A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation
This work provides a more efficient numerical method for simulating ultrashort optical pulses, which is important for researchers in nonlinear optics and related fields.
The authors develop a self-adaptive moving mesh method for the short pulse equation that efficiently captures ultrashort pulses and loop solitons, reducing computational cost compared to existing methods.
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.