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The explicit periodic wave solutions and their limit forms for a generalized b-equation

Research paper by Yi-ren Chen, Wei-bo Ye, Rui Liu

Indexed on: 29 Apr '16Published on: 29 Apr '16Published in: Acta Mathematicae Applicatae Sinica, English Series



Abstract

In this paper, for b ∈ (−∞,∞) and b ≠ −1,−2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux − uxxt + (1+b)u2ux = buxuxx + uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable α. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for α. When α tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = −1 or b = −2, we guess that the equation does not have any one of above solutions.