On traveling wave solutions of the θ-equation of dispersive type

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:421

Article

Ha, Tae Gab¹  Liu, Hailiang² 

[1] Pusan Natl Univ, Dept Math, Pusan 609735, South Korea

[2] Iowa State Univ, Dept Math, Ames, IA 50011 USA

关键词: Dispersive equations;    Traveling wave solutions;    Peakon;    Soliton;   

DOI  :  10.1016/j.jmaa.2014.06.058

来源: Elsevier

【 摘 要 】

Traveling wave solutions to a class of dispersive models, u(t) - u(txx) + uu(x) = theta uu(xxx) + (1 - theta)u(x)u(xx), are investigated in terms of the parameter theta, including two integrable equations, the Camassa Holm equation, theta = 1/3, and the Degasperis Procesi equation, theta = 1/4, as special models. It was proved in H. Liu and Z. Yin (2011) [39] that when 1/2 < theta <= 1 smooth solutions persist for all time, and when 0 <= theta <= 1/2, strong solutions of the theta-equation may blow up in finite time, yielding rich traveling wave patterns. This work therefore restricts to only the range theta is an element of[0,1/2]. It is shown that when theta = 0, only periodic travel wave is permissible, and when theta = 1/2 traveling waves may be solitary, periodic or kink-like waves. For 0 < 1/2, traveling waves such as periodic, solitary, peakon, peaked periodic, cusped periodic, or cusped soliton are all permissible. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free