Heteroclinic structure of parametric resonance in the nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the nonlinear dynamics of parametric resonance in the defocusing nonlinear Schr"odinger equation, revealing a heteroclinic structure and breather solutions that influence recurrence regimes and amplification.

Contribution

It introduces a combined mode truncation and averaging approach to describe the nonlinear stage of modulational instability, uncovering complex heteroclinic structures and breather solutions.

Findings

Identification of heteroclinic structures in the nonlinear Schr"odinger equation

Existence of breather solutions separating Fermi-Pasta-Ulam regimes

Optimal parametric amplification occurs outside linear resonance bandwidth

Abstract

We show that the nonlinear stage of modulational instability induced by parametric driving in the {\em defocusing} nonlinear Schr\"odinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis.