Riemann problems and dispersive shocks in self-focusing media

TL;DR

This paper studies the evolution of discontinuities in focusing nonlinear media, revealing how initial jumps lead to complex dispersive shock structures described by Whitham theory and numerical simulations.

Contribution

It combines numerical simulations with Whitham modulation theory to analyze Riemann problems in focusing nonlinear Schrödinger equations, detailing the formation of dispersive shocks.

Findings

Discontinuous initial conditions lead to dispersive shock waves.

Solutions are characterized by genus-zero, genus-one, or genus-two modulations.

The structure depends on the initial amplitude and wavenumber jumps.

Abstract

The dynamical behavior resulting from an initial discontinuity in focusing media is investigated using a combination of numerical simulations and Whitham modulation theory for the focusing nonlinear Schrodinger equation. Initial conditions with a jump in either or both the amplitude and the local wavenumber are considered. It is shown analytically and numerically that the space-time plane divides into expanding domains in which the solution is described by a slow modulation of genus-zero, genus-one or genus-two solutions, their precise arrangement depending on the specifics of the initial datum