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
Copyright Status
Unknown
Socpus ID
84935859704 (Scopus)
Source API URL
https://api.elsevier.com/content/abstract/scopus_id/84935859704
STARS Citation
Baxter, Mathew; Van Gorder, Robert A.; and Vajravelu, Kuppalapalle, "Several Types Of Similarity Solutions For The Hunter-Saxton Equation" (2015). Scopus Export 2015-2019. 1094.
https://stars.library.ucf.edu/scopus2015/1094