The Investigation of Solutions to the Coupled Schrödinger-Boussinesq Equations

Abstract

The (𝐺^′/𝐺)-expansion method and the symbolic computation system Mathematica are employed to investigate the coupled Schrödinger-Boussinesq equations. The hyperbolic function solutions, trigonometric function solutions, and rational function solutions to the equations are obtained. The decaying properties of several solutions are analyzed.

1. Introduction

In laser and plasma physics, the important problems under interactions between a nonlinear complex Schrödinger field and a real Boussinesq field have been raised. In particular, the study of the coupled Schrödinger-Boussinesq equations has attracted much attention of mathematicians and physicists (see [1–3]). The existence of the global solution of the initial-boundary problem for the equations was investigated in [1]. The existence of a periodic solution for the equations was considered in [2]. Kılıcman and Abazari [3] used the (𝐺^′/𝐺)-expansion method to construct periodic and soliton solutions for the Schrödinger-Boussinesq equations 𝑖𝑢_𝑡+𝑢_𝑥𝑥−𝑎𝑢𝑣=0, 𝑣_𝑡𝑡−𝑣_𝑥𝑥+𝑣_{𝑥𝑥𝑥𝑥}−𝑏(|𝑢|²)_𝑥𝑥=0, where 𝑎 and 𝑏 are real constants. The investigation of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena (see [4–7]).

In this paper, we consider the following coupled Schrödinger-Boussinesq equations: 𝑖𝐸_𝑡+𝐸_𝑥𝑥+𝛽₁𝐸=𝑁𝐸,3𝑁_𝑡𝑡−𝑁_{𝑥𝑥𝑥𝑥}+3𝑁²_𝑥𝑥+𝛽₂𝑁_𝑥𝑥=|𝐸|²_𝑥𝑥, (1) where 𝐸(𝑥,𝑡) is a complex unknown function, 𝑁(𝑥,𝑡) is a real unknown function, and 𝛽₁ and 𝛽₂ are real positive constants. System (1) is known to describe various physical processes in laser and plasma physics, such as formation, Langmuir field amplitude, intense electromagnetic waves, and modulational instabilities (see [8]). The approximate solutions and conservation law for the coupled system (1) have been studied in [9]. In [10], Chen and Xu used the 𝐹-expansion method to obtain a number of periodic wave solutions expressed by various Jacobi elliptic functions for (1). Cai et al. [11] studied same equations by the modified 𝐹-expansion method.

In the present paper, we use the (𝐺^′/𝐺)-expansion method and the symbolic computation system Mathematica to investigate the coupled Schrödinger-Boussinesq system (1). Here, we state that the previous works do not obtain the solutions presented in this paper.

The layout of this paper is as follows. In Section 2, we give the description of the generalized (𝐺^′/𝐺)-expansion method. In Section 3, we apply this method to solve (1). A conclusion will be obtained in Section 4.

2. Brief Description of the (𝐺^′/𝐺)-Expansion Method

To make our presentation self-contained, we recall the (𝐺^′/𝐺)-expansion method. The details can be found in Wang et al.’s work [12].

Step 1. For a given PDE with two independent variables 𝑡 and 𝑥𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,…=0, (2) we convert it into an ODE 𝑃𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…=0. (3) Using travelling transformation 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥−𝑘𝑡. Equation (3) can be integrated as long as all terms contain derivatives where integration constants are considered to be zeros.

Step 2. Suppose that the solution of (3) can be expressed as a polynomial in (𝐺^′/𝐺)𝑢(𝜉)=^𝑛_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖, (4) where 𝐺=𝐺(𝜉) satisfies the second-order ODE with respect to 𝜉. Namely, 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (5) where 𝑎₁,…,𝑎_𝑛≠0, 𝜆, and 𝜇 are constants to be determined later. The positive integer 𝑛 can be determined by balancing the highest-order derivatives with highest-order nonlinear terms appearing in (3). It is easy to check that (5) admits three types of solutions 𝐺^′𝐺=√𝛼2𝑐₁sinh(1/2)√𝛼𝜉+𝑐₂cosh(1/2)√𝛼𝜉𝑐₁cosh(1/2)√𝛼𝜉+𝑐₂sinh(1/2)√−𝜆𝛼𝜉2,√𝛼>0,−𝛼2−𝑐₁sin(1/2)√−𝛼𝜉+𝑐₂cos(1/2)√−𝛼𝜉𝑐₁cos(1/2)√−𝛼𝜉+𝑐₂sin(1/2)√−𝜆−𝛼𝜉2,𝑐𝛼<0,₂𝑐₁+𝑐₂𝜉−𝜆2,𝛼=0, (6) in which 𝛼=𝜆²−4𝜇.

Step 3. By substituting (4) into (3) and using (5), collecting all terms with the same order of (𝐺^′/𝐺) together, the left-hand side of (3) can be written as a polynomial in (𝐺^′/𝐺). Letting each coefficient of this polynomial be zero yields a system of algebraic equations for 𝑎₁,…,𝑎_𝑛, 𝑘, 𝜆, and 𝜇.

Step 4. Since the general solutions of (5) have been known, substituting 𝑎_𝑖,…,𝑎_𝑛, 𝑘, 𝜆, and 𝜇 into (4), we can obtain travelling wave solutions of the nonlinear evolution equation (2).

3. Solutions of the Coupled Schrödinger-Boussinesq Equations

Following the procedure described in Section 2, we adopt the ansatz solution of (1) in the form 𝐸(𝑥,𝑡)=𝑢(𝑥,𝑡)𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)}, (7) where 𝑢(𝑥,𝑡) is a real function, 𝑘, 𝑙 are constants to be determined, and 𝜉₀ is an arbitrary constant. Substituting (7) into (1) yields 𝑢_𝑡+2𝑘𝑢_𝑥𝑢=0,_𝑥𝑥−𝑙+𝑘²−𝛽₁𝑢=𝑁𝑢,3𝑁_𝑡𝑡−𝑁_{𝑥𝑥𝑥𝑥}+3𝑁²_𝑥𝑥+𝛽₂𝑁_𝑥𝑥=𝑢²_𝑥𝑥. (8)(9)(10) We take 𝑢(𝑥,𝑡)=𝑢(𝜉)=𝑢𝑥−2𝑘𝑡+𝜉₁, (11) where 𝜉₁ is an arbitrary constant. Substituting (11) into (9), one gets 𝑁()=𝑢𝑥,𝑡^′′(𝜉)𝑢(𝜉)−𝑙+𝑘²−𝛽₁. (12) Suppose that 𝑁(𝑥,𝑡)=𝑣(𝜉)=𝑣𝑥−2𝑘𝑡+𝜉₁. (13) It follows from (9), (10), (11), and (13) that 𝑢^′′−𝑙+𝑘²−𝛽₁𝑢−𝑢𝑣=0,−𝑣^′′+12𝑘²+𝛽₂𝑣+3𝑣²−𝑢²=0. (14)(15)

Balancing 𝑢^′′ with 𝑢𝑣 in (14) and 𝑣^′′ with 𝑢² in (15) leads to 𝑚=1, 𝑛=2. Thus we can search for the solutions of (14) and (15) in the following forms: 𝑢(𝜉)=𝑎₀+𝑎₁𝐺^′𝐺+𝑎₂𝐺^′𝐺²,𝑎₂𝑣(𝜉)≠0,=𝑏₀+𝑏₁𝐺^′𝐺+𝑏₂𝐺^′𝐺²,𝑏₂≠0. (16)(17)

Substituting (16) and (17) into (14) and (15), using (5), and setting the coefficients of (𝐺^′/𝐺)^𝑖(𝑖=0,…,4) to be zero, we obtain the algebraic system 6𝑎₂−𝑎₂𝑏₂=0,2𝑎₁+10𝑎₂𝜆−𝑎₂𝑏₁−𝑎₁𝑏₂=0,8𝑎₂𝜇+3𝑎₁𝜆+4𝑎₂𝜆²−𝑙𝑏₂−𝑘²𝑏₂+𝛽₁𝑏₂−𝑎₂𝑏₀−𝑎₁𝑏₁−𝑎₀𝑏₂=0,6𝑎₂𝜆𝜇+2𝑎₁𝜇+𝑎₁𝜆²−𝑙𝑏₁−𝑘²𝑏₁+𝛽₁𝑏₁−𝑎₁𝑏₀−𝑎₀𝑏₁=0,2𝑎₂𝜇²+𝑎₁𝜆𝜇−𝑙𝑏₀−𝑘²𝑏₀+𝛽₁𝑏₀−𝑎₀𝑏₀=0,6𝑏₂−3𝑏²₂+𝑎²₂=0,2𝑏₁+10𝑏₂𝜆−6𝑏₁𝑏₂+2𝑎₁𝑎₂=0,8𝑏₂𝜇+3𝑏₁𝜆+4𝑏₂𝜆²−12𝑘²𝑏₂+𝛽₂𝑏₂−3𝑏²₁+6𝑏₀𝑏₂+𝑎²₁+2𝑎₀𝑎₂=0,6𝑏₂𝜆𝜇+2𝑏₁𝜇+𝑏₁𝜆²−12𝑘²𝑏₁−𝛽₂𝑏₁−6𝑏₀𝑏₁+2𝑎₀𝑎₁=0,2𝑏₂𝜇²+𝑏₁𝜆𝜇−12𝑘²𝑏₀−𝛽₂𝑏₀−3𝑏²₀+𝑎²₀=0. (18)

Solving this system with the Mathematica, we find 𝑎₀=±√2𝜆²+2√,𝑎2𝜇₁=±6√𝑎2𝜆,₂=±6√𝑏2,₀=𝜆²𝑏+2𝜇,₁𝑏=6𝜆,₂1=6,𝑘=±63𝛽₂+𝜆²,1−4𝜇𝑙=1212𝛽₁+𝛽₂+𝜆²∓12√2𝜆²−4𝜇±48√2𝜇, (19) or 𝑎₀=±6√𝑎2𝜇,₁=±6√𝑎2𝜆,₂=±6√2,𝑏₀𝑏=6𝜇,₁𝑏=6𝜆,₂1=6,𝑘=±63𝛽₂−𝜆²,1+4𝜇𝑙=1212𝛽₁+𝛽₂−𝜆²±12√2𝜆²+4𝜇∓48√2𝜇, (20) where 𝜆 and 𝜇 are arbitrary constants.

By using (19) and (20), the solutions (16) and (17) are written as 𝑢(𝜉)=±√2𝜆²+2√2𝜇±6√𝐺2𝜆^′𝐺±6√2𝐺^′𝐺²,𝑣(𝜉)=𝜆²𝐺+2𝜇+6𝜆^′𝐺𝐺+6^′𝐺², (21) where 𝜉=𝑥±(1/3)3(𝛽₂+𝜆²−4𝜇)𝑡+𝜉₁, or 𝑢(𝜉)=±6√2𝜇±6√𝐺2𝜆^′𝐺±6√2𝐺^′𝐺²,𝑣(𝜉)𝐺=6𝜇+6𝜆^′𝐺𝐺+6^′𝐺², (22) where 𝜉=𝑥±(1/3)3(𝛽₂−𝜆²+4𝜇)𝑡+𝜉₁.

Substituting general solutions of (5) into (21) and (22), we obtain three types of travelling wave solutions of the coupled Schrödinger-Boussinesq equations as follows.

3.1. The Hyperbolic Function Solutions to (1) If 𝛼=𝜆²−4𝜇>0

Consider 𝐸₁()√𝑥,𝑡=±2𝛼2×3𝑐₁sinh(1/2)√𝛼𝜉+𝑐₂cosh(1/2)√𝛼𝜉𝑐₁cosh(1/2)√𝛼𝜉+𝑐₂sinh(1/2)√𝛼𝜉²−1×𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₁()=𝛼𝑥,𝑡23𝑐₁sinh(1/2)√𝛼𝜉+𝑐₂cosh(1/2)√𝛼𝜉𝑐₁cosh(1/2)√𝛼𝜉𝑒𝑔+𝑐₂sinh(1/2)√𝛼𝜉²−1, (23) where 𝑘=±(1/6)3(𝛽₂+𝛼),𝑙=(1/12)(12𝛽₁+𝛽₂+𝜆²∓12√2𝜆²−4𝜇±48√2𝜇,𝑐₁,𝑐₂ are arbitrary constants, and 𝐸₂()3√𝑥,𝑡=±2𝛼2×𝑐₁sinh(1/2)√𝛼𝜉+𝑐₂cosh(1/2)√𝛼𝜉𝑐₁cosh(1/2)√𝛼𝜉+𝑐₂sinh(1/2)√𝛼𝜉²−1×𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₂()=𝑥,𝑡3𝛼2𝑐₁sinh(1/2)√𝛼𝜉+𝑐₂cosh(1/2)√𝛼𝜉𝑐₁cosh(1/2)√𝛼𝜉+𝑐₂sinh(1/2)√𝛼𝜉²−1, (24) where 𝑘=±(1/6)3(𝛽₂−𝛼),𝑙=(1/12)(12𝛽₁+𝛽₂−𝜆²±12√2𝜆²+4𝜇∓48√2𝜇).

Remark 1. When 𝜉→∞, we find |𝐸₁(𝑥,𝑡)|→±√2𝛼, 𝑁₁(𝑥,𝑡)→𝛼, |𝐸₂(𝑥,𝑡)|→0, 𝑁₂(𝑥,𝑡)→0.

Remark 2. If 𝑐₁≠0,𝑐₂=0, 𝜆>0, 𝜇=0, we find envelope solitary wave solutions for (1). Namely, 𝐸₁(𝑥;𝑡) and 𝑁₁(𝑥;𝑡) become 𝐸₁()√𝑥,𝑡=±22𝜆²3tanh²𝜆2𝜉−1𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₁()=1𝑥,𝑡2𝜆²3tanh²𝜆2𝜉−1. (25)
𝐸₂(𝑥;𝑡) and 𝑁₂(𝑥;𝑡) are turned into 𝐸₂()3√𝑥,𝑡=±22𝜆²tanh²𝜆2𝜉−1𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₂()=3𝑥,𝑡2𝜆²tanh²𝜆2𝜉−1. (26)

3.2. The Trigonometric Function Solutions to (1) If 𝛼<0

Consider 𝐸₃()√𝑥,𝑡=±2𝛼2×3−𝑐₁sin(1/2)√−𝛼𝜉+𝑐₂cos(1/2)√−𝛼𝜉𝑐₁cos(1/2)√−𝛼𝜉+𝑐₂sin(1/2)√−𝛼𝜉²+1×𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₃()𝛼𝑥,𝑡=−23−𝑐₁sin(1/2)√−𝛼𝜉+𝑐₂cos(1/2)√−𝛼𝜉𝑐₁cos(1/2)√−𝛼𝜉+𝑐₂sin(1/2)√−𝛼𝜉²+1, (27) where 𝑘=±(1/6)3(𝛽₂+𝛼),𝑙=(1/12)(12𝛽₁+𝛽₂+𝜆²∓12√2𝜆²−4𝜇±48√2𝜇), and 𝐸₄(𝑥,𝑡)=±3𝛼2×−𝑐₁sin(1/2)√−𝛼𝜉+𝑐₂cos(1/2)√−𝛼𝜉𝑐₁cos(1/2)√−𝛼𝜉+𝑐₂sin(1/2)√−𝛼𝜉²+1×𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₄(𝑥,𝑡)=−3𝛼2−𝑐₁sin(1/2)√−𝛼𝜉+𝑐₂cos(1/2)√−𝛼𝜉𝑐₁cos(1/2)√−𝛼𝜉+𝑐₂sin(1/2)√−𝛼𝜉²+1, (28) where 𝑘=±(1/6)3(𝛽₂−𝛼),  𝑙=(1/12)(12𝛽₁+𝛽₂−𝜆²±12√2𝜆²+4𝜇∓48√2𝜇).

3.3. The Rational Function Solutions to (1) If 𝛼=0

We obtain 𝐸₅()6√𝑥,𝑡=±2𝑐²₂𝑐₁+𝑐₂𝜉²𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₅(𝑥,𝑡)=6𝑐²₂𝑐₁+𝑐₂𝜉², (29) where 𝑘=±(1/6)3(𝛽₂+𝛼),𝑙=(1/12)(12𝛽₁+𝛽₂+𝜆²∓12√2𝜆²−4𝜇±48√2𝜇), and 𝐸₆(𝑥,𝑡)=𝐸_9,10()6√𝑥,𝑡=±2𝑐²₂𝑐₁+𝑐₂𝜉²𝑒^{𝑖(𝑘𝑥+𝑙𝑡+𝜉0)},𝑁₆(𝑥,𝑡)=𝑁_9,10(𝑥,𝑡)=6𝑐²₂𝑐₁+𝑐₂𝜉², (30) where 1𝑘=±63𝛽₂,1−𝛼𝑙=1212𝛽₁+𝛽₂+𝜆²±12√2𝜆²+4𝜇∓48√2𝜇. (31)(32)

4. Conclusion

The (𝐺^′/𝐺)-expansion method is effectively employed to deal with the coupled Schrödinger-Boussinesq equations. The hyperbolic function solutions, the trigonometric function solutions, and the rational function solutions to the equations in the case of 𝛼>0, 𝛼<0, and 𝛼=0 are obtained. In particular, the well-known soliton solutions are only the special case of the hyperbolic-type solutions. We find several properties of solutions when 𝜉→∞.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11171241, 11071177, 11226162), the Key Project of Chinese Ministry of Education (Grant no. 211162), and the Sichuan Province Science Foundation for Youths (no. 2012JQ0011).