Novel symmetric structures and explicit solutions to a coupled Hunter-Saxton equation

Abstract

In the current study, novel symmetric structures to a coupled Hunter-Saxton equation are synthetically investigated. These novel symmetric structures include Lie symmetries, discrete symmetries, nonlocally related systems, and μ-symmetries. Lie symmetries and μ-symmetries are then used to derive explicit invariant solutions. Based on the established optimal system, the coupled Hunter-Saxton equation can be reduced to rich ordinary differential equations by the Lie group transformation. Its group invariant solutions are thus obtained. Discrete symmetries to the coupled Hunter-Saxton equation are constructed utilizing Lie symmetries, which can help calculate new solutions from known explicit solutions. Moreover, nonlocally related systems of the coupled Hunter-Saxton equation are completed, which contain potential systems and inverse potential systems based on conservation laws and Lie symmetries, respectively. Furthermore, without using the group theory, more plentiful similarity reductions and similarity solutions to the coupled Hunter-Saxton equation are produced by employing the direct reduction method. Another class of symmetric structures to the coupled Hunter-Saxton equation explored in this paper are μ-symmetries, which are given by matching an integrable and horizontal one-form μ = Λ x dx + Λ t dt for Lie symmetries. Hence, μ-reductions, explicit solutions and μ-conservation laws can be determined by μ-symmetries. In addition, polynomial solutions are researched by considering the linear invariant subspaces admitted by the coupled Hunter-Saxton equation. Several explicit invariant solutions are described by graphs ultimately.