Exact Solutions for a Higher-Order Boussinesq Equation and a New Higher-Order Boussinesq-Like Equation

Abstract

A kind of higher-order Boussinesq equation is studied in this work. Based on the Bell polynomials theories, bilinear representations of the equation are derived, and diverse interaction solutions are constructed through Hirota’s bilinear method. These include interaction solutions characterized by hyperbolic cosine and cosine functions, as well as interactions between lump and soliton solutions. In addition, generalized bilinear operators are used in order to construct a new higher-order Boussinesq-like equation, while lump and breather solutions are also developed utilizing Hirota’s bilinear technique. For the various explicit solutions obtained in this work, several of them are considered to selected particular values for the relevant parameters in order to plot different kind of three-dimensional surfaces with associated two-dimensional density profiles to give a comprehensive understanding of the evolution for various solutions.