Solitary Waves in a Stochastic Parametrically Forced Nonlinear Schrödinger Equation

SIAM Journal on Applied Dynamical Systems, vol. 24, pp. 3012–3044

Abstract

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 [Formula: see text] for any [Formula: see text], demonstrating persistence of regularity.

Authors 3

1. Manuel V. Gnann

   Delft University of Technology

   Affiliation as printed

   Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands

2. Rik W. S. Westdorp Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, 2300 RA Leiden, The Netherlands

3. Joris van Winden

   Delft University of Technology

   Affiliation as printed

   Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands