A semi-discrete scheme for the stochastic nonlinear Schrödinger equation
The authors consider the multipicative nonlinear Schrödinger equation in the sense of Stratonovich \[ idu+(\Delta u+ \lambda| u|^{2\sigma} u)\,dt= u\circ dW,\quad u(0,x)= u_0(x), \] where \(x\in \mathbb{R}^d\), \(t\geq 0\), \(u(t,x)\in C\), \(\sigma> 0\), \(\lambda=\pm 1\), \(W\) is a real-valued Wiener process depending on \(t\), \(x\). For a fixed positive \(T_0\), they set \(\delta t= T_0/n\) for each natural \(n\) and introduce the scheme defined by the following semi-discrete equation \[ i{u_{l+1}- u_l\over\delta t}+\Delta u_{l+1/2}+ f(| u_l|,| u_{l+1}|) u_{l+1/2}= {\chi_l\over\sqrt{\delta t}} u_{l+1/2},\quad l= 0,\dots, n-1. \] Here \(u_l= u^n_l\), \(l= 0,\dots, n\), is an approximation of \(u(l\delta t)\), \[ u_{l+1/2}:= {1\over 2} (u_l+ u_{l+1}),\quad \chi_l:= {W((l+ 1)\delta t)- W(l\delta t)\over \sqrt{\delta t}},\quad l+ 0,\dots, n-1, \] and \[ f(a,b):= {\lambda\over \sigma+1} {a^{2\sigma+ 2}- b^{2\sigma+ 2}\over a^2- b^2}\text{ if }a\neq b,\;f(a,a)=\lambda a^{2\sigma}. \] For the scheme, the authors prove a convergence theorem.
stochastic partial differential equations
scientific article; zbMATH DE number 2077947