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Mathematical Problems in Engineering
Volume 2015, Article ID 641308, 14 pages
http://dx.doi.org/10.1155/2015/641308
Research Article

Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

College of Mathematics, Chongqing Normal University, Chongqing 401331, China

Received 19 May 2014; Accepted 30 July 2014

Academic Editor: Salvatore Alfonzetti

Copyright © 2015 Weiguo Rui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By using Frobenius’ idea together with integral bifurcation method, we study a third order nonlinear equation of generalization form of the modified KdV equation, which is an important water wave model. Some exact traveling wave solutions such as smooth solitary wave solutions, nonsmooth peakon solutions, kink and antikink wave solutions, periodic wave solutions of Jacobian elliptic function type, and rational function solution are obtained. And we show their profiles and discuss their dynamic properties aim at some typical solutions. Though the types of these solutions obtained in this work are not new and they are familiar types, they did not appear in any existing literatures because the equation + + = is very complex. Particularly, compared with the cited references, all results obtained in this paper are new.