A low regularity exponential-type integrator for the derivative nonlinear Schrödinger equation

Abstract: In this work, we present a first-order unfiltered exponential integrator for the one-dimensional derivative nonlinear Schrödinger equation with low regularity. Our analysis shows that for any $s>\frac12$, the method converges with first-order in $H^s(\mathbb{T})$ for initial data $u_0\in H^{s+1}(\mathbb{T})$. Moreover, we constructed a symmetrized version of this method that performs better in terms of both global error and conservation behavior. To the best of our knowledge, these are the first low regularity integrators for the derivative nonlinear Schrödinger equation. Numerical experiments illustrate our theoretical findings.