Multi solitary waves to stochastic nonlinear Schrödinger equations

In this paper, we present a pathwise construction of multi-soliton solutions for focusing stochastic nonlinear Schrödinger equations with linear multiplicative noise, in both the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}-critical and subcritical cases. The constructed multi-solitons behave asymptotically as a sum of K solitary waves, where K is any given finite number. Moreover, the convergence rate of the remainders can be of either exponential or polynomial type, which reflects the effects of the noise in the system on the asymptotical behavior of the solutions. The major difficulty in our construction of stochastic multi-solitons is the absence of pseudo-conformal invariance. Unlike in the deterministic case (Merle in Commun Math Phys 129:223–240, 1990; Röckner et al. in Multi-bubble Bourgain–Wang solutions to nonlinear Schrödinger equation, arXiv: 2110.04107, 2021), the existence of stochastic multi-solitons cannot be obtained from that of stochastic multi-bubble blow-up solutions in Röckner et al. (Multi-bubble Bourgain–Wang solutions to nonlinear Schrödinger equation, arXiv:2110.04107, 2021), Su and Zhang (On the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations, arXiv:2012.14037v1, 2020). Our proof is mainly based on the rescaling approach in Herr et al. (Commun Math Phys 368:843–884, 2019), relying on two types of Doss–Sussman transforms, and on the modulation method in Côte and Friederich (Commun Partial Differ Equ 46:2325–2385, 2021), Martel and Merle (Ann Inst H Poincaré Anal Non Linéaire 23:849–864, 2006), in which the crucial ingredient is the monotonicity of the Lyapunov type functional constructed by Martel et al. (Duke Math J 133:405-466, 2006). In our stochastic case, this functional depends on the Brownian paths in the noise.