Various Heteroclinic Solutions for the Coupled Schrödinger-Boussinesq Equation

Abstract

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 𝑖𝐸_𝑡+𝐸_𝑥𝑥+𝛽₁𝐸−𝑁𝐸=0,3𝑁_𝑡𝑡−𝑁_{𝑥𝑥𝑥𝑥}+3𝑁²_𝑥𝑥+𝛽₂𝑁_𝑥𝑥−|𝐸|²_𝑥𝑥=0, (1) with the periodic boundary condition 𝐸(𝑥,𝑡)=𝐸(𝑥+𝑙,𝑡),𝑁(𝑥,𝑡)=𝑁(𝑥+𝑙,𝑡), (2) 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 (𝑒^𝑖𝜃0,𝛽₁) is a fixed point of (1), and 𝜃₀ is an arbitrary constant. We consider a small perturbation of the form 𝐸=𝑒^𝑖𝜃0(1+𝜖),𝑁=𝛽₁1+𝜙, (3) where |𝜖(𝑥,𝑡)|≪1, |𝜙(𝑥,𝑡)|≪1. Substituting (3) into (1), we get the linearized equations 𝑖𝜖_𝑡+𝜖_𝑥𝑥−𝛽₁𝜙=0,3𝜙_𝑡𝑡−𝜙_{𝑥𝑥𝑥𝑥}+𝛽₂+2𝛽²₁𝜙_𝑥𝑥−𝜖_𝑥𝑥−𝜖_𝑥𝑥=0. (4) Assume that 𝜖 and 𝜙 have the following forms: 𝜖=𝐺𝑒^{𝑖𝜇𝑛𝑥+𝜎𝑛𝑡}+𝐻𝑒^{−𝑖𝜇𝑛𝑥+𝜎𝑛𝑡},𝜙=𝐶𝑒^{𝑖𝜇𝑛𝑥+𝜎𝑛𝑡}+𝑒^{−𝑖𝜇𝑛𝑥+𝜎𝑛𝑡}, (5) where 𝐺,𝐻 are complex constants, and 𝐶 is a real number; 𝜇_𝑛=2𝜋𝑛/𝑙, and 𝜎_𝑛 is the growth rate of the 𝑛th modes.

Substituting (5) into (4), we have 𝐺𝑖𝜎_𝑛−𝜇²_𝑛=𝛽₁𝐻𝐶,𝑖𝜎_𝑛−𝜇²_𝑛=𝛽₁𝐶,3𝜎²_𝑛−𝜇⁴_𝑛−𝜇²_𝑛𝛽₂+2𝛽²₁𝐶=−𝐺+𝐻𝜈²_𝑛,3𝜎²_𝑛−𝜇⁴_𝑛−𝜇²_𝑛𝛽₂+2𝛽²₁𝐶=−𝐻+𝐺𝜇²_𝑛. (6) Solving (6), we obtain that 𝜎²_𝑛=𝜇²_𝑛𝛽₂+2𝛽²₁−2𝜇⁴_𝑛±√Δ6, (7) with Δ=4𝜇⁸_𝑛+𝜇⁴_𝑛𝛽₂+2𝛽²₁²−4𝜇⁶_𝑛𝛽₂+2𝛽²₁+12𝜇⁴_𝑛𝜇⁴_𝑛+𝜇²_𝑛𝛽₂+2𝛽²₁−2𝛽₁. (8)Obviously, (7) implies that 𝜇²_𝑛(𝛽₂+2𝛽²₁)−2𝜇⁴_𝑛>0; then, 𝜇²_𝑛<𝛽₂+2𝛽²₁2. (9)

3. Various Heterclinic Breather Solutions

Set 𝐸(𝑥,𝑡)=𝑒^{−𝑖𝑎𝑡}𝑢(𝑥,𝑡),𝑁(𝑥,𝑡)=𝑣₀+𝑣(𝑥,𝑡). (10) Substituting (10) into (1), we get 𝑖𝑢_𝑡+𝑢_𝑥𝑥+𝑎+𝛽₁−𝑣₀𝑢=𝑢𝑣,3𝑣_𝑡𝑡−𝑣_{𝑥𝑥𝑥𝑥}+6𝑣₀+𝛽₂𝑣_𝑥𝑥+3𝑣²_𝑥𝑥=|𝑢|²_𝑥𝑥. (11)We can choose 𝑎,𝑣₀ such that 𝑎+𝛽₁−𝑣₀=0.

By using the following transformation𝑔()𝑢=𝑥,𝑡𝑓()()𝑥,𝑡,𝑣=−2ln𝑓𝑥,𝑡_𝑥𝑥. (12) Equation (11) can be reduced into the following bilinear form: 𝑖𝐷_𝑡+𝐷²_𝑥𝑔⋅𝑓=0,3𝐷²_𝑡+6𝑣₀+𝛽₂𝐷²_𝑥−𝐷⁴_𝑥−𝜆𝑓⋅𝑓+𝑔𝑔^∗=0, (13) 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 𝐷^𝑚_𝑥𝐷^𝑛_𝑡 are defined by 𝐷^𝑚_𝑥𝐷^𝑛_𝑡𝑓()()=𝜕𝑥,𝑡⋅𝑔𝑥,𝑡−𝜕𝜕𝑥𝜕𝑥^′𝑚𝜕−𝜕𝜕𝑡𝜕𝑡^′𝑛𝑓(𝑥,𝑡)𝑔𝑥^′,𝑡^′_{𝑥′=𝑥,𝑡′=𝑡}. (14)

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: 𝑔=1+𝑏₁cos𝑝𝑥𝑒^Ω𝑡+𝛾+𝑏₂𝑒^{2Ω𝑡+2𝛾},𝑓=1+𝑏₃cos𝑝𝑥𝑒^Ω𝑡+𝛾+𝑏₄𝑒^{2Ω𝑡+2𝛾}, (15) where 𝑏₁,𝑏₂ are complex numbers and 𝑏₃,𝑏₄ are real numbers. 𝑏_𝑖(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: 𝜆=1,𝑏₁=𝑖Ω+𝑝²𝑖Ω−𝑝²𝑏₃,𝑏₂=𝑖Ω+𝑝²𝑖Ω−𝑝²²𝑏₄,𝑏₄=Ω²+𝑝⁴4Ω²𝑏²₃,3Ω²−𝑝⁴−6𝛽₁+𝛽₂𝑝²Ω²+𝑝⁴=2𝑝⁴. (16) Therefore, we have the heterclinic solution for (1) as: 𝐸()=𝑒𝑥,𝑡^Ω𝑡+𝛾+𝑏₁cos𝑝𝑥+𝑏₂𝑒^Ω𝑡+𝛾𝑏₄2coshΩ𝑡+𝛾+ln𝑏₄+𝑏₃,𝑁()cos𝑝𝑥𝑥,𝑡=𝛽₁+2𝑏₃𝑝²2𝑏₄cos𝑝𝑥coshΩ𝑡+𝛾+ln𝑏₄+𝑏₃𝑏₄2coshΩ𝑡+𝛾+ln𝑏₄+𝑏₃cos𝑝𝑥². (17) 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).

(a)

(b)

Figure 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: 𝑓(𝑥,𝑡)=𝑒^{−𝑝1(𝑥−𝛼𝑡)−𝜂0}+𝑏₃cos𝑝(𝑥+𝛼𝑡)+𝜂₁+𝑏₄𝑒^{𝑝1(𝑥−𝛼𝑡)+𝜂0},𝑔(𝑥,𝑡)=𝑒^−𝑖𝜃𝑒^{−𝑝1(𝑥−𝛼𝑡)−𝜂0}+𝑏₁cos𝑝(𝑥+𝛼𝑡)+𝜂₁+𝑏₂𝑒^{𝑝1(𝑥−𝛼𝑡)+𝜂0}, (18) where the parameters 𝑝,𝑝₁,𝛼,𝜂₀,𝜂₁,𝑏_𝑠(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: 𝑝²=3𝑝²₁𝑝,𝜆=1,₂₁=34𝛼²−14𝛽₂−32𝛽₁,𝛼²=𝛽₂+6𝛽₁²−24𝛽₂+6𝛽₁,𝑏₁=𝑏₃𝑖𝛼−2𝑝₁𝑖𝛼+2𝑝₁,𝑏₂=𝑏₄𝑖𝛼−2𝑝₁²𝑖𝛼+2𝑝₁²,𝑏₃=±2𝑝₁3𝛼²−4𝑝²₁𝑏₄𝑝𝛼²+4𝑝²₁. (19) From (19), we get the restrictive conditions with −√2<𝛽₂+6𝛽₁<0,𝑏₄<0. (20) Denote that (𝑖𝛼−2𝑝₁)/(𝑖𝛼+2𝑝₁)=𝑒^𝑖𝜃0. Then, substituting (10) into (1) and employing (19), we obtain the solution of the coupled Schrödinger-Boussinesq equation as follows: 𝐸(𝑥,𝑡)=𝑒^{𝑖(𝜃0−𝜃)}2−𝑏₄sinh𝑝₁(𝑥−𝛼𝑡)+𝜂₀+ln−𝑏₄+𝑖𝜃₀−𝑏₃cos𝑝(𝑥+𝛼𝑡)+𝜂₁2−𝑏₄sinh𝑝₁(𝑥−𝛼𝑡)+𝜂₀+ln−𝑏₄−𝑏₃cos𝑝(𝑥+𝛼𝑡)+𝜂₁,𝑁(𝑥,𝑡)=𝛽₁−8−𝑏₄𝑏₃𝑝²₁sinh𝑝₁(𝑥−𝛼𝑡)+𝜂₀+ln−𝑏₄cos𝑝(𝑥+𝛼𝑡)+𝜂₁2−𝑏₄sinh𝑝₁(𝑥−𝛼𝑡)+𝜂₀+ln−𝑏₄−𝑏₃cos𝑝(𝑥+𝛼𝑡)+𝜂₁²−2−4−𝑏₄𝑝𝑝₁𝑏₃cosh𝑝₁(𝑥−𝛼𝑡)+𝜂₀+ln−𝑏₄sin𝑝(𝑥+𝛼𝑡)+𝜂₁+4𝑏₄−3𝑏²₃𝑝²₁2−𝑏₄sinh𝑝₁(𝑥−𝛼𝑡)+𝜂₀+ln−𝑏₄−𝑏₃cos𝑝(𝑥+𝛼𝑡)+𝜂₁², (21)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).

(a)

(b)

Figure 2 

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.

Figure 3 

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

(3) Use the following forms of the heterclinic solution [14]: 𝑔=𝑏₁cosh(𝛼𝑡)+𝑏₂cos𝑝𝑥+𝑏₃sinh(𝛼𝑡),𝑓=𝑏₄cosh(𝛼𝑡)+𝑏₅cos𝑝𝑥, (22) where 𝑏₁,𝑏₂,𝑏₃ are complex numbers and 𝑏₄,𝑏₅ are real numbers. 𝑏_𝑖(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: 𝑖𝑏₃𝑏₄𝛼=𝑏₂𝑏₅𝑝²,𝑏₅𝑏₁+𝑏₃𝑖𝛼−𝑝²=𝑏₂𝑏₄𝑖𝛼+𝑝²,𝑏₂𝑏₄𝑖𝛼−𝑝²=𝑏₅𝑏₁−𝑏₃𝑖𝛼+𝑝²,−𝑏²₄+12𝛼²𝑏²₄−2𝑏²₅cos²𝑝𝑥−16𝑏²₅𝑝⁴−4𝑏²₅𝑝²6𝛽₁+𝛽₂+𝑏₁𝑏^∗₁−𝑏₃𝑏^∗₃+2𝑏₂𝑏^∗₂cos²𝑝𝑥=0. (23) Solving (23), we get 𝑏₁=𝑝⁴−𝛼²𝑏₂𝛼2𝛼²+𝑝⁴,𝑏₃√=±𝑖2𝑝²𝑏₂𝛼²+𝑝⁴,𝑏²₄=𝛼²+𝑝⁴𝑏²₅2𝛼². (24) Therefore, we have the heterclinic solution for (1) as 𝐸()=𝑏𝑥,𝑡¹cosh(𝛼𝑡)+𝑏₂cos𝑝𝑥+𝑏₃sinh(𝛼𝑡)𝑏₄cosh(𝛼𝑡)+𝑏₅,𝑁()cos𝑝𝑥𝑥,𝑡=𝛽₁𝑏+2⁵𝑝²𝑏₄cos𝑝𝑥cosh(𝛼𝑡)+𝑏₅𝑏₄cosh(𝛼𝑡)+𝑏₅cos𝑝𝑥². (25) 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→+∞.

(a)

(b)

Figure 4 

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.