Various closed-form heteroclinic breather solutions including classical heteroclinic,
heteroclinic breather and Akhmediev breathers solutions for coupled Schrödinger-Boussinesq equation
are obtained using two-soliton and homoclinic test methods, respectively. Moreover, various
heteroclinic structures of waves are investigated.

1. Introduction

The existence of the homoclinic and heteroclinic orbits is very important for investigating the spatiotemporal chaotic behavior of the nonlinear evolution equations (NEEs). In recent years, exact homoclinic and heterclinic solutions were proposed for some NEEs like nonlinear Schrödinger equation, Sine-Gordon equation, Davey-Stewartson equation, Zakharov equation, and Boussinesq equation [1–7].

The coupled Schrödinger-Boussinesq equation is considered as
(1)iEt+Exx+β1E-NE=0,3Ntt-Nxxxx+3(N2)xx+β2Nxx-(|E|2)xx=0,
with the periodic boundary condition
(2)E(x,t)=E(x+l,t),N(x,t)=N(x+l,t),
where l,β1,β2 are real constants, E(x,t) is a complex function, and N(x,t) is a real function. Equation (1) has also appeared in [8] as a special case of general systems governing the stationary propagation of coupled nonlinear upper-hybrid and magnetosonic waves in magnetized plasma. The complete integrability of (1) was studied by Chowdhury et al. [9], and N-soliton solution, homoclinic orbit solution, and rogue solution were obtained by Hu et al. [10], Dai et al. [11–13], and Mu and Qin [14].

2. Linear Stability Analysis

It is easy to see that (eiθ0,β1) is a fixed point of (1), and θ0 is an arbitrary constant. We consider a small perturbation of the form
(3)E=eiθ0(1+ϵ),N=β1(1+ϕ),
where |ϵ(x,t)|≪1, |ϕ(x,t)|≪1. Substituting (3) into (1), we get the linearized equations
(4)iϵt+ϵxx-β1ϕ=0,3ϕtt-ϕxxxx+(β2+2β12)ϕxx-ϵxx-ϵ¯xx=0.
Assume that ϵ and ϕ have the following forms:
(5)ϵ=Geiμnx+σnt+He-iμnx+σnt,ϕ=C(eiμnx+σnt+e-iμnx+σnt),
where G,H are complex constants, and C is a real number; μn=2πn/l, and σn is the growth rate of the nth modes.

Substituting (5) into (4), we have
(6)G(iσn-μn2)=β1C,H(iσn-μn2)=β1C,(3σn2-μn4-μn2(β2+2β12))C=-(G+H¯)νn2,(3σn2-μn4-μn2(β2+2β12))C=-(H+G¯)μn2.
Solving (6), we obtain that
(7)σn2=μn2(β2+2β12)-2μn4±Δ6,
with
(8)Δ=4μn8+μn4(β2+2β12)2-4μn6(β2+2β12)+12μn4(μn4+μn2(β2+2β12)-2β1).Obviously, (7) implies that μn2(β2+2β12)-2μn4>0; then,
(9)μn2<β2+2β122.

3. Various Heterclinic Breather Solutions

Set
(10)E(x,t)=e-iatu(x,t),N(x,t)=v0+v(x,t).
Substituting (10) into (1), we get
(11)iut+uxx+(a+β1-v0)u=uv,3vtt-vxxxx+(6v0+β2)vxx+3(v2)xx=(|u|2)xx.We can choose a,v0 such that a+β1-v0=0.

By using the following transformation(12)u=g(x,t)f(x,t),v=-2(lnf(x,t))xx.
Equation (11) can be reduced into the following bilinear form:
(13)(iDt+Dx2)g·f=0,(3Dt2+(6v0+β2)Dx2-Dx4-λ)f·f+gg*=0,
where g(x,t) is an unknown complex function and f(x,t) is a real function, g* is conjugate function of g(x,t), and λ is an integration constant. The Hirota bilinear operators DxmDtn are defined by
(14)DxmDtnf(x,t)·g(x,t)=(∂∂x-∂∂x′)m(∂∂t-∂∂t′)n[f(x,t)g(x′,t′)]x′=x,t′=t.

We use three test functions to investigate the variation of the heterclinic solution for the coupled Schrödinger-Boussinesq equation (1).

(1) We seek the following forms of the heterclinic solution:
(15)g=1+b1cos(px)eΩt+γ+b2e2Ωt+2γ,f=1+b3cos(px)eΩt+γ+b4e2Ωt+2γ,
where b1,b2 are complex numbers and b3,b4 are real numbers. bi(i=1,2,3,4),p,Ω,γ will be determined later.

Choosing v0=β1, then a=0. Substituting (15) into the (13), we have the following relations among these constants:
(16)λ=1,b1=iΩ+p2iΩ-p2b3,b2=(iΩ+p2iΩ-p2)2b4,b4=Ω2+p44Ω2b32,(3Ω2-p4-(6β1+β2)p2)(Ω2+p4)=2p4.
Therefore, we have the heterclinic solution for (1) as:
(17)E(x,t)=eΩt+γ+b1cos(px)+b2eΩt+γb4(2cosh(Ωt+γ+lnb4)+b3cos(px)),N(x,t)=β1+2b3p2(2b4cos(px)cosh(Ωt+γ+lnb4)+b3)b4(2cosh(Ωt+γ+lnb4)+b3cos(px))2.
It is easy to see that (E,N)→(1,β1) as t→-∞ and (E,N)→(((iΩ+p2)/(iΩ-p2))2,β1) as t→+∞. After giving some constants in (17), we find that the shape of the heterclinic orbit for Schrödinger-Boussinesq equation likes the hook, and the orbits are heterclinic to two different fixed points (see Figure 1 with β1=1, β2=-2, p=1, and γ=1).

Hook heteroclinic orbits for Schrödinger-Boussinesq equation as t→-∞ (a) and t→+∞ (b).

(2) We take ansatz of extended homoclinic test approach for (13) as follows:
(18)f(x,t)=e-p1(x-αt)-η0+b3cos(p(x+αt)+η1)+b4ep1(x-αt)+η0,g(x,t)=e-iθ(e-p1(x-αt)-η0+b1cos(p(x+αt)+η1)aaaaaa+b2ep1(x-αt)+η0),
where the parameters p,p1,α,η0,η1,bs(s=1,2,3,4) will be determined later, b1 and b2 are complex numbers, and b3 and b4 are real numbers. Substituting (18) into (13) and choosing v0=β1, we get the following relations among the parameters:
(19)p2=3p12,λ=1,p12=34α2-14β2-32β1,α2=(β2+6β1)2-24(β2+6β1),b1=b3(iα-2p1)iα+2p1,b2=b4(iα-2p1)2(iα+2p1)2,b3=±2p1(3α2-4p12)b4pα2+4p12.
From (19), we get the restrictive conditions with
(20)-2<β2+6β1<0,b4<0.
Denote that (iα-2p1)/(iα+2p1)=eiθ0. Then, substituting (10) into (1) and employing (19), we obtain the solution of the coupled Schrödinger-Boussinesq equation as follows: (21)E(x,t)=ei(θ0-θ)2-b4sinh(p1(x-αt)+η0+ln(-b4)+iθ0)-b3cos(p(x+αt)+η1)2-b4sinh(p1(x-αt)+η0+ln(-b4))-b3cos(p(x+αt)+η1),N(x,t)=β1-8-b4b3p12sinh(p1(x-αt)+η0+ln(-b4))cos(p(x+αt)+η1)(2-b4sinh(p1(x-αt)+η0+ln(-b4))-b3cos(p(x+αt)+η1))2-2(-4-b4pp1b3cosh(p1(x-αt)+η0+ln-b4)sin(p(x+αt)+η1)+(4b4-3b32)p12)(2-b4sinh(p1(x-αt)+η0+ln(-b4))-b3cos(p(x+αt)+η1))2,where η0,η1 are arbitrary numbers.

Solution in (21) is a heteroclinic breather wave solution. It is easy to see that (E,N)→(e-i(θ+2θ0),β1) as t→-∞ and (E,N)→(e-iθ,β1) as t→+∞. Given some constants in (21), this kind of the heterclinic orbit likes a spiral, and it is heterclinic to the points(e-i(θ+2θ0),β1) and (e-iθ,β1) (see Figure 2 with β1=-1.5, β2=8, and b4=-4).

Spiral heteroclinic orbits for Schrödinger-Boussinesq equation as t→-∞ (a) and t→+∞ (b).

Note that (e-i(θ+2θ0),β1) and (e-iθ,β1) are two different fixed points of (21), which is a heteroclinic solution (see Figure 3). This wave also contains the periodic wave, and its amplitude periodically oscillates with the evolution of time, which shows that this wave has breather effect. The previous results combined with (21) show that interaction between a solitary wave and a periodic wave with the same velocity α and opposite propagation direction can form a heteroclinic breather flow. This is a new phenomenon of physics in the stationary propagation of coupled nonlinear upper-hybrid and magnetosonic waves in magnetized plasma.

One heteroclinic orbit for Schrödinger-Boussinesq equation as x=0.

(3) Use the following forms of the heterclinic solution [14]:
(22)g=b1cosh(αt)+b2cos(px)+b3sinh(αt),f=b4cosh(αt)+b5cos(px),
where b1,b2,b3 are complex numbers and b4,b5 are real numbers. bi(i=1,2,3,4,5),p,α will be determined later.

We also choose v0=β1 and substitute (22) into (13). We have the following relations among these constants:
(23)ib3b4α=b2b5p2,b5(b1+b3)(iα-p2)=b2b4(iα+p2),b2b4(iα-p2)=b5(b1-b3)(iα+p2),-b42+12α2b42-2b52cos2(px)-16b52p4-4b52p2(6β1+β2)+b1b1*-b3b3*+2b2b2*cos2(px)=0.
Solving (23), we get
(24)b1=(p4-α2)b2α2(α2+p4),b3=±i2p2b2α2+p4,b42=(α2+p4)b522α2.
Therefore, we have the heterclinic solution for (1) as
(25)E(x,t)=b1cosh(αt)+b2cos(px)+b3sinh(αt)b4cosh(αt)+b5cos(px),N(x,t)=β1+2b5p2(b4cos(px)cosh(αt)+b5)(b4cosh(αt)+b5cos(px))2. Giving some special parameters in (25), we see that the shape of the heterclinic orbits likes the arc (see Figure 4 with β1=1, α=3, and p=2). The fixed points are (E,N)→((b1-b3)/b4,β1) as t→-∞ and (E,N)→((b1+b3)/b4,β1) as t→+∞.

Arc Heteroclinic orbit for Schrödinger-Boussinesq equation as t→±∞ at x=10*(2k+1) (a) and x=10*(4k+2) (b), where k=0,1,2,….

4. Conclusion

In this work, by using three special test functions in two-soliton method and homoclinic test method, we obtain three families of heteroclinic breather wave solution heteroclinic to two different fixed points, respectively. Moreover, we investigate different structures of these wave solutions. These results show that the Schrödinger-Boussinesq equation has the variety of heteroclinic structure. As the further work, we will consider whether there exist the spatiotemporal chaos for the coupled Schrödinger-Boussinesq equation or not.

Acknowledgments

This work was supported by Chinese Natural Science Foundation Grant nos. 11161055 and 11061028, as well as Yunnan NSF Grant no. 2008PY034.

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