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Abstract and Applied Analysis
Volume 2014, Article ID 893279, 12 pages
http://dx.doi.org/10.1155/2014/893279
Research Article

Bifurcation Approach to Analysis of Travelling Waves in Nonlocal Hydrodynamic-Type Models

1Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China
2Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Received 7 November 2013; Revised 22 January 2014; Accepted 30 January 2014; Published 18 March 2014

Academic Editor: Wen-Xiu Ma

Copyright © 2014 Jianping Shi and Jibin Li. 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

The paper considers the nonlocal hydrodynamic-type systems which are two-dimensional travelling wave systems with a five-parameter group. We apply the method of dynamical systems to investigate the bifurcations of phase portraits depending on the parameters of systems and analyze the dynamical behavior of the travelling wave solutions. The existence of peakons, compactons, and periodic cusp wave solutions is discussed. When the parameter equals 2, namely, let the isochoric Gruneisen coefficient equal 1, some exact analytical solutions such as smooth bright solitary wave solution, smooth and nonsmooth dark solitary wave solution, and periodic wave solutions, as well as uncountably infinitely many breaking wave solutions, are obtained.