For the Shallow Water Waves: Bilinear-Form and Similarity-Reduction Studies on a Boussinesq-Burgers System

Abstract

Fluid dynamics cooperating with nonlinear science could describe many natural phenomena, e.g., the Boussinesq-Burgers-type equations for the shallow water waves. In this paper, as for the shallow water waves in a lake or near an ocean beach, we study a Boussinesq-Burgers system. Via the Hirota method and symbolic computation, we derive two sets of the bilinear forms, namely, transforming that Boussinesq-Burgers system into two sets of the bilinear form equations. Besides, we also create a set of the similarity reductions for that Boussinesq-Burgers system via the Clarkson-Kruskal direct method, simplifying that Boussinesq-Burgers system to a solvable ordinary differential equation. Our results rely on the variable coefficient in that Boussinesq-Burgers system. We hope that our results could be of some use for the future water-wave studies.