Several Types Of Similarity Solutions For The Hunter-Saxton Equation

Keywords

analytical methods; exact solutions; Hunter-Saxton equation; nonlinear waves; self-similar solutions

Abstract

We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director Geld of a nematic liquid crystal (among other applications). Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the Hunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameter essentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact solutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.

Publication Date

6-1-2015

Publication Title

Communications in Theoretical Physics

Volume

63

Issue

6

Number of Pages

675-681

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1088/0253-6102/63/6/675

Socpus ID

84935859704 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84935859704

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