Group Classification of a Higher-Order Boussinesq Equation

TL;DR

This paper classifies higher-order Boussinesq equations based on their Lie symmetry properties and provides exact solutions for quadratic cases, including elliptic function solutions.

Contribution

It identifies symmetry classes of higher-order Boussinesq equations and derives explicit solutions for quadratic nonlinearities.

Findings

Classified equations by Lie symmetry algebra.

Derived exact solutions including elliptic functions.

Enhanced understanding of solution structures for nonlinear Boussinesq equations.

Abstract

We consider a family of higher-order Boussinesq equations with an arbitrary nonlinearity. We determine the classes of equations so that a certain type of Lie symmetry algebra is admitted in this family. In case of a quadratic nonlinearity we provide several exact solutions, some of which are in terms of elliptic functions.