Article Contents
Article Contents

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

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

 Citation:

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