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

Abstract

In this paper, we study the nonlinear wave solutions of thegeneralized $b$-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 obtainthe following results: 1. If $-\infty 2. If $-\infty 3. If $k=\frac{1+b}{8}$ and $c=\frac{b+1}{2}$, then there aretwo types of explicit nonlinear wave solutions, fractionalpeakon wave solution and fractional blow-up solution.Not only is the existence of these solutions shown, but theirconcrete expressions are presented. We also reveal the relationships amongthese solutions. Besides, the correctness of these solutionsis tested by using the software Mathematica.