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Exact Traveling Wave Solutions and Bifurcations of Classical and Modified Serre Shallow Water Wave Equations

Jibin Li

School of Mathematical Science, Huaqiao University, Quanzhou, Fujian 362021, P. R. China

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

E-mail Address: lijb@zjnu.cn

E-mail Address: lijb@hqu.edu.cn

Corresponding author.

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Guanrong Chen

Department of Electrical Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China

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, and

Jie Song

School of Mathematical Science, Huaqiao University, Quanzhou, Fujian 362021, P. R. China

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https://doi.org/10.1142/S0218127419501530Cited by:2 (Source: Crossref)

Abstract

Using the dynamical systems analysis and singular traveling wave theory developed by Li and Chen [2007] to the classical and modified Serre shallow water wave equations, it is shown that, in different regions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) can be obtained. More than 28 explicit and exact parametric representations are precisely derived. It is demonstrated that, more interestingly, the modified Serre equation has uncountably infinitely many smooth solitary wave solutions and uncountably infinitely many pseudo-peakon solutions. Moreover, it is found that, differing from the well-known peakon solution of the Camassa–Holm equation, the modified Serre equation has four new forms of peakon solutions.

This research was supported in part by the National Natural Science Foundation of China (11162020 and 11871231).

Keywords:

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