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.
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