Thresholds for Existence of dispersion management solitons for general nonlinearities

TL;DR

This paper establishes a threshold for the existence of dispersion management solitons in optical fibers, using variational methods that apply to a broad class of nonlinearities and dispersion profiles.

Contribution

It introduces a direct variational approach to prove the existence of solitons without Lions' concentration compactness, covering a wide range of physically relevant conditions.

Findings

Existence of solitons depends on surpassing a specific power threshold.

Existence results hold under mild conditions on dispersion and nonlinear polarization.

The approach applies to nonlinearities that can change sign, broadening previous results.

Abstract

We prove a threshold phenomenon for the existence of solitary solutions of the dispersion management equation for positive and zero average dispersion for a large class of nonlinearities. These solutions are found as minimizers of nonlinear and nonlocal variational problems which are invariant under a large non-compact group. There exists a threshold such that minimizers exist when the power of the solitons is bigger than the threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. The existence of dispersion managed solitons is shown under very mild conditions on the dispersion profile and the nonlinear polarization of optical active medium, which cover all physically relevant cases for the dispersion profile and a large class of nonlinear polarizations, for example, they are allowed to change sign.