Semiclassical dynamics and coherent soliton condensates in self-focusing nonlinear media with periodic initial conditions

TL;DR

This paper analyzes the small dispersion limit of the focusing nonlinear Schrödinger equation with periodic initial conditions, revealing the formation of a coherent soliton condensate with zero velocity and deriving spectral properties and scaling laws.

Contribution

It provides a comprehensive analytical and numerical study of the spectral and dynamical properties of solutions, introducing new asymptotic expressions and a scaling law for effective solitons.

Findings

Spectral confinement to real and imaginary axes in the semiclassical limit.

Solutions form a zero-velocity coherent soliton condensate.

Number of effective solitons scales inversely with the dispersion parameter.

Abstract

The small dispersion limit of the focusing nonlinear Schrodinger equation with periodic initial conditions is studied analytically and numerically. First, through a comprehensive set of numerical simulations, it is demonstrated that solutions arising from a certain class of initial conditions, referred to as "periodic single-lobe" potentials, share the same qualitative features, which also coincide with those of solutions arising from localized initial conditions. The spectrum of the associated scattering problem in each of these cases is then numerically computed, and it is shown that such spectrum is confined to the real and imaginary axes of the spectral variable in the semiclassical limit. This implies that all nonlinear excitations emerging from the input have zero velocity, and form a coherent nonlinear condensate. Finally, by employing a formal Wentzel-Kramers-Brillouin expansion for the scattering eigenfunctions, asymptotic expressions for the number and location of the bands and gaps in the spectrum are obtained, as well as corresponding expressions for the relative band widths and the number of "effective solitons". These results are shown to be in excellent agreement with those from direct numerical computation of the eigenfunctions. In particular, a scaling law is obtained showing that the number of effective solitons is inversely proportional to the small dispersion parameter.