# New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation

Authors: XiaoHua Liu; CaiXia He
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.


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## Body

## 1. Introduction

For the investigation of traveling wave solutions to nonlinear partial differential equations, which have been the subject of study in various branches of mathematical physical sciences such as physics, biology, and chemistry, many effective methods (see, e.g., [1–12]) have been presented, such as inverse scattering transform method [1], Hirota’s method [2], Backlund and Darboux transformation method [5], the Jacobi (Weierstrass) elliptic function method [7], undetermined coefficient method [9, 10],G′/G-expansion method and others.Following the work of Hirota and Satsuma in [13], recently, considerable attention has been focused on the study of coupled nonlinear partial differential equation (see, e.g., [14–19]) that can be solved exactly. However, less work on the coupled version of the higher KdV equation seems to have been reported.In [20], Guha-Roy has been presented a system of coupled nonlinear wave equations as follows
(1)ut+βu2ux+αv2vx+λuux+γuxxx=0,vt+δ(uv)x+evvx=0,
where the subscripts refer to partial differentiations with respect to the indicated variables, and α,β,γ,δ,λ,e are arbitrary parameters. Equation (1) is the coupled version of combined form of the higher (modified) KdV equation and KdV equation. It is interesting to point out that, as is outlined in Wadati [21], (1) shares properties with the KdV and the modified KdV equation, under certain conditions.Guha-Roy [22] supposed that|ξ|=|x-ct|→+∞ with u(ξ),u′(ξ),u′′(ξ)→0 by transforming tou(ξ)=1/ϕ(ξ) that Weierstrauss elliptic function, some exact solitary wave solutions in special conditions to (1) are obtained. Lu et al. [23] obtain the kink-antikink solitary wave solutions of (1) by using a truncated expansion. However, there has not been found any literature on the general analysis of the existences of the solitary wave solution and the exact expressions of the solitary wave solution with nonzero asymptotic value for (1).In this work, we investigate generally the existence of bell-shaped, kink-shaped solitary wave solutions and periodic wave solutions by using the theory of planar dynamical system under different parameter conditions. It follows that we obtain the new exact expressions of all three bell-shaped solitary wave solutions, and one kink-shaped solitary wave solution with nonzero asymptotic value for (1) by undetermined coefficient methods. These solutions obtained here cannot be deduced from [20, 23].


## 2. Existence of the Bounded Traveling Wave Solutions to (1)

By introducing an analogue of the stream function, Guha-Roy et al. [18] have shown that if one of the solutions of some coupled nonlinear equations is of the traveling wave type, then the other must also exhibit the same form. Keeping this in mind, we choose a new variable ξ=x-ct, wherecis the wave speed, such that u(x,t)=φ(ξ) and v(x,t)=ψ(ξ), substituting them to (1) yields
(2)-cφ′+βφ2φ′+αψ2ψ′+λφφ′+γφ′′′=0,-cψ′+δ(ϕφ)′+eψψ′=0,
where the prime denotes the derivative with respect toξ.Integrating the second equation of (2), we have
(3)φ=kψ+cδ-eψ2δ,
where k is the integration constant treated as an arbitrary parameters. In order to have a regular φ everywhere, we have to impose k=0. It may be noted that φ(ξ) satisfies the following boundary conditions:
(4)limx→+∞φ(ξ)=D+,limx→-∞φ(ξ)=D-,(5)φ′(ξ)⟶0,φ′′(ξ)⟶0,
as |ξ|→∞. Condition (4) is different from those considered by Guha-Roy [20], in Guha-Roy’s, φ(ξ)was found to be vanished in the infinity.Thus, (3) reduces to
(6)ψ=2c-2δφe.
This shows that ψ(ξ) is directly related to φ(ξ). By (6) and the first equation of (2), we get, after rearrangement,
(7)γφ′′+b33φ3+b22φ2-b1φ=g,
where b1=c+(8αδc2/e3), b2=λ+(16αδ2c/e3), b3=β-(8αδ3/e3), and g is an arbitrary integration constant.After the translationφ¯=φ+(b2/2b3), we rewrite (7) to
(8)γφ¯′′+b33(φ¯3+pφ¯+q)=0,
where p=-(12b1b3+3b22)/4b32 and q=(b23+6b1b2b3-12b32g)/4b33.It followsx=φ¯(ξ) andy=φ¯′(ξ) that (8) is equivalent to the two dimensional system
(9)dxdξ=y,dydξ=-b33γ(x3+px+q),
which has the first integral
(10)H(x,y)=y22+b33γ(qx+p2x2+14x4)=h.
System (9) is a four-parameter planar dynamical system depending on the parameter group (γ,b3,p,q). Because of the phase orbits defined by the vector fields of system (9) that determine all traveling wave solutions of (8), we will investigate the phase portraits of (9) in the phase plane (x,y) as the parameters γ,b3,p,q are changed.We point out that here we are considering a physical model where only bounded traveling waves solutions are meaningful, so that we only pay attention to the bounded solutions of system (9).To investigate the equilibrium points of system (9), we need to find all real zeros of the function f(x)=x3+px+q. Suppose that p<0 and Δ=(q/2)2+(p/3)3≤0, clearly, f(x) has three real zeros at most, denoted by x1,x2, and x3. Therefore, system (9) has three equilibrium points at pi(xi,0), i=1,2,3, at most.Let(11)J(xi,0)=(01-β23γf′(xi)0),
where f′(xi)=3xi2+p, i=1,2,3, is the coefficient matrix of the linearized system of (9) at equilibrium point pi(xi,0), i=1,2,3. At this equilibrium point, we obtain the determinant of matrix J(xi,0) which is
(12)detJ(xi,0)=b33γf′(xi),i=1,2,3.
By the theory of planar dynamical systems [24–26] for an equilibrium point of a planar dynamical (Hamiltonian) system, if detJ(xi,0)<0, then the equilibrium pointpiis a saddle point; if detJ(xi,0)>0, then the equilibrium pointpiis a center point; and if detJ(xi,0)=0and the Poincare index of the equilibrium point is zero, then the equilibrium pointpiis a cusp point. So, we have(1)
forb3>0 and Δ<0, there exists three equilibrium points of system (9) at pi(xi,0), i=1,2,3, with x1<x2<x3. The points p1 andp3are center points, p2 is a saddle point. There is two homoclinic orbits to the saddle point p2, in which there exists a family of periodic orbits surrounding the center p1 and p3. The phase portrait is shown in Figures 1(b), 1(c), and 1(e).(2)
Forb3>0 and Δ=0, there exists two equilibrium points of system (9) atpi(xi,0). If q<0, the point p1=p2 is a cusp point, p3 is a center point; if q>0, the point p1 is a center point, p3=p2 is a cusp point. There is a homoclinic orbit to the cusp point p2, in which there exists a family of periodic orbits surrounding the center p1 (or p3). The phase portrait is shown in Figures 1(a) and 1(d).(3)
Forb3<0 and Δ<0, there exists three equilibrium points of system (9) at pi(xi,0) with x1<x2<x3. The points p1 and p3 are saddle points, p2 is a center point. If q≠0, There is a homoclinic orbit to the saddle points p1 and p3, respectively, if q=0. There are two heteroclinic orbits connecting the saddle point p1 and p3, in which there exists a family of periodic orbits surrounding the center p2. The phase portrait is shown in Figures 2(b), 2(c), and 2(e).(4)
Forb3<0andΔ=0, there exists two equilibrium points of system (9). Ifq<0, the point p1=p2is a cusp point,p3is a saddle point; ifq>0, the pointp1is a saddle point,p3=p2is a cusp point. There does not exist bounded orbits. The phase portraits are shown in Figures 2(a) and 2(d).The phase portraits of system (9) forb3>0.
(a)
q<0,Δ=0
(b)
q
<
0
,
Δ
<
0
(c)
q
=
0
,
Δ
<
0
(d)
q
>
0
,
Δ
=
0
(e)
q
>
0
,
Δ
<
0The phase portraits of system (9) forb3<0.
(a)
q<0,Δ=0
(b)
q
<
0
,
Δ
<
0
(c)
q
=
0
,
Δ
<
0
(d)
q
>
0
,
Δ
=
0
(e)
q
>
0
,
Δ
<
0Because orbits cannot be changed by the transformationφ¯=φ+(b2/2b3), it follows that (6), (7), (8), and the above discussion the following.Theorem 1.
Suppose thatb3>0, wave speed c, and integration constantgsatisfy 4b1b3+b22>0; then(1)
whenΔ<0, (1) has two bell-shaped solitary wave solutions and uncountable infinite many periodic traveling wave solutions in the case ofq>0, q=0, andq<0, respectively, (see Figures 1(b), 1(c), and 1(e)).(2)
WhenΔ=0, (1) has one bell-shaped solitary wave solution and uncountably infinite many periodic traveling wave solutions in the case ofq>0andq<0, respectively, (see Figures 1(a), and 1(d)).Theorem 2.
Suppose thatb3<0, wave speed c, and integration constant g satisfy 4b1b3+b22>0; then(1)
whenΔ<0, (1) has one bell-shaped solitary wave solution and uncountably infinite many periodic traveling wave solutions in the case of q>0and q<0, respectively, and two kink-shaped solitary wave solutions and uncountably infinite many periodic traveling wave solutions in the case ofq=0 (see Figures 2(b), 2(c), and 2(e)).(2)
WhenΔ=0, (1) does not exist bounded traveling wave solutions (see Figures 2(a), and 2(d)).


## 3. Exact Explicit Representations of Bell-Shaped and Kink-Shaped Solitary Wave Solutions

According to the discussion in Section2, we assume that (7) has solution with the following form:
(13)φ(ξ)=Aem(ξ+ξ0)(1+em(ξ+ξ0))2+Bem(ξ+ξ0)+D=Asech2((m(ξ+ξ0))/2)4+BAsech2(m(ξ+ξ0)/2)+D,
where A,B,D, and m are undetermined real parameters and ξ0 is arbitrary constant.Substituting (13) and φ′(ξ), φ′′(ξ) into (7), by using the linear independence of ekm(ξ+ξ0), k=0,1,…,6, we obtain that the following algebraic equations with A,B,D, and m:
(14)2b3D3+3b2D2-6b1D-6g=0,γm2+b2D+b3D2-b1=0,(b2+2b3D)A-6γm2(2+B)=0,2b3A2+(2+B)(3b2+6b3D)A-6m2(2+B)2-48m2=0.
Suppose thatDsatisfy
(15)2b3D3+3b2D2-6b1D-6g=0.
By solving the above equations; we have the following.Case 1.
Forb2+2b3D≠0, we have
(16)m=±-b3D2+b2D-b1γ,A=6γm2(2+B)b2+2b3D,2+B=±|b2+2b3D|b22+6b3b1-2b32D2-2b2b3D.Case 2.
Forb2+2b3D=0,b3>0,b1>b2D/2, we have
(17)m=±b1γ-b2D2γ,A=±12b3(2b1-b2D),B=-2.
By (6), (13), and the above conclusions, we can write the exact solutions of (1).Theorem 3.
Forγ>0, ξ=x-ct, ξ0 is an arbitrary constant; suppose that D is real and satisfies 2b3D3+3b2D2-6b1D-6g=0; then(1)
whenA1=b2+2b3D≠0,A2=b3D2+b2D-b1<0, and A3=b22+6b3b1-2b32D2-2b2b3D>0, (1) has the following bell-shaped solitary wave solutions:
(18)φ1±(ξ)=±(6A2|A1|/A3A1)sech2η2+(-1±(|A1|/A3))sech2η+D,ψ1±(ξ)=2c-2δφ1±e,where η=(1/2)-(b3D2+b2D-b1)/γ(ξ+ξ0).(2)
WhenA1=b2+2b3D=0,b3>0 and b1>(b2/2)D, (1) has the following bell-shaped solitary wave solutions
(19)φ2±(ξ)=±3b3(2b1-b2D)sech(2b1-b2D2(ξ+ξ0))+D,ψ2±(ξ)=2c-2δφ2±e.Remark 4.
(
1
)  (φi+,ψi+), i=1,2, denotes the solitary wave solutions taking “+” in expression of (16) and (17), (φi-,ψi-), i=1,2, is similar.
(
2
) Substituting D=0,β=8αδ3/e3 into solutions (φ1+,ψ1+) of expressions (18) yields
(20)φ1+(ξ)=3b1b2sech212b1γ(ξ+ξ0),ψ1+(ξ)=2c-2δφ1+e,
where b1>0 and b2>0 are denoted by (7). Solution (20) is the solitary wave solution (17) and (7) of Guha-Roy [20].
(
3
) Substituting D=0 and λ=-16αδ2c/e3into solutions (φ2±,ψ2±) of expressions (19) yields
(21)φ2±(ξ)=±6b1b3sechb1γ(ξ+ξ0),ψ2±(ξ)=2c-2δφ2±e,
where b1>0 and b3>0 are denoted by (7). Solution (21) is the solitary wave solution (20) and (7) of Guha-Roy [20].
(
4
) Solutions (18) and (19) cannot be obtained by the method used in Guha-Roy [20].
(
5
) For b3>0, q<0, and Δ<0, the roots D1,D2, and D3 of 2b3D3+3b2D2-6b1D-6g=0 satisfy xi=Di+(b2/2b3), i=1,2,3, where x1,x2, and x3 are the roots of f(x)=0. After computation, we know that D1<D2<-(b2/2b3)<D3, b2+2b3D2<0, and b3D22+b2D2-b1<0, so, solution φ1+ in expression (18) is less than D2 and φ1- in expression (18) is over D2. Then, two homoclinic orbits at a saddle point (D2,0) can be denoted by solutions φ1± in (18), in here, φ1+ denotes the left hand homoclinic orbit and φ1- denotes the right hand homoclinic orbit in Figure 1(b). Similarly, we can explain that solutions φ1± denote the two homoclinic orbits in Figure 1(e).
(
6
) For b3<0, q<0, and Δ<0, the roots D1,D2, and D3 of 2b3D3+3b2D2-6b1D-6g=0 satisfy xi=Di+(b2/2b3), i=1,2,3, where x1,x2, and x3 are the roots of f(x)=0. (D1,0) and (D3,0) are saddle points and (D2,0) is a center point. After computation, we know that D1<D2<-(b2/2b3)<D3, b2+2b3D1>0 and b3D12+b2D1-b1<0, by q=A1A3/4b33 yields A3>0, φ1+(ξ)>D1. It follows b2+2b3D3<0 and b3D32+b2D3-b1<0, q=A1A3/4b33 that A3<0, so φ1-(ξ) is unbounded. Therefore, the homoclinic orbit at a saddle point (D1,0) in Figure 2(b) is denoted by φ1+ in expression (18), φ1- is unbounded. Similarly, we can explain that solutions φ1+ denote the homoclinic orbits in Figure 2(e).
(
7
) For b3>0, q=0 and Δ<0, point (D1,0) and (D3,0) are corresponding to the centers p1(--p,0) and p3(-p,0) of system (9), respectively, points (D2,0) is corresponding to a saddle point p2(0,0) of system (9), where D2=-b2/2b3, D2<2b1/b2; then φ2± denotes the two homoclinic orbits in Figure 1(c).
If we suppose that (7) has solutions with the following form:
(22)φ(x-ct)=φ(ξ)=Aem(ξ+ξ0)1+em(ξ+ξ0)+D,
where A,D, and m are undetermined real parameters and ξ0 is arbitrary constant.
Substituting (22) into (7) and using the linear independence of ek(ξ+ξ0), k=0,1,2,3 yields
(23)2b3D3+3b2D2-6b1D-6g=0,γm2+b2D+b3D2-b1=0,2b3A2+(3b2+6b3D)A-6γm2=0,(3b2+6b3D)A-18γm2=0.
By solving the above equations, we obtain
(24)m=±-b22+4b1b32γb3,A=±3b22+12b1b3b32,D=-b22b3±3b22+12b1b32b3.
It follows (22), (24) and (6) thatTheorem 5.
Suppose thatb22+4b1b3>0,b3<0,γ>0,ξ=x-ct,ξ0is an arbitrary constant, andD=-(b2/2b3)±(3b22+12b1b3/2b3), (1) has the following kink-shaped solitary wave solutions:
(25)φ3±(ξ)=(D+b22b3)tanh(12-b22+4b1b32γb3(ξ+ξ0))-b22b3,ψ3±(ξ)=2c-2δφ3±e.Remark 6.
(
1
) In [20], expression (13) can be rewritten as follows:
(26)u(ξ)=-3b1b2(1±12b16(ξ+ξ0)),
whereb22=-6b1b3. Comparing expression (26) withφ3± in expression (25), we know that the solution (13) solved by Guha-Roy [20] is in accordance with solutionsφ3±of expression (25) in the case ofb22=-6b1b3. But the general solutions (25) cannot be solved by Guha-Roy  [20].
(
2
) For b3<0, q=0, and Δ<0, points (D1,0) and (D3,0) are two saddle points of system (9), and (D2,0)is a center point, where D1,3=-(b2/2b3)±(3b22+12b1b3/2b3), D2=-b2/2b3. According to q=0 yields b22+6b1b3-2b32Di2-2b2b3Di=0, i=1,3; thus,φ3+(ξ)and φ3-(ξ) of expression (25) are corresponding to the two heteroclinic orbits in Figure 2(c).
If we suppose that (7) has solutions with the following form:
(27)φ(ξ)=AB+mξ2+D,
where A,B,D, andmare undetermined real parameters andξ0is arbitrary constant.
By using (27) and its derivations, it follows (7) that
(28)b3D2+b2D-b1=0,A=-2B(b2+2b3D)b3,m=B(b22+4b1b3)6γb3,g=b1b2-4b1b3D-b22D6b3.
Further, we obtain the following results.Theorem 7.
Suppose thatb3>0,γ>0, and ξ=x-ct,ξ0 is an arbitrary constant, and D satisfies b3D2+b2D-b1=0 integration constant g=(b1b2-4b1b3D-b22D)/6b3; then (1) has the following bell-shaped solitary wave solutions:
(29)φ4(ξ)=-12(b2+2b3D)γ6γb3+(b22+4b1b3)(ξ+ξ0)2+D,ψ4(ξ)=2c-2δφ4e.Remark 8.
(
1
) By the hypothesisp<0and b3D2+b2D-b1=0, we know that b22+4b1b3>0, and that D1=-b2/2b3+3b22+12b1b3/2b3, D2=-b2/2b3-3b22+12b1b3/2b3. If we takeD=D1in solution (29), solutionφ4(ξ)of expression (29) is corresponding to the homoclinic orbit in Figure 1(d). If we take D=D2in solution (29), solutionφ4(ξ)of expression (29) is corresponding to the homoclinic orbit in Figure 1(a).


## 4. Discussion and Conclusion

In this paper, we obtain all the three bell-shaped and one kink-shaped solitary wave solutions of (1) by using three different undetermined coefficient methods. The conclusions have not been deduced from the method reported by Guha-Roy. The method is simple and can be applied to solve many couple nonlinear equations such as Ito equation, Ito-type equation, and coupled KdV equations.



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