On non-local variational problems with lack of compactness related to non-linear optics

TL;DR

This paper presents a straightforward proof for the existence of solutions to non-local variational problems in non-linear optics, specifically addressing dispersion and diffraction management equations with zero average dispersion.

Contribution

It introduces a direct method to prove existence of solutions without relying on Lions' concentration compactness or Ekeland's variational principle.

Findings

Existence of solutions for dispersion and diffraction management equations.

Solutions are maximizers of invariant non-local variational problems.

Proof avoids complex compactness arguments.

Abstract

We give a simple proof of existence of solutions of the dispersion manage- ment and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local vari- ational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions' concentration compactness argument or Ekeland's variational principle.