Evolution of localized pulses in the defocusing modified Korteweg-de Vries equation theory

TL;DR

This paper develops an analytical modulation theory within the Gurevich-Pitaevskii framework to describe the evolution of localized pulses in the defocusing mKdV equation, especially when dispersive shocks do not form solitons.

Contribution

It introduces solutions to the Whitham modulation equations for quasi-simple dispersive shock waves in the defocusing mKdV context, validated by numerical simulations.

Findings

Analytical solutions accurately describe wave patterns.

Modulation theory effectively captures pulse evolution.

Comparison with simulations confirms theory's precision.

Abstract

In this work, we develop, in the Gurevich-Pitaevskii framework, an analytic theory for the evolution of localized pulses in the defocusing modified Korteweg-de Vries equation theory for situations when a dispersive shock does not eventually transform into a sequence of well-separated solitons. We found solutions to the Whitham modulation equations for the corresponding so-called "quasi-simple" dispersive shock waves and illustrated this solution with concrete examples of an initial pulse. Comparison of the analytical solution with direct numerical simulations showed that the modulation theory provides a very accurate description of the wave pattern even at one wavelength scale.