Complex geometric optics for symmetric hyperbolic systems II: nonlinear theory in one space dimension

TL;DR

This paper develops a nonlinear theory for complex geometric optics solutions in one-dimensional symmetric hyperbolic systems, leveraging the unique coherence condition that simplifies the construction of approximate solutions.

Contribution

It extends complex geometric optics to nonlinear symmetric hyperbolic systems in one dimension, highlighting the role of the naive coherence condition for phase construction.

Findings

The naive coherence condition is always satisfiable in one dimension.

The theory provides a framework for approximate solutions in nonlinear hyperbolic systems.

Extension to higher dimensions remains an open problem.

Abstract

This is the second part of a work aimed to study complex-phase oscillatory solutions of nonlinear symmetric hyperbolic systems. We consider, in particular, the case of one space dimension. That is a remarkable case, since one can always satisfy the \emph{naive} coherence condition on the complex phases, which is required in the construction of the approximate solution. Formally the theory applies also in several space dimensions, but the \emph{naive} coherence condition appears to be too restrictive; the identification of the optimal coherence condition is still an open problem.