Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation

2024/03/07 by Manuel V. Gnann, Gnann, Manuel V., Rik W. S. Westdorp +3
Mathematics · Physics and Astronomy · #35C08 #35Q55 #35Q60 #35R60 #37H30 #60H15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Laser-Matter Interactions and Applications #Nonlinear Photonic Systems #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2403.04625

openalex publication_date 2024/03/07 · openalex created_date 2024/03/09 · openalex updated_date 2026/07/28

Abstract

We study a parametrically forced nonlinear Schrödinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Krüger and Stannat. Additionally, we show well-posedness of the equation in the fractional Bessel space H^s for any s ∈ [0,∞), demonstrating persistence of regularity.

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