- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 170372, 5 pages
The Investigation of Solutions to the Coupled Schrödinger-Boussinesq Equations
1College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
2Department of Basic Courses, Sichuan Finance and Economics Vocational College, Chengdu 610101, China
Received 12 May 2013; Accepted 2 June 2013
Academic Editor: Shaoyong Lai
Copyright © 2013 Xin Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The ()-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.
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 . The existence of a periodic solution for the equations was considered in . Kılıcman and Abazari  used the -expansion method to construct periodic and soliton solutions for the Schrödinger-Boussinesq equations , , where and 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: where is a complex unknown function, is a real unknown function, and and 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 ). The approximate solutions and conservation law for the coupled system (1) have been studied in . In , Chen and Xu used the -expansion method to obtain a number of periodic wave solutions expressed by various Jacobi elliptic functions for (1). Cai et al.  studied same equations by the modified -expansion method.
In the present paper, we use the -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 -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 -Expansion Method
To make our presentation self-contained, we recall the -expansion method. The details can be found in Wang et al.’s work .
Step 1. For a given PDE with two independent variables and we convert it into an ODE Using travelling transformation . 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 where satisfies the second-order ODE with respect to . Namely, where , , and are constants to be determined later. The positive integer 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 in which .
Step 3. By substituting (4) into (3) and using (5), collecting all terms with the same order of together, the left-hand side of (3) can be written as a polynomial in . Letting each coefficient of this polynomial be zero yields a system of algebraic equations for , , , and .
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 where is a real function, , are constants to be determined, and is an arbitrary constant. Substituting (7) into (1) yields We take where is an arbitrary constant. Substituting (11) into (9), one gets Suppose that It follows from (9), (10), (11), and (13) that
Solving this system with the Mathematica, we find or where and are arbitrary constants.
3.1. The Hyperbolic Function Solutions to (1) If
Consider where are arbitrary constants, and where .
Remark 1. When , we find , , , .
Remark 2. If , , , we find envelope solitary wave solutions for (1). Namely, and become
and are turned into
3.2. The Trigonometric Function Solutions to (1) If
Consider where , and where , .
3.3. The Rational Function Solutions to (1) If
We obtain where , and where
The -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 , , and 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 .
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).
- B. L. Guo and L. J. Shen, “The global solution of initial value problem for nonlinear Schrödinger-Boussinesq equation in 3-dimensions,” Acta Mathematicae Applicatae Sinica, vol. 6, no. 1, pp. 11–21, 1990.
- B. Guo and X. Du, “Existence of the periodic solution for the weakly damped Schrödinger-Boussinesq equation,” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 453–472, 2001.
- A. Kılıcman and R. Abazari, “Travelling wave solutions of the Schrödinger-Boussinesq system,” Abstract and Applied Analysis, vol. 2012, Article ID 198398, 11 pages, 2012.
- S. Y. Lai and B. Wiwatanapataphe, “The asymptotics of global solutions for semilinear wave equations in two space dimensions,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B, vol. 18, no. 5, pp. 647–657, 2011.
- S. Lai, Y. H. Wu, and B. Wiwatanapataphee, “On exact travelling wave solutions for two types of nonlinear equations and a generalized KP equation,” Journal of Computational and Applied Mathematics, vol. 212, no. 2, pp. 291–299, 2008.
- S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038–2063, 2010.
- S. Lai and A. Wang, “The well-posedness of solutions for a generalized shallow water wave equation,” Abstract and Applied Analysis, vol. 2012, Article ID 872187, 15 pages, 2012.
- H. Schamel and K. Elsässer, “The application of the spectral method to nonlinear wave propagation,” Journal of Computational Physics, vol. 22, no. 4, pp. 501–516, 1976.
- V. G. Makhankov, “On stationary solutions of Schrödinger equation with a self-consistent potential satisfying Boussinesq's equations,” Physics Letters A, vol. 50, no. 1, pp. 42–44, 1974.
- H. L. Chen and Z. H. Xu, “Periodic wave solutions for the coupled Schrödinger-Boussinesq equations,” Acta Mathematicae Applicatae Sinica, vol. 29, no. 5, pp. 955–960, 2006.
- G. L. Cai, F. Y. Zhang, and L. Ren, “More exact solutions for coupling Schrödinger-Boussinesq equations by a modified F-expansion method,” Mathematica Applicata, vol. 21, no. 1, pp. 90–97, 2008.
- M. Wang, X. Li, and J. Zhang, “The ()-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.