Heteroclinic structure of parametric resonance in the nonlinear Schrödinger equation

Abstract

We show that the nonlinear stage of modulational instability induced by parametric driving in the {\em defocusing} nonlinear Schrödinger 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.

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Cited by in corpus (9)

• Growth rate of modulation instability driven by superregular breathers
• Heteroclinic-structure transition of the pure quartic modulation instability
• Recurrence in the high-order nonlinear Schrödinger equation: a low dimensional analysis
• Structural Phase Transitions of Optical Patterns in Atomic Gases with Microwave Controlled Rydberg Interactions
• Parametric resonance of a Bose-Einstein condensate in a ring trap with periodically driven interactions
• Stabilization of unsteady nonlinear waves by phase space manipulation
• Stable-fixed-point description of square-pattern formation in driven two-dimensional Bose-Einstein condensates
• Modulational instability in optical fibers with randomly-kicked normal dispersion
• Femtosecond Optical Superregular Breathers