Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse

TL;DR

This paper develops a Hamiltonian-based theory for dispersive shock wave edges, revealing a symmetry and providing a new method to estimate the number of solitons generated from localized pulses.

Contribution

It introduces a Hamiltonian framework for dispersive shock wave edge dynamics and proposes a symmetry conjecture, offering a novel approach to soliton count estimation.

Findings

Edge motion obeys Hamiltonian mechanics.

A Hopf-like equation describes background flow evolution.

New asymptotic formula for soliton number from pulses.

Abstract

The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow that interacts with edge wave packets or edge solitons. A conjecture about existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In case of localized simple wave pulses propagating through a quiescent medium this theory provided a new approach to derivation of an asymptotic formula for the number of solitons produced eventually from such a pulse.