A B Aceves ldquoControl of opticalsoliton interactionsrdquo Optical Fiber Technology vol 1 no 3 pp187ndash217 1995

[35] X Geng and Y Lv ldquoDarboux transformation for an integrablegeneralization of the nonlinear Schrodinger equationrdquo Nonlin-ear Dynamics vol 69 no 4 pp 1621ndash1630 2012

[36] S Kumar K Singh and R K Gupta ldquoCoupled Higgs fieldequation and Hamiltonian amplitude equation lie classicalapproach and (GrsquoG)-expansion methodrdquo Pramana vol 79 no1 pp 41ndash60 2012

[37] M Inc A I Aliyu A Yusuf andD Baleanu ldquoDispersive opticalsolitons and modulation instability analysis of Schrodinger-Hirota equation with spatio-temporal dispersion and Kerr lawnonlinearityrdquo Superlattices andMicrostructures vol 113 pp 319ndash327 2018

[38] Y Yildirim ldquoOptical solitons to SchrodingerHirota equation inDWDM system with trial equation integration architecturerdquoOptik vol 18 pp 275ndash281 2019

[39] A Biswas Y Yildirim E Yasar et al ldquoDispersive optical solitonswith SchrodingerndashHirota model by trial equation methodrdquoOptik - International Journal for Light and Electron Optics vol162 pp 35ndash41 2018

[40] A A Al Qarni M A Banaja and H O Bakodah ldquoNumericalanalyses optical solitons in dual core couplers with kerr lawnonlinearityrdquo Applied Mathematics vol 06 no 12 pp 1957ndash1967 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 4: Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe › 2019 › 2960912.pdfMathematicalProblemsinEngineering [] M. Younis, H. U. Rehman,

4 Mathematical Problems in Engineering

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 1 Comparison of the exact and approximate solution for Example 1 for minus20 le 119909 le 20

Table 1 The absolute error for Example 1 when 119909 = 20119905 119887 = 0120574 = 0 120574 = 06 120574 = 100 00 00 0001 0000010261373 000003209069 00000637830002 0000021764973 000006266170 00001259991303 0000034488427 000009173715 00001866700204 0000048407918 000011934200 00002458179205 0000063498224 000014550173 000030346539

Table 2 The absolute error for Example 2 when 119909 = 20119905 119887 = 01120574 = 0 120574 = 06 120574 = 100 547003times10minus7 75968times10minus7 87637times10minus701 0000013623790 000003154594 00000649859302 0000028096531 000006216823 00001291215103 0000043940472 000009113126 00001915515404 0000061129371 000011845995 00002522976405 0000079635465 000014417984 000031138194

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 2 Comparison of the exact and approximate solution for Example 2 for minus20 le 119909 le 20

Mathematical Problems in Engineering 5

Table 3 The absolute error for Example 3 when 119909 = 20119905 119887 = 06120574 = 0 120574 = 06 120574 = 100 000014373490 000017576127 00001956321901 000014417980 000011583001 00000927555102 000014701256 000005817085 00000075892103 000015221500 000000366844 00001054147304 000015976708 000004859351 00002007345805 000016964674 000009833252 000029356299

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 3 Comparison of the exact and approximate solution for Example 3 for minus20 le 119909 le 20

Example 3 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 06 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 3 and Figure 3

5 Conclusion

In this work an Improved Adomian DecompositionMethod (IADM) has been proposed to solve the nonlinearSchrodinger-Hirota equation in presence of severalHamiltonian type perturbation terms The obtainedresults possess high precision and converged to the exactsolution with less computational efforts It was noticedthat a high accuracy of results is obtained when is thecoefficient of 3OD 120574 = 0 and within the range 0 gt 120574 119887 lt 1(119887 is the coefficient of spatio-temporal dispersion term) ascommented also in [40] It is also clear from Figures 1ndash3 thatthe complete correspondence of the two solutions is indeedremarkable Thus the IADM is an efficient method fornonlinear Schrodinger equations since it shows a high levelof accuracy with lesser computational efforts as compared toother numerical methods This topic is still open for further

researches for different values of parameters and other typesof soliton solutions

Data Availability

All the data used for the numerical simulations and compar-ison purpose have been reported in the tables included andvisualized in the graphical illustrations and nothing is left

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y Kodama and A Hasegawa ldquoIV theoretical foundation ofoptical-soliton concept in fibersrdquo Progress in Optics vol 30 pp205ndash259 1992

[2] V E Zakharov and S Wabnitz Optical Solitons TheoreticalChallenges and Industrial Perspectives Springer New York NYUSA 1999

[3] M J Ablowitz and H Segur Solitons and the Inverse ScatteringTransform SIAM Philadelphia Pa USA 1981

[4] I Bernstein E Zerrad Q Zhou A Biswas and N MelikechildquoDispersive optical solitons with Schrodinger-Hirota equationby traveling wave hypothesisrdquo Optoelectronics and AdvancedMaterials ndash Rapid Communications vol 9 no 5-6 pp 792ndash7972015

[5] G P Agarwal Nonlinear Fiber Optics Academic Press 2001

6 Mathematical Problems in Engineering

[6] M Younis H U Rehman and F Tahir ldquoOptical Gaussonsand dark solitons in directional couplers with spatiotemporaldispersionrdquo Optical and Quantum Electronics vol 49 no 122018

[7] A Biswas ldquoStochastic perturbation of optical solitons inSchrodingerndashHirota equationrdquo Optics Communications vol239 no 4-6 pp 461ndash466 2004

[8] R K Dowluru and P R Bhima ldquoInfluences of third-orderdispersion on linear birefringent optical soliton transmissionsystemsrdquo Journal of Optics (India) vol 40 no 3 pp 132ndash1422011

[9] P DGreen andA Biswas ldquoBright and dark optical solitonswithtime-dependent coefficients in a non-Kerr lawmediardquoCommu-nications inNonlinear Science andNumerical Simulation vol 15no 12 pp 3865ndash3873 2010

[10] A J Mohamad Jawad M D Petkovic and A Biswas ldquoSolitonsolutions of BURgers equations and perturbed BURgers equa-tionrdquoAppliedMathematics and Computation vol 216 no 11 pp3370ndash3377 2010

[11] Y Kodama and A Hasegawa ldquoNonlinear pulse propagationin a monomode dielectric guiderdquo IEEE Journal of QuantumElectronics vol 23 no 5 pp 510ndash524 1987

[12] Y Kodama M Romagnoli S Wabnitz andMMidrio ldquoRole ofthird-order dispersion on soliton instabilities and interactionsin optical fibersrdquoOptics Expresss vol 19 no 3 pp 165ndash167 1994

[13] G-D Lin Y-T Gao X-L Gai and D-X Meng ldquoExtendeddouble Wronskian solutions to the WHItham-Broer-Kaupequations in shallow waterrdquo Nonlinear Dynamics vol 64 no1-2 pp 197ndash206 2011

[14] Q Cao T Zhang K Djidjeli G Price and E H Twizell ldquoSoli-ton solutions of a class of generalized nonlinear Schrodingerequationsrdquo Applied Mathematics vol 12 no 4 pp 389ndash3981997

[15] A K Sarma ldquoA comparative study of soliton switching in a two-and three-core coupler with TOD and IMDrdquoOptik vol 120 no8 pp 390ndash394 2009

[16] S Tinggen ldquoPropagation characteristics of dark soliton studyin optical fibers with slowly decreasing dispersionrdquo ChineseJournal of Computational Physics vol 13 pp 115ndash118 1996

[17] M RMiura Backlund Transformation Springer-Verlag BerlinGermany 1978

[18] RHirota ldquoExact solution of theKorteweg-de-Vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[19] R Hirota The Direct Method in Soliton Theory CambridgeUniversity Press Cambridge UK 2004

[20] W Malfliet and W Hereman ldquoThe tanh method I Exactsolutions of nonlinear evolution and wave equationsrdquo PhysicaScripta vol 54 no 6 pp 563ndash568 1996

[21] A M Wazwaz ldquoThe tanh method for traveling wave solutionsof nonlinear equationsrdquoAppliedMathematics andComputationvol 154 no 3 pp 713ndash723 2004

[22] S A El-Wakil and M A Abdou ldquoNew exact travelling wavesolutions using modified extended tanh-function methodrdquoChaos Solitons amp Fractals vol 31 no 4 pp 840ndash852 2007

[23] R I Nuruddeen ldquoElzaki decomposition method and its appli-cations in solving linear and nonlinear schrodinger equationsrdquoSohag Journal of Mathematics vol 4 no 2 pp 31ndash35 2017

[24] A M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[25] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[26] H O Bakodah M A Banaja A A Alqarni et al ldquoOpticalsolitons in birefringent fibers with adomian decompositionmethodrdquo Journal of Computational andTheoretical Nanosciencevol 12 no 12 pp 5846ndash5853 2015

[27] A A Qarni M A Banaja H O Bakodah M Mirzazadehand A Biswas ldquoOptical solitons with coupled nonlinearschrodingerrsquos equation in birefringent nano-fibers by adomiandecompositionmethodrdquo Journal of Computational andTheoret-ical Nanoscience vol 13 no 8 pp 5493ndash5498 2016

[28] A A Al Qarni M A Banaja H O Bakodah A A Alshaery FB Majid and A Biswas ldquoOptical solitons in birefringent fibersA numerical studyrdquo Journal of Computational and TheoreticalNanoscience vol 13 no 11 pp 9001ndash9013 2016

[29] A Al-Shareef A A Al Qarni S Al-Mohalbadi and HO Bakodah ldquoSoliton solutions and numerical treatment ofthe nonlinear schrodingerrsquos equation using modified adomiandecomposition methodrdquo Journal of Applied Mathematics andPhysics vol 04 no 12 pp 2215ndash2232 2016

[30] H O Bakodah A A Al Qarni M A Banaja Q Zhou S PMoshokoa andA Biswas ldquoBright anddark thirring optical soli-tons with improved adomian decomposition methodrdquo Optikvol 130 pp 1115ndash1123 2017

[31] M A Banaja A A Al Qarni H O Bakodah Q Zhou S PMoshokoa and A Biswas ldquoThe investigate of optical solitons incascaded systemby improved adomiandecomposition schemerdquoOptik vol 130 pp 1107ndash1114 2017

[32] G Adomian ldquoSolution of physical problems by decompositionrdquoComputers amp Mathematics with Applications vol 27 no 9-10pp 145ndash154 1994

[33] A Hasegawa and Y Kodama Solitons in Optical Communica-tions Oxford University Press Oxford UK 1995

[34] S Wabnitz Y Kodama and A B Aceves ldquoControl of opticalsoliton interactionsrdquo Optical Fiber Technology vol 1 no 3 pp187ndash217 1995

[35] X Geng and Y Lv ldquoDarboux transformation for an integrablegeneralization of the nonlinear Schrodinger equationrdquo Nonlin-ear Dynamics vol 69 no 4 pp 1621ndash1630 2012

[36] S Kumar K Singh and R K Gupta ldquoCoupled Higgs fieldequation and Hamiltonian amplitude equation lie classicalapproach and (GrsquoG)-expansion methodrdquo Pramana vol 79 no1 pp 41ndash60 2012

[37] M Inc A I Aliyu A Yusuf andD Baleanu ldquoDispersive opticalsolitons and modulation instability analysis of Schrodinger-Hirota equation with spatio-temporal dispersion and Kerr lawnonlinearityrdquo Superlattices andMicrostructures vol 113 pp 319ndash327 2018

[38] Y Yildirim ldquoOptical solitons to SchrodingerHirota equation inDWDM system with trial equation integration architecturerdquoOptik vol 18 pp 275ndash281 2019

[39] A Biswas Y Yildirim E Yasar et al ldquoDispersive optical solitonswith SchrodingerndashHirota model by trial equation methodrdquoOptik - International Journal for Light and Electron Optics vol162 pp 35ndash41 2018

[40] A A Al Qarni M A Banaja and H O Bakodah ldquoNumericalanalyses optical solitons in dual core couplers with kerr lawnonlinearityrdquo Applied Mathematics vol 06 no 12 pp 1957ndash1967 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 5: Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe › 2019 › 2960912.pdfMathematicalProblemsinEngineering [] M. Younis, H. U. Rehman,

Mathematical Problems in Engineering 5

Table 3 The absolute error for Example 3 when 119909 = 20119905 119887 = 06120574 = 0 120574 = 06 120574 = 100 000014373490 000017576127 00001956321901 000014417980 000011583001 00000927555102 000014701256 000005817085 00000075892103 000015221500 000000366844 00001054147304 000015976708 000004859351 00002007345805 000016964674 000009833252 000029356299

t=05 =00 t=05 =1t=05 =06

exact qnumerical q

exact qnumerical q

exact qnumerical q

Figure 3 Comparison of the exact and approximate solution for Example 3 for minus20 le 119909 le 20

Example 3 We consider Schrodinger-Hirota equation withspatiotemporal dispersion equation (6) with 119887 = 06 and thefollowing three special cases

Case 1 Let 120574 = 00Case 2 Let 120574 = 06Case 3 Let 120574 = 10

The result and the profiles of this example are presentedin Table 3 and Figure 3

5 Conclusion

In this work an Improved Adomian DecompositionMethod (IADM) has been proposed to solve the nonlinearSchrodinger-Hirota equation in presence of severalHamiltonian type perturbation terms The obtainedresults possess high precision and converged to the exactsolution with less computational efforts It was noticedthat a high accuracy of results is obtained when is thecoefficient of 3OD 120574 = 0 and within the range 0 gt 120574 119887 lt 1(119887 is the coefficient of spatio-temporal dispersion term) ascommented also in [40] It is also clear from Figures 1ndash3 thatthe complete correspondence of the two solutions is indeedremarkable Thus the IADM is an efficient method fornonlinear Schrodinger equations since it shows a high levelof accuracy with lesser computational efforts as compared toother numerical methods This topic is still open for further

researches for different values of parameters and other typesof soliton solutions

Data Availability

All the data used for the numerical simulations and compar-ison purpose have been reported in the tables included andvisualized in the graphical illustrations and nothing is left

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] Y Kodama and A Hasegawa ldquoIV theoretical foundation ofoptical-soliton concept in fibersrdquo Progress in Optics vol 30 pp205ndash259 1992

[2] V E Zakharov and S Wabnitz Optical Solitons TheoreticalChallenges and Industrial Perspectives Springer New York NYUSA 1999

[3] M J Ablowitz and H Segur Solitons and the Inverse ScatteringTransform SIAM Philadelphia Pa USA 1981

[4] I Bernstein E Zerrad Q Zhou A Biswas and N MelikechildquoDispersive optical solitons with Schrodinger-Hirota equationby traveling wave hypothesisrdquo Optoelectronics and AdvancedMaterials ndash Rapid Communications vol 9 no 5-6 pp 792ndash7972015

[5] G P Agarwal Nonlinear Fiber Optics Academic Press 2001

6 Mathematical Problems in Engineering

[6] M Younis H U Rehman and F Tahir ldquoOptical Gaussonsand dark solitons in directional couplers with spatiotemporaldispersionrdquo Optical and Quantum Electronics vol 49 no 122018

[7] A Biswas ldquoStochastic perturbation of optical solitons inSchrodingerndashHirota equationrdquo Optics Communications vol239 no 4-6 pp 461ndash466 2004

[8] R K Dowluru and P R Bhima ldquoInfluences of third-orderdispersion on linear birefringent optical soliton transmissionsystemsrdquo Journal of Optics (India) vol 40 no 3 pp 132ndash1422011

[9] P DGreen andA Biswas ldquoBright and dark optical solitonswithtime-dependent coefficients in a non-Kerr lawmediardquoCommu-nications inNonlinear Science andNumerical Simulation vol 15no 12 pp 3865ndash3873 2010

[10] A J Mohamad Jawad M D Petkovic and A Biswas ldquoSolitonsolutions of BURgers equations and perturbed BURgers equa-tionrdquoAppliedMathematics and Computation vol 216 no 11 pp3370ndash3377 2010

[11] Y Kodama and A Hasegawa ldquoNonlinear pulse propagationin a monomode dielectric guiderdquo IEEE Journal of QuantumElectronics vol 23 no 5 pp 510ndash524 1987

[12] Y Kodama M Romagnoli S Wabnitz andMMidrio ldquoRole ofthird-order dispersion on soliton instabilities and interactionsin optical fibersrdquoOptics Expresss vol 19 no 3 pp 165ndash167 1994

[13] G-D Lin Y-T Gao X-L Gai and D-X Meng ldquoExtendeddouble Wronskian solutions to the WHItham-Broer-Kaupequations in shallow waterrdquo Nonlinear Dynamics vol 64 no1-2 pp 197ndash206 2011

[14] Q Cao T Zhang K Djidjeli G Price and E H Twizell ldquoSoli-ton solutions of a class of generalized nonlinear Schrodingerequationsrdquo Applied Mathematics vol 12 no 4 pp 389ndash3981997

[15] A K Sarma ldquoA comparative study of soliton switching in a two-and three-core coupler with TOD and IMDrdquoOptik vol 120 no8 pp 390ndash394 2009

[16] S Tinggen ldquoPropagation characteristics of dark soliton studyin optical fibers with slowly decreasing dispersionrdquo ChineseJournal of Computational Physics vol 13 pp 115ndash118 1996

[17] M RMiura Backlund Transformation Springer-Verlag BerlinGermany 1978

[18] RHirota ldquoExact solution of theKorteweg-de-Vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[19] R Hirota The Direct Method in Soliton Theory CambridgeUniversity Press Cambridge UK 2004

[20] W Malfliet and W Hereman ldquoThe tanh method I Exactsolutions of nonlinear evolution and wave equationsrdquo PhysicaScripta vol 54 no 6 pp 563ndash568 1996

[21] A M Wazwaz ldquoThe tanh method for traveling wave solutionsof nonlinear equationsrdquoAppliedMathematics andComputationvol 154 no 3 pp 713ndash723 2004

[22] S A El-Wakil and M A Abdou ldquoNew exact travelling wavesolutions using modified extended tanh-function methodrdquoChaos Solitons amp Fractals vol 31 no 4 pp 840ndash852 2007

[23] R I Nuruddeen ldquoElzaki decomposition method and its appli-cations in solving linear and nonlinear schrodinger equationsrdquoSohag Journal of Mathematics vol 4 no 2 pp 31ndash35 2017

[24] A M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[25] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[26] H O Bakodah M A Banaja A A Alqarni et al ldquoOpticalsolitons in birefringent fibers with adomian decompositionmethodrdquo Journal of Computational andTheoretical Nanosciencevol 12 no 12 pp 5846ndash5853 2015

[27] A A Qarni M A Banaja H O Bakodah M Mirzazadehand A Biswas ldquoOptical solitons with coupled nonlinearschrodingerrsquos equation in birefringent nano-fibers by adomiandecompositionmethodrdquo Journal of Computational andTheoret-ical Nanoscience vol 13 no 8 pp 5493ndash5498 2016

[28] A A Al Qarni M A Banaja H O Bakodah A A Alshaery FB Majid and A Biswas ldquoOptical solitons in birefringent fibersA numerical studyrdquo Journal of Computational and TheoreticalNanoscience vol 13 no 11 pp 9001ndash9013 2016

[29] A Al-Shareef A A Al Qarni S Al-Mohalbadi and HO Bakodah ldquoSoliton solutions and numerical treatment ofthe nonlinear schrodingerrsquos equation using modified adomiandecomposition methodrdquo Journal of Applied Mathematics andPhysics vol 04 no 12 pp 2215ndash2232 2016

[30] H O Bakodah A A Al Qarni M A Banaja Q Zhou S PMoshokoa andA Biswas ldquoBright anddark thirring optical soli-tons with improved adomian decomposition methodrdquo Optikvol 130 pp 1115ndash1123 2017

[31] M A Banaja A A Al Qarni H O Bakodah Q Zhou S PMoshokoa and A Biswas ldquoThe investigate of optical solitons incascaded systemby improved adomiandecomposition schemerdquoOptik vol 130 pp 1107ndash1114 2017

[32] G Adomian ldquoSolution of physical problems by decompositionrdquoComputers amp Mathematics with Applications vol 27 no 9-10pp 145ndash154 1994

[33] A Hasegawa and Y Kodama Solitons in Optical Communica-tions Oxford University Press Oxford UK 1995

[34] S Wabnitz Y Kodama and A B Aceves ldquoControl of opticalsoliton interactionsrdquo Optical Fiber Technology vol 1 no 3 pp187ndash217 1995

[35] X Geng and Y Lv ldquoDarboux transformation for an integrablegeneralization of the nonlinear Schrodinger equationrdquo Nonlin-ear Dynamics vol 69 no 4 pp 1621ndash1630 2012

[36] S Kumar K Singh and R K Gupta ldquoCoupled Higgs fieldequation and Hamiltonian amplitude equation lie classicalapproach and (GrsquoG)-expansion methodrdquo Pramana vol 79 no1 pp 41ndash60 2012

[37] M Inc A I Aliyu A Yusuf andD Baleanu ldquoDispersive opticalsolitons and modulation instability analysis of Schrodinger-Hirota equation with spatio-temporal dispersion and Kerr lawnonlinearityrdquo Superlattices andMicrostructures vol 113 pp 319ndash327 2018

[38] Y Yildirim ldquoOptical solitons to SchrodingerHirota equation inDWDM system with trial equation integration architecturerdquoOptik vol 18 pp 275ndash281 2019

[39] A Biswas Y Yildirim E Yasar et al ldquoDispersive optical solitonswith SchrodingerndashHirota model by trial equation methodrdquoOptik - International Journal for Light and Electron Optics vol162 pp 35ndash41 2018

[40] A A Al Qarni M A Banaja and H O Bakodah ldquoNumericalanalyses optical solitons in dual core couplers with kerr lawnonlinearityrdquo Applied Mathematics vol 06 no 12 pp 1957ndash1967 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 6: Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe › 2019 › 2960912.pdfMathematicalProblemsinEngineering [] M. Younis, H. U. Rehman,

6 Mathematical Problems in Engineering

[6] M Younis H U Rehman and F Tahir ldquoOptical Gaussonsand dark solitons in directional couplers with spatiotemporaldispersionrdquo Optical and Quantum Electronics vol 49 no 122018

[7] A Biswas ldquoStochastic perturbation of optical solitons inSchrodingerndashHirota equationrdquo Optics Communications vol239 no 4-6 pp 461ndash466 2004

[8] R K Dowluru and P R Bhima ldquoInfluences of third-orderdispersion on linear birefringent optical soliton transmissionsystemsrdquo Journal of Optics (India) vol 40 no 3 pp 132ndash1422011

[9] P DGreen andA Biswas ldquoBright and dark optical solitonswithtime-dependent coefficients in a non-Kerr lawmediardquoCommu-nications inNonlinear Science andNumerical Simulation vol 15no 12 pp 3865ndash3873 2010

[10] A J Mohamad Jawad M D Petkovic and A Biswas ldquoSolitonsolutions of BURgers equations and perturbed BURgers equa-tionrdquoAppliedMathematics and Computation vol 216 no 11 pp3370ndash3377 2010

[11] Y Kodama and A Hasegawa ldquoNonlinear pulse propagationin a monomode dielectric guiderdquo IEEE Journal of QuantumElectronics vol 23 no 5 pp 510ndash524 1987

[12] Y Kodama M Romagnoli S Wabnitz andMMidrio ldquoRole ofthird-order dispersion on soliton instabilities and interactionsin optical fibersrdquoOptics Expresss vol 19 no 3 pp 165ndash167 1994

[13] G-D Lin Y-T Gao X-L Gai and D-X Meng ldquoExtendeddouble Wronskian solutions to the WHItham-Broer-Kaupequations in shallow waterrdquo Nonlinear Dynamics vol 64 no1-2 pp 197ndash206 2011

[14] Q Cao T Zhang K Djidjeli G Price and E H Twizell ldquoSoli-ton solutions of a class of generalized nonlinear Schrodingerequationsrdquo Applied Mathematics vol 12 no 4 pp 389ndash3981997

[15] A K Sarma ldquoA comparative study of soliton switching in a two-and three-core coupler with TOD and IMDrdquoOptik vol 120 no8 pp 390ndash394 2009

[16] S Tinggen ldquoPropagation characteristics of dark soliton studyin optical fibers with slowly decreasing dispersionrdquo ChineseJournal of Computational Physics vol 13 pp 115ndash118 1996

[17] M RMiura Backlund Transformation Springer-Verlag BerlinGermany 1978

[18] RHirota ldquoExact solution of theKorteweg-de-Vries equation formultiple Collisions of solitonsrdquo Physical Review Letters vol 27no 18 pp 1192ndash1194 1971

[19] R Hirota The Direct Method in Soliton Theory CambridgeUniversity Press Cambridge UK 2004

[20] W Malfliet and W Hereman ldquoThe tanh method I Exactsolutions of nonlinear evolution and wave equationsrdquo PhysicaScripta vol 54 no 6 pp 563ndash568 1996

[21] A M Wazwaz ldquoThe tanh method for traveling wave solutionsof nonlinear equationsrdquoAppliedMathematics andComputationvol 154 no 3 pp 713ndash723 2004

[22] S A El-Wakil and M A Abdou ldquoNew exact travelling wavesolutions using modified extended tanh-function methodrdquoChaos Solitons amp Fractals vol 31 no 4 pp 840ndash852 2007

[23] R I Nuruddeen ldquoElzaki decomposition method and its appli-cations in solving linear and nonlinear schrodinger equationsrdquoSohag Journal of Mathematics vol 4 no 2 pp 31ndash35 2017

[24] A M Wazwaz ldquoA sine-cosine method for handling nonlinearwave equationsrdquo Mathematical and Computer Modelling vol40 no 5-6 pp 499ndash508 2004

[25] M Wang Y Zhou and Z Li ldquoApplication of a homogeneousbalance method to exact solutions of nonlinear equations inmathematical physicsrdquo Physics Letters A vol 216 no 1ndash5 pp67ndash75 1996

[26] H O Bakodah M A Banaja A A Alqarni et al ldquoOpticalsolitons in birefringent fibers with adomian decompositionmethodrdquo Journal of Computational andTheoretical Nanosciencevol 12 no 12 pp 5846ndash5853 2015

[27] A A Qarni M A Banaja H O Bakodah M Mirzazadehand A Biswas ldquoOptical solitons with coupled nonlinearschrodingerrsquos equation in birefringent nano-fibers by adomiandecompositionmethodrdquo Journal of Computational andTheoret-ical Nanoscience vol 13 no 8 pp 5493ndash5498 2016

[28] A A Al Qarni M A Banaja H O Bakodah A A Alshaery FB Majid and A Biswas ldquoOptical solitons in birefringent fibersA numerical studyrdquo Journal of Computational and TheoreticalNanoscience vol 13 no 11 pp 9001ndash9013 2016

[29] A Al-Shareef A A Al Qarni S Al-Mohalbadi and HO Bakodah ldquoSoliton solutions and numerical treatment ofthe nonlinear schrodingerrsquos equation using modified adomiandecomposition methodrdquo Journal of Applied Mathematics andPhysics vol 04 no 12 pp 2215ndash2232 2016

[30] H O Bakodah A A Al Qarni M A Banaja Q Zhou S PMoshokoa andA Biswas ldquoBright anddark thirring optical soli-tons with improved adomian decomposition methodrdquo Optikvol 130 pp 1115ndash1123 2017

[31] M A Banaja A A Al Qarni H O Bakodah Q Zhou S PMoshokoa and A Biswas ldquoThe investigate of optical solitons incascaded systemby improved adomiandecomposition schemerdquoOptik vol 130 pp 1107ndash1114 2017

[32] G Adomian ldquoSolution of physical problems by decompositionrdquoComputers amp Mathematics with Applications vol 27 no 9-10pp 145ndash154 1994

[33] A Hasegawa and Y Kodama Solitons in Optical Communica-tions Oxford University Press Oxford UK 1995

[34] S Wabnitz Y Kodama and A B Aceves ldquoControl of opticalsoliton interactionsrdquo Optical Fiber Technology vol 1 no 3 pp187ndash217 1995

[35] X Geng and Y Lv ldquoDarboux transformation for an integrablegeneralization of the nonlinear Schrodinger equationrdquo Nonlin-ear Dynamics vol 69 no 4 pp 1621ndash1630 2012

[36] S Kumar K Singh and R K Gupta ldquoCoupled Higgs fieldequation and Hamiltonian amplitude equation lie classicalapproach and (GrsquoG)-expansion methodrdquo Pramana vol 79 no1 pp 41ndash60 2012

[37] M Inc A I Aliyu A Yusuf andD Baleanu ldquoDispersive opticalsolitons and modulation instability analysis of Schrodinger-Hirota equation with spatio-temporal dispersion and Kerr lawnonlinearityrdquo Superlattices andMicrostructures vol 113 pp 319ndash327 2018

[38] Y Yildirim ldquoOptical solitons to SchrodingerHirota equation inDWDM system with trial equation integration architecturerdquoOptik vol 18 pp 275ndash281 2019

[39] A Biswas Y Yildirim E Yasar et al ldquoDispersive optical solitonswith SchrodingerndashHirota model by trial equation methodrdquoOptik - International Journal for Light and Electron Optics vol162 pp 35ndash41 2018

[40] A A Al Qarni M A Banaja and H O Bakodah ldquoNumericalanalyses optical solitons in dual core couplers with kerr lawnonlinearityrdquo Applied Mathematics vol 06 no 12 pp 1957ndash1967 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 7: Numerical Solution of Dispersive Optical Solitons …downloads.hindawi.com › journals › mpe › 2019 › 2960912.pdfMathematicalProblemsinEngineering [] M. Younis, H. U. Rehman,

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


Younis majeed ppt final (1)

Younis majeed ppt final (1)

Introduction to Solitons - BLTP JINR Home Pagetheor.jinr.ru/~shnir/Solitons/Lecture1.pdf · Introduction to Solitons Institute of Theoretical Physics and Astronomy Vilnius, 2013 University

Introduction to Solitons - BLTP JINR Home Pagetheor.jinr.ru/~shnir/Solitons/Lecture1.pdf · Introduction to Solitons Institute of Theoretical Physics and Astronomy Vilnius, 2013 University

Michelle Younis - PCA Certification

Michelle Younis - PCA Certification

Professor Ihab Younis, M.D .

Professor Ihab Younis, M.D .

Younis spain project homework half term

Younis spain project homework half term

Falaco Solitons

Falaco Solitons

Arthropods Attacks I IHAB YOUNIS, M.D.

Arthropods Attacks I IHAB YOUNIS, M.D.

CUTANEOUS VASCULITIS IHAB YOUNIS,MD

CUTANEOUS VASCULITIS IHAB YOUNIS,MD

ELECTRON SELF-LOCALIZATION, SOLITONS, AND ...lptms.u-psud.fr/membres/brazov/solitons-review,web.pdfELECTRON SELF-LOCALIZATION, SOLITONS, AND PERIODIC SUPERSTRUCTURES IN QUASI ONE-DIMENSIONAL

ELECTRON SELF-LOCALIZATION, SOLITONS, AND ...lptms.u-psud.fr/membres/brazov/solitons-review,web.pdfELECTRON SELF-LOCALIZATION, SOLITONS, AND PERIODIC SUPERSTRUCTURES IN QUASI ONE-DIMENSIONAL

Shoaib Younis

Shoaib Younis

Bright Solitons in Bose-Einstein Condensatesjosselin-garnier.org/wp-content/uploads/2013/12/review_soliton_bec.pdf · Bright Solitons in Bose-Einstein Condensates 3 2 Bright solitons

Bright Solitons in Bose-Einstein Condensatesjosselin-garnier.org/wp-content/uploads/2013/12/review_soliton_bec.pdf · Bright Solitons in Bose-Einstein Condensates 3 2 Bright solitons

Prepared & presented by Tayseer Younis

Prepared & presented by Tayseer Younis

SECTION C: SOLITONS - University of Sheffield › kul › sectionB.pdf · SECTION C: SOLITONS Course text: Solitons: an introduction P.G. Drazin & R.S. Johnson ... Scattering & inverse

SECTION C: SOLITONS - University of Sheffield › kul › sectionB.pdf · SECTION C: SOLITONS Course text: Solitons: an introduction P.G. Drazin & R.S. Johnson ... Scattering & inverse

Introduction to Solitons - BLTP JINR Home Pagetheor.jinr.ru/~shnir/Solitons/Lecture2.pdf · Introduction to Solitons Institute of Theoretical Physics and Astronomy Vilnius, 2013 University

Introduction to Solitons - BLTP JINR Home Pagetheor.jinr.ru/~shnir/Solitons/Lecture2.pdf · Introduction to Solitons Institute of Theoretical Physics and Astronomy Vilnius, 2013 University

Oscillating Spinor Solitons

Oscillating Spinor Solitons

Optical Solitons fin - prashub.com

Optical Solitons fin - prashub.com

Solitonstraining.hepi.tsu.ge › rtn › activities › sources › Solitons by AKh.pdfcalled nontopological solitons in contrast to another class of solitons, which interpolates between

Solitonstraining.hepi.tsu.ge › rtn › activities › sources › Solitons by AKh.pdfcalled nontopological solitons in contrast to another class of solitons, which interpolates between

12-May-151 Khan Younis Waste Water Treatment Plant Management Arrangements Khan Younis Waste Water Treatment Plant Management Arrangements.

12-May-151 Khan Younis Waste Water Treatment Plant Management Arrangements Khan Younis Waste Water Treatment Plant Management Arrangements.

Solitons : An Introduction

Solitons : An Introduction

Awais Younis Complaint

Awais Younis Complaint