ANGULAR VECTOR SOLITONS OF THE COUPLED … · OF THE COUPLED NONLINEAR SCHRÖDINGER EQUATIONS WITH...

Romanian Reports in Physics, Vol. 67, No. 2, P. 375–385, 2015

ANGULAR VECTOR SOLITONS OF THE COUPLED NONLINEAR SCHRÖDINGER EQUATIONS

WITH SPATIALLY MODULATED NONLINEARITIES

RUO-YUN CAO1, WEI-PING ZHONG1,*, DUMITRU MIHALACHE2 1 Shunde Polytechnic, Department of Electronic and Information Engineering,

Guangdong Province, Shunde 528300, China 2“Horia Hulubei” National Institute for Physics and Nuclear Engineering, P.O. Box MG-6,

RO-077125, Bucharest-Magurele, Romania *Corresponding author: [email protected]

Received December 24, 2014

Abstract. We present the angular vector soliton solutions of the coupled (2+1)-dimensional nonlinear Schrödinger (NLS) equations via a similarity transformation that is connected with the stationary NLS equation. Then we investigate the transverse spatial distributions of the controllable vector soliton clusters. We obtain exact angular vector soliton solutions that are constructed with the help of Whittaker special functions. We find that these solitons can be effectively controlled by the modulation depth, the topological charge, and the radial quantum number. Our results show that, for integer or fractional topological charge, the intensity profiles of these vector solitons exhibit various forms, such as the vortex-ring shapes and either symmetric or asymmetric necklace-ring patterns.

Keywords: coupled (2+1)-dimensional NLS equations, angular vector soliton, symmetric necklace- ring soliton, asymmetric necklace-ring soliton.

1. INTRODUCTION

Solitons are self-reinforcing nonlinear wave packets that maintain their main features while they propagate in nonlinear media, due to an exact balance between linear and nonlinear effects in these media. Such robust localized structures can exist in a large variety of physical settings: from water waves and plasma physics to nonlinear photonics and Bose-Einstein condensates; see, for example, Refs. [1–34].

Vector solitons, known as multiple-component self-trapped structures, are soliton clusters that consist of more than one field component. They are supported not only by self-interaction of the same physical fields, but also by cross interactions between fields of different types [35–36]. The simplest type of vector soliton is the two-component one that is governed by the integrable Manakov model [35, 37].

Ruo-Yun Cao, Wei-Ping Zhong, Dumitru Mihalache 2 376

In the past decade, two-component angular vector solitons in (2+1)-dimensional [(2+1)D] settings have drawn significant attention in nonlinear optics and photonics [38–40], as well as in other fields of physics. We note that the intensities of each of the components display the “necklace” pattern, that is, azimuthally modulated beams with ring-type intensity profiles, see e. g. Refs. [40–41]. Such a necklace-ring soliton is shown to stabilize during propagation in nonlinear media due to the balance between the compression-like nonlinear effect and the repulsive force between the neighboring petals [38–39]. It is important to note that the total intensity of the angular vector soliton (i.e., the composite solitary wave) exhibits a vortex structure, namely its intensity is zero at the center position [38–39]. There has been a great deal of theoretical and experimental investigations of a series of evolution models based on the generic nonlinear Schrödinger (NLS) equation. Many analytical methods have been developed to obtain the soliton solutions of various coupled one-dimensional (1D) NLS equations with different types of nonlinearities. The propagation dynamics and the excitation of such solitons were studied in detail; see, for example, Refs. [42–43].

In this paper, we investigate both analytically and numerically two-component angular vector solitons in two-dimensional (2D) Kerr-type media with spatially modulated nonlinearities. The main objective of our research is twofold. First, we present exact analytical solutions for two-component angular vector solitons by using the separation of variables and the similarity transformation. For a special external potential, two-component angular vector soliton solutions are constructed with the help of Whittaker special functions. The solutions to be given below have not been studied before, to the best of our knowledge. Second, by making use of numerical simulations, we find that the angular vector solitary waves are stable to propagation. The numerical simulations thus have confirmed the validity of the obtained analytical solutions.

The paper is organized as follows. In Sec. 2, by making use of the method of separation of variables and the similarity transformation, we present the 2D coupled NLS equations with spatially modulated nonlinearities and a special external potential. We give the scheme to reduce the 2D model to the related stationary NLS equation with constant coefficients. We find that the exact solution of the two-component angular vector model is constructed through the well-known Whittaker special functions. In Sec. 3, we concentrate on some representative examples to illustrate our exact solutions. The typical patterns of these exact solutions are displayed for different sets of three free parameters: the modulated depth, the topological charge, and the radial direction quantum number. The unique properties of the solution patterns are also analyzed. Section 3 is also devoted to some numerical simulations of the nonlinear evolution equations in order to confirm the stability and the validity of analytical solutions. Finally, in Sec. 4, the conclusions are summarized.

3 Angular vector solitons of coupled nonlinear Schrödinger equations 377

2. THEORETICAL MODEL AND EXACT SOLITARY-WAVE SOLUTIONS

We consider a vector beam consisting of two mutually incoherent optical components propagating in a bulk nonlinear Kerr medium. In polar coordinates, 2D coupled NLS equations with spatially modulated nonlinearities describing the propagation of the beam along z direction for the complex envelopes ( )ϕ,, rzE j can

be written in the following dimensionless form [38–39], [44, 45]：

( ) ( )2 2 2 2

2 2 21

1 1 1i 02

j j j jj j j

j

E E E Er E E U r E

z r r r rχ

ϕ =

⎛ ⎞∂ ∂ ∂ ∂+ + + + + =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∑ , (1)

where j = 1 and 2, 22 yxr += , and ϕ ( )xy1tan−= is the azimuthal angle.

Here ( )rχ ( 0> ) represents the nonlinearity coefficient which, as well as the external potential U(r), are to be determined, and are functions of the radial coordinate r. Our goal is to construct special types of localized solutions of the 2D coupled NLS equations (1) using an incoherent superposition of the necklace beam components such that ( ) 0,,lim =

∞→ϕrzE jr

. The separation of variables for Eq. (1),

( ) ( ) ( ) i, , e kzj jE z r u rϕ ϕ −= Φ (2)

leads to the following two equations:

2

22

d0

dj

jmϕΦ

+ Φ = , (3a)

( )2 2

22 2

1 d 1 d 02 d d

u u m u u u ku U r ur r r r

χ⎛ ⎞

+ − + + + =⎜ ⎟⎝ ⎠

, (3b)

together with the self-consistency condition 12

1

2=Φ∑

=jj . The total intensity of

angular two-component vector solitons is ( )2ruI = , which depends on the radial

coordinate only. Here k denotes the propagation constant, m is the topological charge. The topological charge may be a fractional (noninteger) quantity; this issue has been theoretically discussed in Ref. [46] and has been experimentally demonstrated, see e.g., Refs. [47–49]. Obviously, Eq. (3a) has an analytical solution: ( ) ( ) ( )ϕϕϕ mBmA jjj sincos +=Φ , (4)


with complex coefficients Aj and Bj satisfying the conditions ( )∑=

∗ =2

1

0Rej

jjBA and

∑ ∑= =

==2

1

2

1

221

j jjj BA . Then we obtain the following relations for the complex

coefficients Aj and Bj [40]:

( ) 2121 1 −

+= qA , 1 1iB qA= , (5a) 12 qAA = , 2 1iB A= ± , (5b) where [ ]0,1q∈ is called the modulation depth of the beam. From Eq. (4) and Eq. (5), it is easy to find out that the intensities of each component of angular vector solitons are modulated by the azimuthal angle φ and the modulation depth q, while the composite solitons display the typical vortex structure [40].

To obtain the exact solution of Eq. (3b), we define the similarity

transformation [50]:

( ) ( ) ( ),u r X V Rρ= (6)

where ( ) ( )2dR r r rρ= ∫ , ( )Xρ is the amplitude of the soliton, assumed to be a

real function, and X = X(r) is the similarity variable to be determined later. In the expression (6), V(R) obeys the stationary NLS equation:

2

32

1 d 02 d

VEV VR

+ + = , (7)

where E denotes the eigenvalue of the corresponding nonlinear differential equation. The reduction of the coupled NLS equations (1) to Eq. (3b) helps one to obtain families of exact solutions, as Eq. (7) is solvable depending on the value of the eigenvalue E, which can be either positive or negative. In the case when the eigenvalue is negative (E < 0), Eq. (7) possesses the well-known bright solitary wave solution: ( ) ( )2 sech 2V R E E R= − − . (8)

The substitution of Eq. (6) into Eq. (3b) leads to Eq. (7); thus one obtains a set of partial differential equations (PDEs) for the unknown functions ( )rχ , ( )rX , U(r), and ρ = ρ (X). Solving these PDEs, we find the following solutions: ( ) 2 2

rr ERχ ρ= − , 2X r= , where ρ(X) obeys the following differential equation:

22 2

2 2

d 1 d 0d d 2 4

rU k ER mX X X X Xρ ρ ρ

⎡ ⎤+ −+ + − =⎢ ⎥

⎣ ⎦. (9)


Now introducing another useful transformation ( ) ( ) XXFX =ρ , the external potential U(r) and the real constant k are chosen as follows: ( ) 2 22 rU r r ER= − + and 2 1k n m= − − . Then, we finally obtain

( )22

2 2

1 4 2d 1 1 2 2 2 0,d 4

mF k m n FX X X

⎛ ⎞−+ + −⎜ ⎟+ − + + =⎜ ⎟⎝ ⎠

(10)

where n (= 0, 1, 2, …) is called the radial quantum number. It is worth mentioning that Eq. (10) is the standard Whittaker differential

equation [51], whose solution is given by F(X) = c1W(n, m + 1, X) + c2M(n, m + 1, X), where W and M are the Whittaker special functions and the two integration constants c1 and c2 satisfy the condition c1c2 > 0. Thus, collecting all these exact solutions together, we finally obtain the analytical vector soliton solutions of Eq. (1) in the form:

( ) ( ) ( ) ( )

( ) ( )

2 21 2

1 2

2 ,1 , ,1 , cos sin

×sech 2 e ,

j j j

i m n z

E cW n m r c M n m r A m B mr

ER

ψ ϕ ϕ

+ −

− ⎡ ⎤⎡ ⎤= + + + + ×⎣ ⎦⎣ ⎦

−

(11)

where Aj and Bj are determined by Eq. (5) and R(r) is given by

( )( ) ( ) 2

2 21 2

d

, 1, , 1,

r rR rc W n m r c M n m r

=⎡ ⎤+ + +⎣ ⎦∫ .

It is worth mentioning that in order to avoid introducing singularities of the function R(r), we should appropriately choose the integration constants c1, c2, and the two free parameters n and m.

We first investigate some key features of the nonlinearity parameter ( )rχ and the external potential U(r). Figure 1 shows the radial profiles of ( )rχ and U(r) for c1 = c2 = 1, the eigenvalue E = –100, m = 0.5, and for several values of n. In this case, the eigenvalue E should be a negative quantity and thus the nonlinearity parameter ( )rχ is a positive number (see, for example, [52, 53]); Fig. 1a illustrates the shapes of the nonlinearity parameter ( )rχ for two values of n (n = 1, 2). The external potential ( )rU corresponding to the nonlinearity parameter shown in Fig. 1a is plotted in Fig. 1b for the same values of n. It can be seen that U(r) initially decreases with increasing the radial coordinate r, before reaching a minimum value; then it increases to a maximum value, and finally it monotonously decreases. Such a potential is readily realizable in experimental settings, see e.g., [54].


(a)

(b)

Fig. 1 – Color online: a) the nonlinearity parameter ( )rχ ; b) the external potential U(r) for different values of n. The other parameters are given in the text.

3. THE ANALYSIS AND DISCUSSION OF RESULTS

In this section, we show by means of some typical examples that the intensity 2uI = of the two-soliton complex can be controlled by varying the three free parameters n, m, and the modulation depth q, which describe the obtained exact solution (11). Throughout this paper we take the values of the parameters c1 = c2 = 1 and E = –100. We thus have 21 III += , where 2

11 ψ=I and 222 ψ=I ; here I

is the total intensity distribution, whereas I1 and I2 are the intensities of the two components of the vector soliton, respectively.

First, our interest will be in the effect of the topological charge m in Eq. (11) for each component of the composite (vector) soliton. According to Eq. (11), there exist two distinct types of soliton families, namely, either symmetric or asymmetric necklace-ring soliton solutions, which depend on the parameter m being either an integer number or a fractional one.

An interesting example with m being an integer number would be for q = 0 and different non-negative integer numbers n. For this case, we see the occurrence of either a symmetrical necklace-ring pattern or a vortex-ring soliton.

In Fig. 2, we display the vector soliton profiles for m=10 in the transverse (x, y) space. As shown in the top row of Fig. 2, with the parameters n = 0 and m = 10, each component of the single-layer soliton is modulated, displaying the occurrence of 20 “petals” in the azimuthal direction, while the total intensity displays the single-layer vortex-ring pattern. When we choose n = 1, the solitons exhibit two layers in the radial direction, see the middle row in Fig. 2. However, when n = 2, there exist three layers with radial symmetry, see the bottom row in Fig. 2.


We find that the intensity is the strongest one in the innermost layer. The number of the layers in the radial direction is determined by the parameter n and the number of soliton “petals” is fixed by the parameter m. There exist (n+1) layers in the radial direction and 2m petals in each azimuthal layer. Thus these solitons are characterized by the occurrence of 2m (n+1) “petals”.

Fig. 2 – Color online – intensity profiles of the symmetric vector solitons with m = 10 and q = 0, shown for n = 0, 1, 2, from top to bottom.

Next, we consider the field distribution of each component of the two- component vector solitons in the case 1→q , to show how they behave in the (x, y) transverse space. One finds that, with the increase of the parameter q, each compo-nent changes its structure from the spiky pattern to the modulated vortex- ring.

However, when q = 1, the solitons become the so-called vortex-ring solitons. When one keeps the same setup and parameters as in Fig. 2, except for q = 0.9, the solutions given by Eq. (11) are displayed in Fig. 3. Each soliton component exhibits a symmetric sawtooth ring structure in the azimuthal direction; we clearly see both single-layer ring solitons (for n = 0) and multi-layer ring solitons (for n = 2 and n = 4).


Fig. 3 – Color online – symmetric vector soliton profiles. The setup is the same as in Fig. 2, except for

q = 0.9 and n = 0, 2, 4, from top to bottom.

In Fig. 4, we show some typical examples of asymmetric sawtooth necklace-ring and vortex-ring soliton solutions, generated for the case when q = 0.9, m = 5/2, and for different values of n. The soliton structures are quite similar to those shown in Fig. 3, excepting the fact that the two components of the vector soliton have asymmetric field distributions in the transverse plane.

For n = 0, each component of the single-layer soliton is shown in the left and the middle panels of the top row of Fig. 4; we see the typical asymmetric sawtooth necklace-ring structure. The optical intensity is strong on one side of the ring, and weak on the other side of the ring, while the total intensity displays the vortex-ring field profile. Similarly, one obtains two-layer solitons for n = 1, which are shown in the middle row of Fig. 4. Another typical example of such solitons is displayed in the bottom row of Fig. 4 when the parameter n = 2, thus exhibiting three-layer field distributions. The innermost layer has the weakest optical intensity, while the outermost shows the strongest one.

We have performed direct numerical simulations of the nonlinear evolution equation (1) with added white noise on the initial distribution, in order to investigate the stability of such vector solitons. We used the standard beam-propagation method [54–55] to perform numerical simulations of the propagation of vector solitons. The numerical simulations confirmed the stability of the analytical solution (11).


Fig. 4 – Color online – intensity profiles of two-component asymmetric vector solitons for m = 5/2 and q = 0.9 for n = 0, 1, 2, from top to bottom.

Fig. 5 – Color online – comparison of the analytical solution with the numerical simulation at z = 80.

Top row: analytical solution of Eq. (11). Bottom row: numerical simulation of Eq. (1). The initial field distribution is given by Eq. (11) with an added %10 white noise. The setup and parameters are

the same as in Fig. 2, except for m = 4 and n = 3.


A typical example of such a behavior is displayed in Fig. 5 for q = 0, i.e., the same setup and parameters as in Fig. 2, except for m = 4 and n = 3. As expected, the two-component vector solitons can stably propagate over a long distance (z = 80).

Although we have shown here the results of the stability simulations only for a single example of soliton solution of Eq. (1), similar conclusions hold for other sets of the soliton parameters q, m, and n as well.

4. CONCLUSIONS

In conclusion, we have studied the unique features of families of two-component angular vector solitons of the coupled (2+1)-dimensional nonlinear Schrödinger equations with spatially modulated nonlinearities and a special external potential. The unique dynamics of such soliton families is investigated in detail, through observing the various soliton shape patterns. Based on the obtained exact solutions, we have found that these angular two-component vector solitons can be well described through three free parameters n, m, and q, that is, the radial quantum number, the topological charge (vorticity number), and the modulation depth. Numerical simulations have been also performed, in order to compare with the exact analytical solutions and to confirm the stability and robustness of the angular vector solitons.

Acknowledgments. This work was supported by the National Natural Science Foundation of China under grant No. 61275001 and by the Natural Science Foundation of Guangdong Province, China, under Grant No. 1414050004503. D.M. was supported by the Romanian Ministry of Education and Research, Project PN-II-ID-PCE-2011-3-0083.

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ANGULAR VECTOR SOLITONS OF THE COUPLED … · OF THE COUPLED NONLINEAR SCHRÖDINGER EQUATIONS WITH...

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