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

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.