Solving the Regularized Schamel Equation by the Singular Planar Dynamical System Method and the Deep Learning Method

Abstract

In this paper, two methods are applied to solve the regularized Schamel equation. Firstly, by using the singular planar dynamical system method, we discover its peakon structure which was not reported before. We also derive some new explicit traveling wave solutions of this equation, including various solitary wave solutions, periodic wave solutions and compactons. Especially, for the first time, we find the W-shape solitary wave solutions and W-shape periodic wave solutions of the equation. Then, in order to discover more wave phenomena, a deep learning framework is introduced to solve complicated initial boundary value problems of this equation. In comparison with the exact solutions given previously, our deep learning framework is reliable and highly accurate in capturing the dynamical behavior of the traveling wave solutions of the equation. As an application, we use it to solve a specific initial boundary value problem and obtain a new data-driven solution.