Abstract
An integrable 2-component Camassa-Holm (2-CH) shallow water system is studied by using integral bifurcation method together with a translation-dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution are obtained. Further more, their dynamic behaviors are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.
Highlights
In this paper, employing the integral bifurcation method together with a translation-dilation transformation, we will study an integrable 2-component Camassa-Holm 2-CH shallow water system 1 as follows: mt εmux1 2 εumx σ ε ρ2 x 0, ρt ε 2 ρu x1.1 which is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed 2, as well as water waves moving overJournal of Applied Mathematics an underlying shear flow 3
0, 1.1 which is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed 2, as well as water waves moving over Journal of Applied Mathematics an underlying shear flow 3
The constant κ denotes the speed of the water current which is related to the shallow water wave speed
Summary
In this paper, employing the integral bifurcation method together with a translation-dilation transformation, we will study an integrable 2-component Camassa-Holm 2-CH shallow water system 1 as follows: mt εmux. By using the integral bifurcation method , we will investigate different kinds of new traveling wave solutions of 1.1 and their dynamic properties under the translationdilation transformation u x, t v 1 φ x − vt , ρ x, t v 1 ψ x − vt. The integral bifurcation method possessed some advantages of the bifurcation theory of the planar dynamic system and auxiliary equation method see 33, and references cited therein , it is combined with computer method and useful for many nonlinear partial diffential equations PDEs including some PDEs with high power terms, such as K m, n equation By using this method, we will obtain some new traveling wave solutions of 1.1. The other two cases can be discussed, here we omit them
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