# The Soliton Solutions and Long-Time Asymptotic Analysis for a General Coupled KdV Equation

Authors: Changhao Zhang; Guiying Chen
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

A general coupled KdV equation, which describes the interactions of two long waves with different dispersion relation, is considered. 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 the asymptotic behavior of the two-soliton solution. Some figures are displayed to illustrate the process of elastic collision.


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## Body

## 1. Introduction

In the soliton theory, finding the soliton solutions for the nonlinear partial equations is becoming more and more important since the soliton solutions can describe many complex physical phenomena [1]. Many effective approaches have been proposed, such as the inverse scattering transformation method [2], the Bäcklund transformation method [3], the Darboux transformation method [4–6], the Hirota bilinear method [7–11], and the Riemann-Hilbert method [12]. Among them, the Hirota bilinear method is not only direct but also effective for investigating the soliton solutions.In the past decades, the coupled Korteweg-de Vries (KdV) equations have been investigated widely and many integrable coupled KdV equations are found. For example, Gurses and Karasu [13] showed that the following coupled KdV equation was integrable and admitted recursion operator and a bi-Hamiltonian structure:
(1)ut+uxxx−6uux−6vx=0,vt−2vxxx+6uvx=0.In fact, this equation is Lax integrable, and the Lax pair was firstly given by Drinfeld and Sokolov [14] and then by Bogoyavlenskii [15] and Karasu and Yurduşen [16] independently. Subsequently, this equation was also derived by Satsuma and Hirota [17] as one case of the four-reduction of the KP Hierarchy. Moreover, Karasu and Yurduşen [16] proposed a Bäcklund transformation and some explicit solutions of Equation (1). As far as we know, the soliton solutions and the collision between two solitons have not been investigated. So in this paper, we investigate the following general coupled KdV equation:
(2)ut+uxxx−6uux+αvx=0,vt−2vxxx+6uvx=0,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) are obtained based on the Hirota’s direct method. In “Asymptotic Analysis on Two-Soliton Solution,” the asymptotic behaviors are studied to prove that the two-soliton collision is elastic. Finally, conclusions are given in “Conclusion.”


## 2. The Bilinear Form and Soliton Solutions

We implement the following dependent variable transformation to Equation (2):
(3)u=−2logϕxx,v=gϕ,where g and ϕ are functions of x and t. Then, the following bilinear equations of Equation (2) are obtained as follows:
(4)DxDt+Dx4ϕ·ϕ=αϕg,Dt−2Dx3g·ϕ=0,where the D-operators [18] are defined by
(5)DxmDtnax,t·bx,t=∂∂x−∂∂x′m∂∂t−∂∂t′nax,tbx′,t′x=x′,t=t′,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 expand functions g and ϕ in power series of a small parameter ε as
(6)g=εg1+ε2g2+ε3g3+⋯,(7)ϕ=1+εϕ1+ε2ϕ2+ε3ϕ3+⋯,where gi and ϕii=1,2,3,⋯ are functions of x and t. Substituting Equations (6) and (7) into the bilinear Equation (4) and collecting the coefficients of parameter ε, we have
(8)ε1:2ϕ1xt+ϕ1xxxx=αg1,g1t−2g1xxx=0,(9)ε2:2ϕ2xt+ϕ2xxxx=−DxDt+Dx4ϕ1·ϕ1+αg2+ϕ1g1,g2t−2g2xxx=−Dt−2Dx3g1·ϕ1,(10)ε3:2ϕ3xt+ϕ3xxxx=−DxDt+Dx4ϕ1·ϕ2+ϕ2·ϕ1+αg3+ϕ1g2+ϕ2g1,g3t−2g3xxx=−Dt−2Dx3g1·ϕ2+g2·ϕ1,(11)ε4:2ϕ4xt+ϕ4xxxx=−DxDt+Dx4ϕ1·ϕ3+ϕ2·ϕ2+ϕ3·ϕ1+αg4+ϕ1g3+ϕ2g2+ϕ3g1,g4t−2g4xxx=−Dt−2Dx3g1·ϕ3+g2·ϕ2+g3·ϕ1,(12)ε5:2ϕ5xt+ϕ5xxxx=−DxDt+Dx4ϕ1·ϕ4+ϕ2·ϕ3+ϕ3·ϕ2+ϕ4·ϕ1+αg5+ϕ1g4+ϕ2g3+ϕ3g2+ϕ4g1,g5t−2g5xxx=−Dt−2Dx3g1·ϕ4+g2·ϕ3+g3·ϕ2+g4·ϕ1.


### 2.1. One-Soliton Solution

To obtain the one-soliton solution for the general coupled KdV Equation (2), set
(13)g1=eη1,where η1=k1x+ω1t+δ1. Substituting it into Equation (8), we have ω1=2k13 and
(14)ϕ1=b1eη1,b1=16αk14.Furthermore, from Equation (9), we have
(15)g2=0,ϕ2=b11e2η1,b11=1288α2k18.Assuminggj=ϕj=0,j=3,4,⋯, it is easy to see that Equation (10) and other equations from the coefficients of parameter ε are satisfied automatically. So we obtain the following one-soliton solution for the general coupled KdV Equation (2) by setting ε=1:(16)u=−2log1+ϕ1+ϕ2xx,v=g11+ϕ1+ϕ2.Figures1(a) and 1(b) demonstrate the soliton structures of one-solutions ux,t and vx,t, respectively, for parameters k1=1/2,α=1.Figure 1
Time and space evolution of the antibell soliton solutionux,t and bell soliton solution vx,t in Equation (16) for parameters k1=1/2,α=1. (a) ux,t. (b) vx,t.
(a)(b)


### 2.2. Two-Soliton Solution

Likewise, to arrive the two-soliton solution, we set(17)g1=eη1+eη2,where ηi=kix+ωit+δi, i=1,2. Plugging Equation (17) into Equation (8) yields ωi=2ki3, i=1,2, and
(18)ϕ1=b1eη1+b2eη2,with bi=1/6α/ki4,i=1,2.From Equations (9) and (18), we have
(19)g2=a12eη1+η2,ϕ2=b11e2η1+b22e2η2+b12eη1+η2,with
(20)a12=16αk1−k22k12−k22k14k24,b11=1288α2k18,b22=1288α2k28,b12=136α2k14+k24k14k24k12+k22k1+k22.Similarly, from Equations (10), (18), and (19), g3 and ϕ3 can be derived as
(21)g3=a112e2η1+η2+a122eη1+2η2,ϕ3=b112e2η1+η2+b122eη1+2η2,with(22)a112=1288α2k1−k22k18k1+k22,a122=1288α2k1−k22k28k1+k22,b112=11728α3k1−k22k18k24k1+k22,b122=11728α3k1−k22k14k28k1+k22.Moreover, if we plug the above obtainedg1,g2,g3 and ϕ1,ϕ2,ϕ3 into Equation (11), we get
(23)g4=0,ϕ4=b1122e2η1+2η2,with b1122=1/82944α4k1−k24/k18k28k1+k24.Assuminggj=ϕj=0,j=5,6,⋯, it is observed that the left equations are satisfied automatically. By setting ε=1, we obtain the two-soliton solutions for the coupled KdV Equation (2). We i4llustrate the structures of u and v in Figures 2(a) and 2(b).Figure 2
(a, d) Time evolution of the 2-soliton solutionsu and v with parameters k1=1/2,k2=1/3,α=1, respectively. (b, e) Soliton interaction 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 while their 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) ux,t. (b) ux,·. (c) densityofux,t. (d) vx,t. (e) vx,·. (f) densityofvx,t.
(a)(b)(c)(d)(e)(f)


## 2.1. One-Soliton Solution

To obtain the one-soliton solution for the general coupled KdV Equation (2), set
(13)g1=eη1,where η1=k1x+ω1t+δ1. Substituting it into Equation (8), we have ω1=2k13 and
(14)ϕ1=b1eη1,b1=16αk14.Furthermore, from Equation (9), we have
(15)g2=0,ϕ2=b11e2η1,b11=1288α2k18.Assuminggj=ϕj=0,j=3,4,⋯, it is easy to see that Equation (10) and other equations from the coefficients of parameter ε are satisfied automatically. So we obtain the following one-soliton solution for the general coupled KdV Equation (2) by setting ε=1:(16)u=−2log1+ϕ1+ϕ2xx,v=g11+ϕ1+ϕ2.Figures1(a) and 1(b) demonstrate the soliton structures of one-solutions ux,t and vx,t, respectively, for parameters k1=1/2,α=1.Figure 1
Time and space evolution of the antibell soliton solutionux,t and bell soliton solution vx,t in Equation (16) for parameters k1=1/2,α=1. (a) ux,t. (b) vx,t.
(a)(b)


## 2.2. Two-Soliton Solution

Likewise, to arrive the two-soliton solution, we set(17)g1=eη1+eη2,where ηi=kix+ωit+δi, i=1,2. Plugging Equation (17) into Equation (8) yields ωi=2ki3, i=1,2, and
(18)ϕ1=b1eη1+b2eη2,with bi=1/6α/ki4,i=1,2.From Equations (9) and (18), we have
(19)g2=a12eη1+η2,ϕ2=b11e2η1+b22e2η2+b12eη1+η2,with
(20)a12=16αk1−k22k12−k22k14k24,b11=1288α2k18,b22=1288α2k28,b12=136α2k14+k24k14k24k12+k22k1+k22.Similarly, from Equations (10), (18), and (19), g3 and ϕ3 can be derived as
(21)g3=a112e2η1+η2+a122eη1+2η2,ϕ3=b112e2η1+η2+b122eη1+2η2,with(22)a112=1288α2k1−k22k18k1+k22,a122=1288α2k1−k22k28k1+k22,b112=11728α3k1−k22k18k24k1+k22,b122=11728α3k1−k22k14k28k1+k22.Moreover, if we plug the above obtainedg1,g2,g3 and ϕ1,ϕ2,ϕ3 into Equation (11), we get
(23)g4=0,ϕ4=b1122e2η1+2η2,with b1122=1/82944α4k1−k24/k18k28k1+k24.Assuminggj=ϕj=0,j=5,6,⋯, it is observed that the left equations are satisfied automatically. By setting ε=1, we obtain the two-soliton solutions for the coupled KdV Equation (2). We i4llustrate the structures of u and v in Figures 2(a) and 2(b).Figure 2
(a, d) Time evolution of the 2-soliton solutionsu and v with parameters k1=1/2,k2=1/3,α=1, respectively. (b, e) Soliton interaction 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 while their 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) ux,t. (b) ux,·. (c) densityofux,t. (d) vx,t. (e) vx,·. (f) densityofvx,t.
(a)(b)(c)(d)(e)(f)


## 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
(24)u=−2log1+ϕ1+ϕ2+ϕ3+ϕ4xx,v=g1+g2+g31+ϕ1+ϕ2+ϕ3+ϕ4,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 fixedη1, note that η2=k2/k1η1+2k2k22−k12t−k2/k1δ1+δ2, then arrive at the following:
(i)
Solitons-1 before collision (t→−∞, η2~−∞)
(25)u1~−16k12β1β12+4β1+8β12+8β1+82,v1~8eη1β12+8β1+8,where β1=b1eη1(ii)
Solitons-1 after collision (t→+∞, η2→+∞)
(26)u1~−16k12β1′β1′2+4β1′+8β1′2+8β1′+82,v1~8eη1+A12β1′2+8β1′+8,where β1′=b1eη1+B12,A12=a122,andB12=k1−k22/k1+k22.For fixedη2, note that η1=k1/k2η2+2k1k12−k22t−k1/k2δ2+δ1, and then we arrive at the following:
(i)
′ Solitons-2 before collision (t→−∞, η1→+∞)
(27)u2~−16k22β2′β2′2+4β2′+8β2′2+8β2′+82,v2~8eη2+A21β2′2+8β2′+8,where β2′=b2eη2+B21,A21=a112,B21=k1−k22/k1+k22(ii)
′ Solitons-2 after collision (t→+∞, η1~−∞)
(28)u2~−16k22β2β22+4β2+8β22+8β2+82,v2~8eη2β22+8β2+8,where β2=b2eη2.The above asymptotic analysis can also be seen in Figure3. Comparing the asymptotic expressions Equation (25) with Equation (26) and Equation (27) with Equation (28), we find that the amplitudes and velocities remain the same, but the phases are changed. To illustrate the collision process exactly, the graphs are presented in Figure 2, which shows that the collisions of the two-soliton waves are exactly elastic.Figure 3
Two-soliton collisions in the coupled KdV system.


## 4. Conclusion

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



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