A first-order Fourier integrator for the nonlinear Schrödinger equation on \(\mathbb T\) without loss of regularity

In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schr\"odinger 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 \(\gamma >\frac32\). That is, up to some fixed time \(T\), there exists some constant \(C=C(\|u\|_{L^\infty([0,T]; H^{\gamma})})>0\), such that $$ \|u^n-u(t_n)\|_{H^\gamma(\mathbb T)}\le C \tau, $$ 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 $$ \|u^n-u(t_n)\|_{H^1(\mathbb T)}\le C\tau^{\frac12-}, $$ where \(C= C(\|u_0\|_{H^1(\mathbb T)})>0\).