Strang Splitting Methods Applied to a Quasilinear Schrödinger Equation - Convergence and Dynamics

We study numerically a class of quasilinear Schrödinger equations using the Strang splitting method. For these particular models, we can prove convergence of our approximation by adapting the work of Lubich [30] for a Lie theoretic approach to the continuous time approximation and Sobolev-based well-posedness results of the second author with J. Metcalfe and D. Tataru in order to model small initial data solutions of finite time intervals [32, 33]. The scheme is symplectic and is stable within a range of parameters, motivated by the analysis in [33], where the analysis is done purely in Sobolev spaces H s for s sufficiently large. In addition the Strang splitting method converges in the order τ2 for time-step τ. We study numerically a 1d model that is a modified version of the superfluid thin film equation as in [29] and discussed in [37]. This can be seen as a leading order approximation to the relativistic Schrödinger models for ultra-short pulse lasers as studied in [14, 15]. We verify the order of the method and find interesting singularities that arise for large data solutions.