Research ArticleThe Soliton Solutions and Long-Time Asymptotic Analysis for aGeneral Coupled KdV Equation

Changhao Zhang1 and Guiying Chen 2

1School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China2School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong 252059, China

Correspondence should be addressed to Guiying Chen; chenguiying@lcu.edu.cn

Received 7 January 2021; Accepted 22 April 2021; Published 7 May 2021

Academic Editor: Laurent Raymond

Copyright © 2021 Changhao Zhang and Guiying Chen. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

A general coupled KdV equation, which describes the interactions of two long waves with different dispersion relation, isconsidered. By employing the Hirota’s bilinear method, the bilinear form is obtained, and the one-soliton solution and two-soliton solution are constructed. Moreover, the elasticity of the collision between two solitons is proved by analyzing theasymptotic behavior of the two-soliton solution. Some figures are displayed to illustrate the process of elastic collision.

1. Introduction

In the soliton theory, finding the soliton solutions for thenonlinear partial equations is becoming more and moreimportant since the soliton solutions can describe manycomplex physical phenomena [1]. Many effective approacheshave been proposed, such as the inverse scattering transfor-mation method [2], the Bäcklund transformation method[3], the Darboux transformation method [4–6], the Hirotabilinear method [7–11], and the Riemann-Hilbert method[12]. Among them, the Hirota bilinear method is not onlydirect but also effective for investigating the soliton solutions.

In the past decades, the coupled Korteweg-de Vries(KdV) equations have been investigated widely and manyintegrable coupled KdV equations are found. For example,Gurses and Karasu [13] showed that the following coupledKdV equation was integrable and admitted recursion opera-tor and a bi-Hamiltonian structure:

ut + uxxx − 6uux − 6vx = 0,vt − 2vxxx + 6 uvx = 0:

(ð1Þ

In fact, this equation is Lax integrable, and the Laxpair was firstly given by Drinfeld and Sokolov [14] and

then by Bogoyavlenskii [15] and Karasu and Yurduşen[16] independently. Subsequently, this equation was alsoderived by Satsuma and Hirota [17] as one case of thefour-reduction of the KP Hierarchy. Moreover, Karasuand Yurduşen [16] proposed a Bäcklund transformationand some explicit solutions of Equation (1). As far as weknow, the soliton solutions and the collision between twosolitons have not been investigated. So in this paper, weinvestigate the following general coupled KdV equation:

ut + uxxx − 6uux + αvx = 0,vt − 2vxxx + 6 uvx = 0,

(ð2Þ

which is just the coupled KdV Equation (1) for α = −6:The rest of this paper is organized as follows. In “The

Bilinear Form and Soliton Solutions,”, the bilinear form,the one-soliton, and two-soliton of Equation (2) areobtained based on the Hirota’s direct method. In “Asymp-totic Analysis on Two-Soliton Solution,” the asymptoticbehaviors are studied to prove that the two-solitoncollision is elastic. Finally, conclusions are given in“Conclusion.”

HindawiAdvances in Mathematical PhysicsVolume 2021, Article ID 5569909, 6 pageshttps://doi.org/10.1155/2021/5569909

2. The Bilinear Form and Soliton Solutions

We implement the following dependent variable transforma-tion to Equation (2):

u = −2 log ϕð Þ½ �xx,

v = gϕ,

8<: ð3Þ

where g and ϕ are functions of x and t: Then, the followingbilinear equations of Equation (2) are obtained as follows:

DxDt +D4x

� �ϕ · ϕ = αϕg,

Dt − 2D3x

� �g · ϕ = 0,

(ð4Þ

where the D-operators [18] are defined by

Dmx D

nt a x, tð Þ · b x, tð Þ = ∂

∂x−

∂∂x′

� �m ∂∂t

−∂∂t ′

� �n

a x, tð Þb x′, t ′� �

x=x′,t=t′,��

ð5Þ

where both m and n are integers.In order to apply the perturbation method to Equation

(4) to find the soliton solutions of Equation (2), we expandfunctions g and ϕ in power series of a small parameterε as

g = εg1 + ε2g2 + ε3g3+⋯, ð6Þ

ϕ = 1 + εϕ1 + ε2ϕ2 + ε3ϕ3+⋯, ð7Þwhere gi and ϕiði = 1, 2, 3,⋯Þ are functions of x and t.Substituting Equations (6) and (7) into the bilinearEquation (4) and collecting the coefficients of parameterε, we have

ε1 : 2 ϕ1xt + ϕ1xxxxð Þ = αg1,g1t − 2g1xxx = 0,

ð8Þ

ε2 : 2 ϕ2xt + ϕ2xxxxð Þ = − DxDt +D4x

� �ϕ1 · ϕ1 + α g2 + ϕ1g1ð Þ,

g2t − 2g2xxx = − Dt − 2D3x

� �g1 · ϕ1,

ð9Þ

ε3 : 2 ϕ3xt + ϕ3xxxxð Þ = − DxDt +D4x

� �ϕ1 · ϕ2 + ϕ2 · ϕ1ð Þ

+ α g3 + ϕ1 g2 + ϕ2 g1ð Þ,

g3t − 2g3xxx = − Dt − 2D3x

� �g1 · ϕ2 + g2 · ϕ1ð Þ, ð10Þ

ε4 : 2 ϕ4xt + ϕ4xxxxð Þ = − DxDt +D4x

� �ϕ1 · ϕ3 + ϕ2 · ϕ2 + ϕ3 · ϕ1ð Þ

+ α g4 + ϕ1 g3 + ϕ2 g2 + ϕ3 g1ð Þ,

g4t − 2g4xxx = − Dt − 2D3x

� �g1 · ϕ3 + g2 · ϕ2 + g3 · ϕ1ð Þ, ð11Þ

ε5 : 2 ϕ5xt + ϕ5xxxxð Þ = − DxDt +D4x

� �ϕ1 · ϕ4 + ϕ2 · ϕ3ð

+ ϕ3 · ϕ2 + ϕ4 · ϕ1Þ + α g5 + ϕ1 g4ð+ ϕ2 g3 + ϕ3 g2 + ϕ4 g1Þ,

g5t − 2g5xxx = − Dt − 2D3x

� �g1 · ϕ4 + g2 · ϕ3 + g3ð

· ϕ2 + g4 · ϕ1Þ:ð12Þ

2.1. One-Soliton Solution. To obtain the one-soliton solu-tion for the general coupled KdV Equation (2), set

g1 = eη1 , ð13Þ

where η1 = k1x + ω1t + δ1. Substituting it into Equation(8), we have ω1 = 2k13 and

ϕ1 = b1 eη1 , b1 =16

α

k14 : ð14Þ

Furthermore, from Equation (9), we have

g2 = 0,

ϕ2 = b11 e2 η1 , b11 =1288

α2

k18 :

ð15Þ

Assuming gj = ϕj = 0, ðj = 3, 4,⋯Þ, it is easy to seethat Equation (10) and other equations from the coeffi-cients of parameter ε are satisfied automatically. So weobtain the following one-soliton solution for the generalcoupled KdV Equation (2) by setting ε = 1 :

u = −2 log 1 + ϕ1 + ϕ2ð Þ½ �xx,

v = g11 + ϕ1 + ϕ2

:

8<: ð16Þ

Figures 1(a) and 1(b) demonstrate the soliton struc-tures of one-solutions uðx, tÞ and vðx, tÞ, respectively,for parameters k1 = 1/2, α = 1.

2.2. Two-Soliton Solution. Likewise, to arrive the two-solitonsolution, we set

g1 = eη1 + eη2 , ð17Þ

where ηi = kix + ωit + δi, i = 1, 2. Plugging Equation (17) intoEquation (8) yields ωi = 2ki3, i = 1, 2, and

ϕ1 = b1 eη1 + b2 eη2 , ð18Þ

with bi = 1/6 ðα/ki4Þ, i = 1, 2.From Equations (9) and (18), we have

g2 = a12eη1+η2 ,ϕ2 = b11 e2 η1 + b22 e2 η2 + b12 eη1+η2 ,

ð19Þ

2 Advances in Mathematical Physics

with

a12 =16α k1 − k2ð Þ2 k21 − k22

� �k1

4k24 ,

b11 =1288

α2

k18 , b22 =

1288

α2

k28 ,

b12 =136

α2 k41 + k42� �

k41 k42 k21 + k22� �

k1 + k2ð Þ2 :ð20Þ

Similarly, from Equations (10), (18), and (19), g3 and ϕ3can be derived as

g3 = a112e2 η1+η2 + a122eη1+2 η2 ,ϕ3 = b112e2 η1+η2 + b122eη1+2 η2 ,

ð21Þ

with

a112 =1288

α2 k1 − k2ð Þ2k81 k1 + k2ð Þ2

,

a122 =1288

α2 k1 − k2ð Þ2k82 k1 + k2ð Þ2

,

u (x,t)

10 5020

0

t

0

X

–10–20

–0.25

–0.2

–0.1

0

–0.15

–0.05

–50

(a)

0.25

v (x,t)

0.5

0.15

0.1

0.05

020

100

0

xt

50

–10–20 –50

100

0–10

(b)

Figure 1: Time and space evolution of the antibell soliton solution uðx, tÞ and bell soliton solution vðx, tÞ in Equation (16) for parametersk1 = 1/2, α = 1. (a) uðx, tÞ. (b) vðx, tÞ.

3Advances in Mathematical Physics

t x0

0

100

–100–200

–0.20

–0.10

–0.15

–0.05

200 100

u (x,t)

500

–50–100

(a)

–150 –100 –50

–0.05

–0.10

–0.15

–0.20

0

t = 0t = –150t = 150

u (x,•)

50

x

100 150

(b)

200

100

100

0

0

xDensity of u (x,t)

t

–100

–50 50–100

–200

(c)

0.20

0.15

0.05

0.10

0–200

–1000

100200 100

500 xt

v (x,t)

–50–100

(d)

0.20

0.15

0.10

0.05

50 50 100 1500–100 –50

v (x,•)

t = 0t = –150t = 150

(e)

200

100

100

0

0

xDensity of v (x,t)

t

–100

–50 50–100

–200

(f)

Figure 2: (a, d) Time evolution of the 2-soliton solutions u and v with parameters k1 = 1/2, k2 = 1/3, α = 1, respectively. (b, e) Solitoninteraction shots of u and v at t = −150, t = 0, and t = 150. Two solitons travel from right to left and pass through each other whiletheir shapes well-maintained, implying perfect elasticity of the collision. (c, f) Density profile of the collision progress of u and v,showing the velocity keep invariable and phase shift after interaction. (a) uðx, tÞ. (b) uðx, ·Þ. (c) density of uðx, tÞ. (d) vðx, tÞ. (e) vðx, ·Þ.(f) density of vðx, tÞ.

4 Advances in Mathematical Physics

b112 =1

1728α3 k1 − k2ð Þ2

k81 k42 k1 + k2ð Þ2 ,

b122 =1

1728α3 k1 − k2ð Þ2

k41 k82 k1 + k2ð Þ2 : ð22Þ

Moreover, if we plug the above obtained g1, g2, g3 andϕ1, ϕ2, ϕ3 into Equation (11), we get

g4 = 0,ϕ4 = b1122e2 η1+2 η2 ,

ð23Þ

with b1122 = 1/82944 ðα4ðk1 − k2Þ4/k81 k82 ðk1 + k2Þ4Þ.Assuming gj = ϕj = 0, ðj = 5, 6,⋯Þ, it is observed that the

left equations are satisfied automatically. By setting ε = 1,we obtain the two-soliton solutions for the coupled KdVEquation (2). We i4llustrate the structures of u and v inFigures 2(a) and 2(b).

3. Asymptotic Analysis on Two-Soliton Solution

Now we analyze the two-soliton solution of Equation (2)with long-time asymptotic method. Note that the two-soliton solution can be written as

u = −2 log 1 + ϕ1 + ϕ2 + ϕ3 + ϕ4ð Þ½ �xx,

v = g1 + g2 + g31 + ϕ1 + ϕ2 + ϕ3 + ϕ4

,ð24Þ

where ϕ1, ϕ2, ϕ3, ϕ4 and g1, g2, g3, g4 are shown in Equations(17)–(23). Without loss of generality, suppose that k2 > k1 > 0.

For fixed η1, note that η2 = ðk2/k1Þη1 + 2k2ðk22 − k21Þt −ðk2/k1Þδ1 + δ2, then arrive at the following:

(i) Solitons-1 before collision (t→ −∞, η2 ~ −∞)

u1 ~ −16 k21β1 β2

1 + 4β1 + 8� �

β21 + 8β1 + 8

� �2 ,

v1 ~8 eη1

β21 + 8β1 + 8

,ð25Þ

where β1 = b1 eη1

(ii) Solitons-1 after collision (t→ +∞, η2 → +∞)

u1 ~ −16 k21β1′ β1′

2 + 4β1′ + 8� �

β1′2 + 8β1′ + 8

� �2 ,

v1 ~8 e η1+A12ð Þ

β1′2 + 8β1′ + 8

,

ð26Þ

where β1′ = b1 eðη1+B12Þ, A12 = a122, andB12 = k1 − k2

2/k1 + k2

2.

For fixed η2, note that η1 = ðk1/k2Þη2 + 2k1ðk21 − k22Þt −ðk1/k2Þδ2 + δ1, and then we arrive at the following:

(i) ′ Solitons-2 before collision (t→ −∞, η1 → +∞)

u2 ~ −16 k22β2′ β2′

2 + 4β2′ + 8� �

β2′2 + 8β2′ + 8

� �2 ,

v2 ~8 e η2+A21ð Þ

β2′2 + 8β2′ + 8

,

ð27Þ

where β2′ = b2 eðη2+B21Þ, A21 = a112, B21 = k1 − k2

2/k1 +k2

2

(ii) ′ Solitons-2 after collision (t→ +∞, η1 ~ −∞)

u2 ~ −16 k22β2 β2

2 + 4β2 + 8� �

β22 + 8β2 + 8

� �2 ,

v2 ~8 eη2

β22 + 8β2 + 8

,ð28Þ

where β2 = b2 eη2 .

The above asymptotic analysis can also be seen inFigure 3. Comparing the asymptotic expressions Equation(25) with Equation (26) and Equation (27) with Equation(28), we find that the amplitudes and velocities remain thesame, but the phases are changed. To illustrate the collisionprocess exactly, the graphs are presented in Figure 2, whichshows that the collisions of the two-soliton waves are exactlyelastic.

Interactionrange

u2

v2

u1

v1

u1

v18eη1+A12

8eη2+A12

(𝛽1′2+8𝛽1′+8)2

𝛽1′2+8𝛽1′+8

𝛽2′2+8𝛽2′+8

–16 k12 (𝛽1′

2+4𝛽1′+ 8)

–16 k22 (𝛽2′

2+4𝛽2′+8)

u2

v28eη2

8eη2

𝛽2+8𝛽2+8

𝛽1+8𝛽1+8

t

Solitons-1

Solitons-2

Solitons-2

Solitons-1

2

2

–16 k22 (𝛽2 +4𝛽2+8) 2

–16 k1 (𝛽1+4𝛽1+8) 22

(𝛽2+8𝛽2+8)2

(𝛽1+8𝛽1+8)2

2

2(𝛽2′

2+8𝛽2′+8)2

Figure 3: Two-soliton collisions in the coupled KdV system.

5Advances in Mathematical Physics

4. Conclusion

In conclusion, we studied a general coupled KdV Equation(2)via the Hirota’s bilinear method. We first constructedthe bilinear form and then the one-soliton solution andtwo-soliton solution. Furthermore, the asymptotic analysisis given to prove that the collision of the two-soliton solu-tions is elastic.

Data Availability

All data, models, and code generated or used during thestudy appear in the submitted article.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Authors’ Contributions

Changhao Zhang is responsible for methodology, data cura-tion, software, and validation. Guiying Chen is responsiblefor conceptualization, writing-original draft, visualization,writing (review and editing), supervision, and project admin-istration. All authors have read and agreed to the publishedversion of the manuscript.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under grant No. 62071053 and PHDResearch Foundation of Liaocheng University under grantNo. 318051431. The authors would like to express theirsincere thanks to the editors and reviewers for the kindcomments and valuable suggestions.

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