Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities

TL;DR

This paper studies solitary wave solutions in a nonlocal nonlinear Schrödinger equation with saturated nonlinearities, relevant for fiber-optic systems, using variational methods to prove existence despite non-standard nonlinearity conditions.

Contribution

It introduces a novel variational approach to establish ground state solutions for a nonlocal NLS with saturated nonlinearities, overcoming challenges posed by non-strict sub-additivity.

Findings

Existence of ground state solutions proven.

Application of a specialized Ekeland's variational principle.

Addresses challenges of saturated nonlocal nonlinearities.

Abstract

A nonlinear Schr\"odinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational approach is that a saturated nonlocal nonlinearity does not satisfy uniformly the so-called strict sub-additivity condition. This is overcome by applying a special version of Ekeland's variational principle.