Strang Splitting Methods for a quasilinear Schr\"odinger equation - Convergence, Instability and Dynamics

TL;DR

This paper investigates the convergence and instability of Strang splitting methods applied to quasilinear Schrödinger equations, revealing conditions for stability, explaining numerical blow-up, and connecting numerical and analytical solution behaviors.

Contribution

It provides the first convergence analysis for small data solutions and links numerical instability to the breakdown of regularity in quasilinear Schrödinger equations.

Findings

Convergence established for small initial data solutions.

Linear instability explains blow-up in large data cases.

Numerical tests confirm theoretical insights on modified superfluid thin film equations.

Abstract

We study the Strang splitting scheme for quasilinear Schr\"odinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to analytical breakdown of regularity of solutions to quasilinear Schr\"odinger equations. Numerical tests are performed for a modified version of the superfluid thin film equation.