Propagation of instability fronts in modulationally unstable systems

TL;DR

This paper develops an asymptotic method to analyze the propagation of instability fronts in focusing nonlinear media, specifically applied to the nonlinear Schrödinger equation, with results validated against numerical solutions.

Contribution

It introduces a novel asymptotic approach for predicting the motion of instability fronts in nonlinear wave systems, applicable to specific initial distributions.

Findings

Analytical results match numerical solutions well.

Method effectively predicts front propagation in nonlinear Schrödinger systems.

Applicable to initial distributions with zero initial phase.

Abstract

We study evolution of pulses propagating through focusing nonlinear media. Small disturbance on the smooth initial non-uniform background leads to formation of the region of strong nonlinear oscillations. We develop here an asymptotic method for finding the law of motion of the front of this region. The method is applied to the focusing nonlinear Schroedinger equation for the particular cases of Talanov and Akhmanov-Sukhorukov-Khokhlov initial distributions with zero initial phase. The approximate analytical results agree well with the exact numerical solutions for these two problems.