A first-order Fourier integrator for the nonlinear Schrödinger equation on $\mathbb{T}$ without loss of regularity

Abstract

In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first order accuracy in $H^{\gamma }$ for any initial data belonging to $H^{\gamma }$, for any $\gamma >\frac{3}{2}$. That is, up to some fixed time $T$, there exists some constant $C=C(\Vert u{\Vert }_{L^{\infty }([0,T];H^{\gamma })})>0$, such that $$\Vert u^n-u(t_n){\Vert }_{H^{\gamma }(\mathbb{T})}\le C\tau ,$$ where $u^n$ denotes the numerical solution at $t_n=n\tau$. Moreover, the mass of the numerical solution $M(u^n)$ verifies $$\left|M(u^n)-M(u_0)\right|\le C{\tau }^5.$$ In particular, our scheme dose not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if $u_0\in H^1(\mathbb{T})$, we rigorously prove that $$\Vert u^n-u(t_n){\Vert }_{H^1(\mathbb{T})}\le C{\tau }^{\frac{1}{2}-},$$ where $C=C(\Vert u_0{\Vert }_{H^1(\mathbb{T})})>0$.

18pages, 2figures