The explicit nonlinear wave solutions of the generalized $b$-equation

Liu Rui

doi:10.3934/cpaa.2013.12.1029

Article Contents

2013, Volume 12, Issue 2: 1029-1047. Doi: 10.3934/cpaa.2013.12.1029

This issue Previous Article Global attractors for strongly damped wave equations with subcritical-critical nonlinearities Next Article Existence of a rotating wave pattern in a disk for a wave front interaction model

The explicit nonlinear wave solutions of the generalized $b$-equation

Liu Rui^1,

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

Received: December 31, 2010

Revised: June 30, 2012

Published: February 2013

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Abstract

In this paper, we study the nonlinear wave solutions of the generalized $b$-equation involving two parameters $b$ and $k$. Let $c$ be constant wave speed, $c_5=$ $\frac{1}{2}(1+b-\sqrt{(1+b)(1+b-8k)})$, $c_6=\frac{1}{2}(1+b+\sqrt{(1+b)(1+b-8k)})$. We obtain the following results:

1. If $-\infty < k < \frac{1+b}{8}$ and $c\in (c_5, c_6)$, then there are three types of explicit nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic peakon wave solution and hyperbolic blow-up solution.

2. If $-\infty < k < \frac{1+b}{8}$ and $c=c_5$ or $c_6$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.

3. If $k=\frac{1+b}{8}$ and $c=\frac{b+1}{2}$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.

Not only is the existence of these solutions shown, but their concrete expressions are presented. We also reveal the relationships among these solutions. Besides, the correctness of these solutions is tested by using the software Mathematica.

Keywords:

Mathematics Subject Classification: Primary: 34A20, 34C35, 35B65; Secondary: 58F05, 76B25.

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