# Solitary Solutions to the Fractional Generalized Hirota–Satsuma-Coupled KdV Equations

Authors: Jing Liu; Zhao Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, the complete discrimination system method is used to the fractional generalized Hirota–Satsuma-coupled KdV equations, which depicts the dispersive long wave widely applied in fluid mechanics. First, by the wave transformation, the fractional generalized Hirota–Satsuma-coupled KdV equation can be transformed into ordinary differential equations. Second, a series of new solutions are constructed by the discrimination system method of the fourth order polynomial, these solutions include kink solitary wave solutions, Jacobian elliptic function double periodic solutions, trigonometric function solution, solitary wave solutions, two-period solutions of elliptic function, and implicit solutions formula solutions. Finally, the comparison between our results and others’ works is also given.


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## Body

## 1. Introduction

In this paper, we study the fractional generalized Hirota–Satsuma-coupled KdV equations as follows [1–4]:(1)Dtαu=14Dx3βu+3uDxβu+3Dxβ−v2+ω,Dtαv=−12Dx3βv−3uDxβv,Dtαω=−12Dx3βω−3uDxβω,where u=ut,x,v=vt,x, and ω=ωt,x are unknown functions. Dtαu and Dxβu stand for the conformable fractional derivatives.In 1981, the Hirota–Satsuma KdV equation was first proposed by Hirota–Satsuma [5]; this equation is widely used to describe complex phenomena in physics, such as electrodynamics, heat transfer, dynamics of light pulses, solid-state physics, nonlinear optics, etc. So far, many effective methods have been proposed to study it [6–9]. In the study [10], authors obtained the exact solutions expressed by the Jacobian elliptic functions. In the study [11, 12], the authors constructed new exact wave soliton solutions via the direct algebraic method. In the study [13], it is proved that (1) has four kinds of soliton solutions that employed the tanh function method. In the study [14], Fan et al. constructed double periodic wave solutions and proved its degenerate to the soliton solutions at a limit condition. In the study [15], Ali studied (1) multiple traveling wave solutions. Various initial value systems are considered by the extended homogeneous balance method [16, 17]. Homotopy analysis method [18] and homotopy perturbation method [19] are used to study (1). Some articles investigated the fractional partial differential equation by utilizing the bifurcation, stability analysis, traveling wave solutions, and phase portraits (see [16, 20–30]). In 1996, Yang [31] et al. introduced the complete discriminant system for solving higher-order polynomial equations. Moreover, the method was widely used in the solution of the partial differential equation by Liu [32]. In this paper, in order to obtain a multiform solution and further explore the physical properties of this equation, we intend to investigate equation (1) in accordance with the complete discrimination system method.The paper is arranged as follows: In Section2, we use the wave transformation (1) into ordinary differential equations (ODEs) and obtain the solitary solutions. In Section 3, we draw three-dimensional and two-dimensional graphics of some solutions. In Section 4, we present the conclusion.


## 2. Solitary Solutions of (1)

We use the following wave transformation:(2)ut,x=Uξ,vt,x=Vξ,ωt,x=Wξ,ξ=kxββ−ctαα,in which α>0 and β>0.Substituting (2) into (1), we obtain as follows:(3)−ckU′=14k3U′′′+3kUU′+3k−V2+W′,−E=−12k3V′′′−3kUV′,−ckW′=−12k3W′′′−3kUW′.Aimed at eliminating the coupling relationship of the equations above, the assumption is made as follows:(4)U=a1V2+b1V+c1,W=a2V+b2,where a1,a2,b1,b2, and c1 are constants.Substituting equation (4) into (3) with a once integration followed, and we can find that equation (3) shares the same following result:(5)k2V″=−2a1V3−3b1V2+2c−3c1V+d1,where d1 is the integral constant.By multiplying (5) by V′ and integrating, (5) is obtained as follows:(6)k2V′2=−a1V4−2b1V3+2c−3c1V2+d1V+d2,where d2 is the integral constant.Whena1<0 and k≠0, we assume that(7)X=−a1k21/4V+b12a1,ξ1=−a1k21/4ξ.Substituting (7) into (6), we obtain as follows:(8)Xξ12=FX=X4+P2X2+P1X+P0,where P2=c−3c1a1+3b12/2a1k2−a1/k21/2,P1=−b13/a12k2+2b1c−3c1/k2a1−d1/k2−a1/k21/4,P0=3b14/16a13k2+b12c−3c1/2a12k2−b1d1/2k2a1+d2/k2Whena1>0 and k≠0, we assume that(9)X=a1k21/4V+b12a1,ξ1=a1k21/4ξ.Substituting (7) into (6), we obtain as follows:(10)Xξ12=−FX=−X4+P2X2+P1X+P0,where P2=c−3c1a1+3b12/2a1k2−a1/k21/2,P1=−b13/a12k2+2b1c−3c1/k2a1−d1/k2−a1/k21/4,P0=3b14/16a13k2+b12c−3c1/2a12k2−b1d1/2k2a1+d2/k2Here, we only consider the case of equation solution whena1<0 and k≠0, and write (6) in following integral form:(11)±ξ1−ξ0=∫dXX4+P2X2+P1X+P0.The complete discriminant system corresponds to the quartic polynomial as follows:(12)D1=4,D2=−P2,D3=−2P23+8P2P0−9P12,D4=−P23P12+4P24P0+36P2P12P0−32P22P02−274P14+64P03,E2=9P22−32P2P0.According to the complete discriminant system, the classification of all solutions of (1) can be obtained, and there are the following nine cases.Cases 1.
SupposeD2<0,D3=0,D4=0. FX has a pair of conjugated complex roots, i.e., FX=X−x12+x222, where x1 and x2 are real numbers, x2>0. The solutions of (4) can be presented as follows:(13)V1=12P2tan12P2−a1k21/4ξ−ξ0.Cases 2.
SupposeD2=0,D3=0,D4=0. FX=0 possesses a quadruple real root. Setting FX=X4, the solutions of (6) can be presented as follows:(14)V2=−−a1k21/4ξ−ξ0−1.Cases 3.
SupposeD2>0,D3=0,D4=0, E2>0, there are two differences double real roots, FX=X−x32X−x42, x3 and x4 are constants, and x3>x4. Then, x3=P01/4, x3=−x4, P22=2P0, and P0>0.
WhenX>x3 or X<x4, the solutions of (6) can be presented as follows:(15)V31=−P01/4cothP01/4−a1k21/4ξ−ξ0−1−P01/4.
Whenx4<X<x3, the solutions of (6) take the following form:(16)V32=−P01/4tanhP01/4−a1k21/4ξ−ξ0−1−P01/4.Cases 4.
SupposeD2>0,D3>0,D4=0, E2>0, there are double real roots and two single roots. FX=X−x52X−x6X−x7, where x5,x6, and x7 are real numbers. When x5>x6,X>x6, or when x5<x7,X<x7, the solutions of (6) can be presented as follows:(17)±−a1k21/4ξ−ξ0=1x5−x6x5−x7lnX−x6x5−x7−x5−x6X−x72X−x5.
Whenx5>x6,X<x7, or when x5<x7,X<x6, the solutions of (6) can be presented as follows:(18)±−a1k21/4ξ−ξ0=1x5−x6x5−x7lnX−x6x7−x5−x6−x5X−x72X−x5.
Whenx7<x5<x6, or when x5<x7, X<x6, the solutions of (6) can be presented as follows:(19)±−a1k21/4ξ−ξ0=1x6−x5x5−x7arcsinX−x6x5−x7+x5−x6X−x72X−x5x6−x7.Cases 5.
Suppose thatD2>0,D3=0,D4=0, E2=0. So, P0=−243/256,P1=−27/8, P2=−27/8 or P0=−243/256,P1=27/8, P2=−27/8. There are triple real roots and a single real root, FX=X−x83X−x9, where x8=3/4,x9=−9/4 or x8=−3/4,x9=9/4.
WhenX>3/4 or when X<−9/4, the solutions of (6) take the following form:(20)V51=43−a1/k21/2ξ−ξ02−4+34.
Meanwhile, we derive extra solutions of (6) as follows:(21)V52=−43−a1/k21/2ξ−ξ02−4−34.Cases 6.
SupposeD2D3<0,D4=0, we have FX=X−x102X−x112+x122, so the solutions of (6) take the following form:(22)V6=−e±x10−x112+x122−a1/k21/4ξ−ξ0−γ+x10−x112+x1222−γe±x10−x112+x122−a1/k21/4ξ−ξ0−γ2−1,where γ=x10−2x11/x10−2x112+x122; this solution is a solitary wave solution.Cases 7.
SupposeD4>0,D3>0,D1>0, then FX=0 admits four different roots. We denote FX=X−x13X−x14X−x15X−x16, where x13,x14,x15, x16 are real numbers and x13>x14>x15>x16. The solutions of (6) take the following form:(23)V71=x14x13−x16sn2x13−x15x14−x16/2−a1/k21/4ξ−ξ0,m−x13x14−x16x13−x16sn2x13−x15x14−x16/2−a1/k21/4ξ−ξ0,m−x14−x16,V72=x16x14−x15sn2x13−x15x14−x16/2−a1/k21/4ξ−ξ0,m−x15x14−x15x14−x15sn2x13−x15x14−x16/2−a1/k21/4ξ−ξ0,m−x14−x16.Cases 8.
SupposeD4<0,D3D2≥0, then we denote FX=X−x17X−x18X−x192+x202, x19,x20 are real numbers and x17>x18. Taking the transformation X=ε1cosφ+ε2/ε3cosφ+ε4, where ε1=1/2x17+x18ε3−1/2x17−x18ε4, ε2=1/2x17+x18ε4−1/2x17−x18ε3, ε3=x17−x19−x20/m1, ε4=x17−x19−x20m1, E1=x202+x17−x19x18−x19/x20x17−x18, m1=E1±E12+1, m02=1/1+m12, the solutions of (6) take the following form:(24)V8=ε1cn−2x20m1x17−x18/2m0m1−a1/k21/4ξ−ξ0,m0+ε2ε3cn−2x20m1x17−x18/2m0m1−a1/k21/4ξ−ξ0,m0+ε4,where this solution is a two-period solution of the elliptic function.Cases 9.
SupposeD4>0,D3D2≤0, then we denote FX=X−x212+x222X−x232+x242, x21, x22, x23, and x24 are real numbers, and x22≥x24>0. Taking the transformation X=σ1tanφ+σ2/σ3tanφ+σ4, where σ1=x21σ3+x22σ4, σ2=x21σ4−x22σ3, σ3=−x22−x24/m2, σ4=x21−x23, E2=x21−x232+x222+x242/2x22x24, m2=E2±E22+1, m32=m22−2/m2, the solutions of (6) takes the following form:(25)V9=σ1snη−a1/k21/4ξ−ξ0,m3+σ2cnη−a1/k21/4ξ−ξ0,m3σ3snη−a1/k21/4ξ−ξ0,m3+σ4cnη−a1/k21/4ξ−ξ0,m3,where η=x24σ32+σ42m22σ32+σ42/σ32+σ42.Remark 1.
In the previous discussion, we only gave the solutions whena1<0 and k≠0. When a1>0 and k≠0, we can also use the complete discriminant system to get the corresponding solutions.Remark 2.
In this article, only the solutions of equationV is obtained, we can easily obtain the solutions of V and W according to the relationship between V and U and W in (4).


## 3. Graphical Representation of the Obtained Solutions

By employing Maple software, we draw some of the obtained solutions in Figures1 and 2.Figure 1
SolutionV1 for (13) with a1=−1, b1=0,c=−1,c1=−1,d2=4,k=1,β=2,α=1,ξ0=0.1. (a) Three-dimensional and (b) two-dimensional.
(a)(b)Figure 2
SolutionV32 for (16) with a1=−1, b1=0,c=1,c1=−1,d2=4,k=1,β=2,α=1,ξ0=2. (a) Three-dimensional and (b) two-dimensional.
(a)(b)


## 4. Conclusion

In the paper, the complete discrimination system method of the fourth order polynomial is applied to find the solution for equation (1). Compared with the existing literature([1–4]), the Jacobian function solutions and implicit solutions are also obtained. The results show that the complete discrimination system method is a powerful mathematical tool of an efficient way to obtain solutions of many forms using one method to solve the fractional generalized Hirota–Satsuma-coupled KdV equation. As a result, the complete discrimination system method can be thought of as a powerful, robust, and versatile mathematical approach for finding analytical solutions to nonlinear PDEs; it is also a promising method to solve other nonlinear equations.



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