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Application of exp-function method to Dullin-Gottwald-Holm equation. (English) Zbl 1173.35666
Summary: We adopt the Exp-function method and the traveling-wave transformation to study the so-called DGH equation, as a result a number of exact solutions of this equation have been found. The family of solution including some exact solutions such as solitary wave pattern, periodic traveling-wave solution, kink-wave solution and new bounded-wave solutions. We explain a physical meaning of some solutions.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35B10Periodic solutions of PDE
35Q51Soliton-like equations
35C05Solutions of PDE in closed form
35A20Analytic methods, singularities (PDE)
References:
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[3]Camassa, R.; Holm, D. D.; Hyman, J. M.: A new integrable shallow water equation, Adv. appl. Mech. 31, 1-6 (1994) · Zbl 0808.76011
[4]Dullin, H. R.; Gottwald, G. A.; Holm, D. D.: On asymptotically equivalent shallow water wave equations, Phys. rev. Lett. 190, No. 1 – 2, 1-8 (2001) · Zbl 1050.76008 · doi:10.1016/j.physd.2003.11.004
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[6]Shang, Y. D.: New solitary wave solutions with compact support for the KdV-like K(m,n) equations with fully nonlinear dispersion, Appl. math. Comput. 173, 1121-1126 (2006) · Zbl 1088.65094 · doi:10.1016/j.amc.2005.04.058
[7]Shang, Y. D.: New exact special solutions with solitary patterns for Boussinesq-like B(m,n) equations with fully nonlinear dispersion, Appl. math. Comput. 173, 1137-1142 (2006) · Zbl 1092.35021 · doi:10.1016/j.amc.2005.04.059
[8]He, Ji-Huan; Wu, Xu-Hong: Exp-function method for nonlinear wave equations, Chaos solitons fract. 30, 700-708 (2006) · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020