A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation

TL;DR

This paper introduces a novel self-adaptive moving mesh numerical method for the short pulse equation, leveraging its hodograph link to the sine-Gordon equation to efficiently capture ultrashort optical pulses and complex solutions.

Contribution

It develops a structure-preserving, self-adaptive mesh scheme based on the hodograph transformation, improving computational efficiency for ultrashort pulse simulations.

Findings

Successfully captures ultrashort pulses with fewer computational resources

Able to simulate exotic solutions like loop solitons

Demonstrates high accuracy and efficiency in numerical experiments

Abstract

The short pulse equation was introduced by Schaefer--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 Schroedinger equation does not possess, have drawn much attention. In such a region, existing numerical methods turn out to require very fine numerical mesh, and accordingly are computationally expensive. In this paper, we establish a new efficient numerical method by combining the idea of the hodograph transformation and the structure-preserving numerical methods. The resulting scheme is a self-adaptive moving mesh scheme that can successfully capture not only the ultrashort pulses but also exotic solutions such as loop soliton solutions.