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Exact solutions of travelling wave model via dynamical system method. (English) Zbl 1470.35341

Summary: By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrödinger-Boussinesq equations are studied. Based on this method, the bounded exact travelling wave solutions are obtained which contain solitary wave solutions and periodic travelling wave solutions. The solitary wave solutions and periodic travelling wave solutions are expressed by the hyperbolic functions and the Jacobian elliptic functions, respectively. The results show that the presented findings improve the related previous conclusions. Furthermore, the numerical simulations of the solitary wave solutions and the periodic travelling wave solutions are given to show the correctness of our results.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C07 Traveling wave solutions
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Guo, B. L.; Shen, L. J., The global solution of initial value problem for nonlinear Schrödinger-Boussinesq equation in 3-dimensions, Acta Mathematicae Applicatae Sinica, 6, 1, 11-21, (1990) · Zbl 0705.35129
[2] Kilicman, A.; Abazari, R., Travelling wave solutions of the Schrödinger-Boussinesq system, Abstract and Applied Analysis, 2012, (2012) · Zbl 1253.65162
[3] Guo, B.; Du, X., Existence of the periodic solution for the weakly damped Schrödinger–Boussinesq equation, Journal of Mathematical Analysis and Applications, 262, 2, 453-472, (2001) · Zbl 1040.35114
[4] Makhankov, V. G., On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq’s equation, Physics Letters A, 50, 1, 42-44, (1974)
[5] Huang, X., The investigation of solutions to the coupled Schrödinger-Boussinesq equations, Abstract and Applied Analysis, 2013, (2013) · Zbl 1297.35013
[6] Schamel, H.; Elsässer, K., The application of the spectral method to nonlinear wave propagation, Journal of Computational Physics, 22, 4, 501-516, (1976) · Zbl 0344.65055
[7] Farah, L. G.; Pastor, A., On the periodic Schrödinger-Bousinesq system, Journal of Mathematical Analysis and Applications, 368, 1, 330-349, (2010) · Zbl 1190.35208
[8] Chen, H. L.; Xu, Z. H., Periodic wave solutions for the coupled Schrödinger-Bousinesq equations, Acta Mathematicae Applicatae Sinica, 29, 5, 955-960, (2006)
[9] Cai, G. L.; Zhang, F. Y.; Ren, L., More exact solutions for coupling Schrödinger-Bousinesq equations by a modified Fexpansion method, Mathematica Applicata, 21, 1, 90-97, (2008)
[10] Jibin, L., Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions, (2013), Beijing, China: Science Press, Beijing, China
[11] Li, J.; Dai, H.-H., On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical System Approach, (2007), Beijing, China: Science Press, Beijing, China
[12] Byrd, P. F.; Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, (1971), Berlin, Germany: Springer, Berlin, Germany · Zbl 0213.16602
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