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

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Abstract

In this paper, for b ∈ (−∞,∞) and b ≠ −1,−2, we investigate the explicit periodic wave solutions for the generalized b-equation u t + 2ku x − u xxt + (1+b)u 2 u x = bu x u xx + uu xxx , 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.

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Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510640, China

    Yi-ren Chen, Wei-bo Ye & Rui Liu

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  1. Yi-ren Chen
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  2. Wei-bo Ye
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  3. Rui Liu
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Correspondence to Rui Liu.

Additional information

Supported by the National Natural Science Foundation of China (No.11401222), Natural Science Foundation of Guangdong Province (No.S2012040007959) and The Fundamental Research Funds for the Central Universities (No.2014ZZ0064) and Pearl River Science and Technology Nova Program of Guangzhou.

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Chen, Yr., Ye, Wb. & Liu, R. The explicit periodic wave solutions and their limit forms for a generalized b-equation. Acta Math. Appl. Sin. Engl. Ser. 32, 513–528 (2016). https://doi.org/10.1007/s10255-016-0581-x

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  • DOI: https://doi.org/10.1007/s10255-016-0581-x

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