On the degenerate Cauchy problem for a nonlinear variational wave system, Part I: The same wave speed case

TL;DR

This paper studies a nonlinear wave system from liquid crystal modeling, proving local existence of smooth solutions for a degenerate Cauchy problem using a partial hodograph transformation and iterative methods.

Contribution

It introduces a novel approach with a partial hodograph transformation to establish local existence for a degenerate nonlinear wave system.

Findings

Established local existence of smooth solutions

Constructed classical solutions via transformation

Applied iterative methods in weighted spaces

Abstract

We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the same wave speed case with initial data on the parabolic degenerating line. By introducing a partial hodograph transformation, we establish the local existence of smooth solutions in a weighted metric space based on the iteration method. A classical solution of the primary problem is constructed by converting the solution in the partial hodograph variables to that in the original variables.