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 $$L^{2}$$ -critical and subcritical cases. The constructed multi-solitons behave asymptotically as sum K solitary waves, where is any given finite number. Moreover, convergence rate remainders can be either exponential or polynomial type, which reflects effects noise system on asymptotical behavior solutions. major difficulty our absence pseudo-conformal invariance. Unlike deterministic case (Merle Commun Math Phys 129:223–240, 1990; Röckner et al. Multi-bubble Bourgain–Wang to equation, arXiv: 2110.04107 , 2021), existence cannot obtained from that multi-bubble blow-up (Multi-bubble arXiv:2110.04107 Su Zhang (On rough equations, arXiv:2012.14037v1 2020). Our proof mainly based rescaling approach Herr (Commun 368:843–884, 2019), relying two types Doss–Sussman transforms, modulation method Côte Friederich Partial Differ Equ 46:2325–2385, Martel Merle (Ann Inst H Poincaré Anal Non Linéaire 23:849–864, 2006), crucial ingredient monotonicity Lyapunov type functional by (Duke J 133:405-466, 2006). case, depends Brownian paths noise.