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Different physical structures of solutions for a generalized Boussinesq wave equation. (English) Zbl 1177.35202

Summary: A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B10 Periodic solutions to PDEs
35C05 Solutions to PDEs in closed form
35Q51 Soliton equations
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[1] Whitham, G. B., Linear and Nonlinear Waves (1974), Interscience Publishers: Interscience Publishers New York · Zbl 0373.76001
[2] Yan, Z., New families of solitons with compact support for Boussinesq-like \(B(m, n)\) equations with fully nonlinear dispersion, Chaos Solitons Fractals, 14, 1151-1158 (2003) · Zbl 1038.35082
[3] Kaya, D., The exact and numerical solitary-wave solutions for generalized modified Boussinesq equation, Phys. Lett. A, 348, 244-250 (2006) · Zbl 1195.35264
[4] Elgarayhi, A.; Elhanbaly, A., New exact traveling wave solutions for the two-dimensional KdV-Burgers and Boussinesq equations, Phys. Lett. A, 343, 85-89 (2005) · Zbl 1181.35231
[5] T. Jiang, S.D. Sharma, Proceedings of the 19th Colloquium on Ship and Ocean Technology, Duisburg, 1998; T. Jiang, S.D. Sharma, Proceedings of the 19th Colloquium on Ship and Ocean Technology, Duisburg, 1998
[6] Nwogu, O., Form of Boussinesq equations for Nearshore wave propagation, J. Waterw. Port Coast. Ocean Eng., 119, 618-638 (1993)
[7] Wazwaz, A. M., Nonlinear variants of the improved Boussinesq equation with compact and noncompact structures, Comput. Math. Appl., 49, 565-574 (2005) · Zbl 1070.35043
[8] Lai, S. Y.; Wu, Y. H., The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation, Discrete Contin. Dyn. Syst. Ser. B, 3, 3, 401-408 (2003) · Zbl 1124.35335
[9] Lai, S. Y.; Wu, Y. H., The global solution of an initial boundary value problem for damped Boussinesq equations, Commun. Pure Appl. Anal., 3, 2, 319-328 (2004) · Zbl 1069.35055
[10] Saucez, P.; Wouwer, A. V.; Schiesser, W. E.; Zegeling, P., Method of lines study of nonlinear dispersive waves, J. Comput. Appl. Math., 168, 413-423 (2004) · Zbl 1052.65084
[11] Dag, I.; Saka, B.; Irk, D., Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J. Comput. Appl. Math., 190, 532-547 (2006) · Zbl 1086.65094
[12] Allan, F. M.; Al-Khaled, K., An approximation of the analytic solution of the shock wave equation, J. Comput. Appl. Math., 192, 301-309 (2006) · Zbl 1091.65104
[13] Biao, L.; Chen, Y.; Zhang, H. Q., Exact travelling wave solutions for a generalized Zakharov-Kuznetsov equation, Appl. Math. Comput., 146, 653-666 (2003) · Zbl 1037.35070
[14] Qiang, L. X.; Jiang, S., The \(\sec_q - \tanh_q\)-method and its applications, Phys. Lett. A, 298, 253-258 (2002)
[15] Qiang, L. X.; Jiang, S.; Fan, W. B.; Liu, W. M., Soliton solutions in linear magnetic field and time-dependent laser field, Commun. Nonlinear Sci. Numer. Simul., 9, 361-365 (2004) · Zbl 1109.78321
[16] Wazwaz, A. M., Compactons and solitary wave solutions for the Boussinesq wave equation and its generalized form, Appl. Math. Comput., 182, 1, 529-535 (2006) · Zbl 1106.65092
[17] Wazwaz, A. M., The sine-cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations, Appl. Math. Comput., 173, 150-164 (2006) · Zbl 1089.65111
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