A robust numerical integrator for the short pulse equation near criticality

Abstract

Abstract The short pulse equation was introduced by Schafer–Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the conventional nonlinear Schrodinger equation does not possess, have drawn much attention. In such a region, the solutions can become quite close to a singularity, and thus existing numerical methods cease to work stably, or even if they do, they require high computational cost. In this paper, we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure-preserving methods. The resulting scheme successfully works even when the singularity occurs, and thus gives a highly robust method for resolving near singular solutions of interest. It is confirmed by numerical experiments, and some new insights about derivative blow-up are obtained.