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 $ \mu $-symmetries. Lie symmetries and $ \mu $-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 $ \mu $-symmetries, which are given by matching an integrable and horizontal one-form $ \mu =\varLambda_{x}dx+\varLambda_{t}dt $ for Lie symmetries. Hence, $ \mu $-reductions, explicit solutions and $ \mu $-conservation laws can be determined by $ \mu $-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.