Interactions of Dispersive Shock Waves

Published

August 2, 2007

Author(s)

Mark Hoefer, Mark J. Ablowitz

Abstract

Interactions and collisions of dispersive shock waves in defocusing (repulsive) nonlinear Schrodinger-type systems are investigated analytically and numerically. Two canonical cases are considered. In one case, two counter-propagating dispersive shock waves collide, interact, and eventually exit the interaction region with larger amplitudes and altered speeds. In the other case, a fast dispersive shock overtakes a slower one, giving rise to an interaction. Eventually, the two merge into a single dispersive shock wave. In both cases, the interaction region is described by a modulated, quasi-periodic two-phase solution of the nonlinear Schrodinger equation. The boundaries between the background density, dispersive shock waves, and their interaction region are calculated by solving the Whitham modulation equations. These asymptotic results are in excellent agreement with full numerical simulations. It is further shown that the interactions of two dispersive shock waves are directly analogous to the interactions of two classical shock waves, the only difference being the generation of an intermediate interaction region in the dispersive case.

Citation

Physica D-Nonlinear Phenomena

Volume

236

Issue

1

Pub Type

Journals

Download Paper

https://doi.org/10.1016/j.physd.2007.07.017

Local Download

Keywords

Bose-Einstein condensates, collisionless shock waves, dispersive shock waves, nonlinear optics, shock waves, zero dispersion limit

Citation

Hoefer, M. and Ablowitz, M. (2007), Interactions of Dispersive Shock Waves, Physica D-Nonlinear Phenomena, [online], https://doi.org/10.1016/j.physd.2007.07.017, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=32653 (Accessed October 22, 2025)