Gap solitons in almost periodic one-dimensional structures

TL;DR

This paper proves the existence of gap solitons in one-dimensional almost periodic nonlinear Schrödinger equations, with implications for photonic crystals, using advanced variational methods and analysis of energy functionals.

Contribution

It introduces a novel variational approach to establish gap solitons in almost periodic structures, extending previous results to more general nonlinear Schrödinger equations.

Findings

Existence of nontrivial finite energy solutions in almost periodic 1D Schrödinger equations.

Demonstration of gap solitons in certain photonic crystals for all forbidden frequencies.

Development of a specialized analysis of the energy functional on the generalized Nehari manifold.

Abstract

We consider almost periodic stationary nonlinear Schr\"odinger equations in dimension $1$. Under certain assumptions we prove the existence of nontrivial finite energy solutions in the strongly indefinite case. The proof is based on a carefull analysis of the energy functional restricted to the so-called generalized Nehari manifold, and the existence and fine properties of special Palais-Smale sequences. As an application, we show that certain one dimensional almost periodic photonic crystals possess gap solitons for all prohibited frequencies.