# Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

Authors: Tianyong Han; Zhao Li
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.


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## Body

## 1. Introduction

The coupled system is composed of two or more differential equations (include ordinary differential equations, partial differential equations fractional partial differential equations, and stochastic partial differential equations) [1–3]. It is a very important class of mathematical and physical equations. In recent years, coupled systems have been widely studied by scholars because they come from physics, chemistry, communication, and engineering [4–8]. Among them, constructing the exact traveling wave solution of this kind of coupled system is a very important topic. Many meaningful methods have been proposed to solve the exact solutions of coupled systems, including Lie symmetry analysis [9], the method of dynamical systems [10, 11], Fan subequation method [12], generalized Jacobi elliptic function expansion method [13], extended modified auxiliary equation mapping method [14], and extended modified auxiliary equation mapping method [15].The fractional coupled Boussinesq equations [16, 17] are a very important coupled system, which is usually used to simulate nonlinear shallow water surface wave phenomena.
(1)Dtαut,x+Dxβvt,x=0,Dtαvt,x+aDxβu2t,x−bDxxx3βut,x=0,0<α,β≤1,b≠0.In [16], Muhannad et al. constructed the traveling wave solutions of fractional coupled Boussinesq equations by using the modified extended Tanh method. In [17], Khatun et al. obtained some explicit solutions of fractional coupled Boussinesq equations by the double G′/G,1/G expansion method. The work of references [16, 17] is based on the Jumarie’s modified Riemann-Liouville derivative to study fractional coupled Boussinesq equations. Unfortunately, many literatures [18–20] have reported that the Jumarie’s modified Riemann-Liouville derivative do not satisfy the chain rule and Leibniz formula. Therefore, it is urgent to find a new fractional derivative that can not only satisfy the chain rule but also obey Leibniz formula. In [21], Khalil et al. gave the definition and properties of the fractional derivative named conformable fractional derivative, which satisfy the above two conditions. The main purpose of this paper is to attain the exact traveling wave solutions of fractional coupled Boussinesq equations in the sense of conformable fractional derivative by using the complete discrimination system method [22–24].Next, we review the definition of conformable fractional derivatives.Definition 1.
Letf:0,∞⟶R. Then, the conformable fractional derivative of f of order α is defined as
(2)Dtαft=limε⟶0ft+εt1−α−ftε,∀t∈0,+∞,α∈0,1.The function f is α-conformable differentiable at a point t if the limit in Equation (2) exists.Theorem 2.
Assume thatf,g:0,∞⟶R be differentiable and also α differentiable functions, then chain rule holds
(3)Dtαf∘gt=t1−αgtα−1g′tDtαftt=gt.The structure of this paper is as follows. In Section2, we simplify Equation (1) to nonlinear ordinary differential equations by fractional traveling wave transformation. Then, the complete discrimination system is used to construct the classification of all single traveling wave solutions of fractional coupled Boussinesq equations. In Section 3, we give a summary.


## 2. Exact Solutions of System (1)

Now, we introduce the transformation(4)ξ=xββ−ctαα,where c is constant.Substituting (4) into Equation (1) and integrating it with respect to ξ, we obtain
(5)c1−cu+v=0,c2−cv+au2−bu′′=0,where c1 and c2 are integral constants.From the first equation of Equation (5), we have
(6)v=cu−c1.Substituting (6) into the second equation of Equation (5), we obtain
(7)c2+c1c−c2u+au2−bu′′=0.Multiplying Equation (8) by u′ and integrating it with respect to ξ, we obtain
(8)u′2=2a3bu3−c2bu2+2c2+2c1cbu+2c3,where c3 is the integral constant.Supposeϕ=2a/3b1/3u, b2=−c2/b2a/3b−2/3, b1=2c2+2c1c/b2a/3b−1/3, and b0=2c3. Hence, Equation (8) can be changed to
(9)ϕ′2=ϕ3+b2ϕ2+b1ϕ+b0.Assume thatfϕ=ϕ3+b2ϕ2+b1ϕ+b0, Δ=−272b23/27+b0−b0b1/32−4b1−b22/33, and D1=b1−b22/3. Then, Equation (9) can be written in the following integral form:
(10)±2a3b1/3ξ−ξ0=∫dϕϕ3+b2ϕ2+b1ϕ+b0,where ξ0 is the integration constant. Then, we will obtain the classification of all solutions of Equation (10).Case 1.
IfΔ=0 and D1<0, then fϕ=0 has a double real root and a single real root. Denote Fϕ=ϕ−γ12ϕ−γ2, where γ1≠γ2. When ϕ>γ2, we have
(11)ξ−ξ0=∫dϕϕ−γ12ϕ−γ2=1γ1−γ2lnϕ−γ2−γ1−γ2ϕ−γ2+γ1+γ2,γ1>γ2,2γ2−γ1arctanϕ−γ2γ2−γ1,γ1<γ2.
Then, the solution of Equation (8) is
(12)u1t,x=2a3b−1/3γ1−γ2tanh2γ1−γ222a3b1/3xββ−ctαα−ξ0+γ2,γ1>γ2,u2t,x=2a3b−1/3γ1−γ2coth2γ1−γ222a3b1/3xββ−ctαα−ξ0+γ2,γ1>γ2,u3t,x=2a3b−1/3−γ1+γ2tan2−γ1+γ222a3b1/3xββ−ctαα−ξ0+γ2,γ1<γ2.Remark 3.
Whena=3, b=1, c=5, β=1/3, c1=5, c2=3, c3=−2, and ξ0=0, three-dimensional diagram and two-dimensional diagram of the solution u1t,x of Equation (1) are drawn in Figure 1, respectively.Case 2.
IfΔ=0 and D1=0, then fϕ=0 has a triple real root. Denoting Fϕ=ϕ−γ3, then we have
(13)u4t,x,z=42a3b−2/3xββ−ctαα−ξ0−2+γ.Case 3.
IfΔ>0 and D1<0, then fv=0 has three different real roots, γ1, γ2, and γ3, and γ1<γ2<γ3.
Ifγ1<ϕ<γ3, taking the transformation ϕ=γ1+γ2−γ1sin2ζ, then we obtain
(14)±ξ−ξ0=∫dϕfϕ=∫2γ2−γ1sinζcosζdζγ3−γ1γ2−γ1sinζcosζ1−m2sin2ζ=2γ3−γ1∫dζ1−m2sin2ζ,where m2=γ2−γ1/γ3−γ1.Figure 1
The solutionu1t,x at a=3, b=1, c=5, β=1/3, c1=5, c2=3, c3=−2, and ξ0=0.
(a)
Perspective view of the wave(b)
The wave alongxFrom the definition of Jacobi function and (14), we have
(15)ϕ=γ1+γ2−γ1sn2γ3−γ12ξ−ξ0,m.Then, the solution of the corresponding Equation (8) is
(16)u5t,x=2a3b−1/3γ1+γ2−γ1sn2γ3−γ122a3b1/3xββ−ctαα−ξ0,m.Ifϕ>γ3, take the transformation ϕ=−γ2sin2ζ+γ3/cos2ζ. Similarly, the solution of the corresponding Equation (8) is
(17)u6t,x=2a3b−1/3γ3−γ2sn2γ3−γ1/22a/3b1/3xβ/β−ctα/α−ξ0,mcn2γ3−γ1/22a/3b1/3xβ/β−ctα/α−ξ0,m.where m2=γ2−γ1/γ3−γ1.Case 4.
IfΔ<0, then fϕ=0 has only real root. Denote Fϕ=ϕ−γϕ2+pϕ+q, where p2−4q<0.
Ifϕ>γ, taking the transformation ϕ=γ+γ2+pγ+qtan2ζ/2, then we obtain
(18)ξ−ξ0=∫dϕϕ−γϕ2+pϕ+q=∫γ2+pγ+qtanζ/2/cos2ζ/2dζγ2+pγ+q3/4tanζ/2/cos2ζ/21−m2sin2ζ=1γ2+pγ+q1/4∫dζ1−m2sin2ζ,where m2=1/21−γ+p/2/γ2+pγ+q.From the definition of Jacobi function and (18), we have cnγ2+pγ+q1/4ξ−ξ0,m=cosζ.
(19)cosζ=2γ2+pγ+qϕ−γ+γ2+pγ+q−1.Then, ifϕ>γ, the solution of the corresponding Equation (8) is
(20)ϕ7ξ=γ+2γ2+pγ+q1+cnγ2+pγ+q1/4ξ−ξ0,m−γ2+pγ+q.Similarly, the solution of the corresponding Equation (8) is
(21)u7t,x=2a3b−1/3γ+2γ2+pγ+q1+cnγ2+pγ+q1/42a/3b−1/3xβ/β−ctα/α−ξ0,m−γ2+pγ+q.(22)v1t,x=2a3b−1/3cγ1−γ2tanh2γ1−γ222a3b1/3xββ−ctαα−ξ0+γ2+c1,γ1>γ2,v2t,x=2a3b−1/3cγ1−γ2coth2γ1−γ222a3b1/3xββ−ctαα−ξ0+γ2+c1,γ1>γ2,v3t,x=2a3b−1/3c−γ1+γ2tan2−γ1+γ222a3b1/3xββ−ctαα−ξ0+γ2+c1,γ1<γ2,v4t,x,z=4c2a3b−2/3xββ−ctαα−ξ0−2+cγ+c1,v5t,x=2a3b−1/3cγ1+γ2−γ1sn2γ3−γ122a3b1/3xββ−ctαα−ξ0,m+c1,v6t,x=2a3b−1/3cγ3−γ2sn2γ3−γ1/22a/3b1/3xβ/β−ctα/α−ξ0,mcn2γ3−γ1/22a/3b1/3xβ/β−ctα/α−ξ0,m+c1,v7t,x=2a3b−1/3cγ+2γ2+pγ+q1+cnγ2+pγ+q1/42a/3b−1/3xβ/β−ctα/α−ξ0,m−γ2+pγ+q+c1.Remark 4.
In the time-space fractional coupled Boussinesq equations, the unknown functionsu and v satisfy the relationship v=cu−c1. In the paper, the traveling wave solutions of Equation (1) have been obtained by the complete discrimination system method, and we can easily obtain the solutions of v of Equation (1) by using (6).Remark 5.
The solutions obtained in references [16, 17] mainly focus on hyperbolic function solutions and trigonometric function solutions. However, in this paper, not only the trigonometric function solution and hyperbolic function solution but also the Jacobian function solution and implicit function solution are obtained. Therefore, a new solution is obtained in this paper.


## 3. Conclusion

The fractional coupled Boussinesq equations, which are usually used to simulate nonlinear shallow water surface wave phenomena, are studied by the complete discrimination system method. A series of new exact solutions are obtained, including rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Compared with the existing literature [16, 17], the implicit function solutions and Jacobian function solutions obtained in the paper are new solutions. Moreover, the complete discrimination system method can also be used to find the exact traveling wave solutions of other coupled systems. In future research work, we will focus on the exact traveling wave solution of more complex coupled systems.



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