New Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method

Abstract

In this paper, we use general Riccati equation to construct new solitary wave solutions of the Van der Waals normal form, which is one of the most famous models for natural and industrial granular materials. It is very important to understand the static and dynamic characteristics of this models in many application fields. We solve the general Riccati equation through different function transformation, and many new hyperbolic function solutions are obtained. Then, it is substituted into the Van der Waals normal form as an auxiliary equation. Abundant types of solitary-wave solutions are obtained by choosing different coefficient in the general Riccati equation, and some of them have not been found in other documents. The results show that the analysis method we used is very simple and effective for dealing with nonlinear models.

Keywords

• Riccati equation
• Van der Waals normal form
• nonlinear evolution equation
• solitary wave solution
• auxiliary equation