# Wave-mean flow interactions in dispersive hydrodynamics: a new take on the old problem

**Date**: Monday 11 March 2019, 15:00 – 16:00**Location**: Mathematics Level 8, MALL 1 & 2, School of Mathematics**Type**: Seminars, Applied Mathematics**Cost**: Free

#### Professor Gennady El, Northumbria University.

The interaction of waves with a mean flow is a fundamental and longstanding problem in fluid mechanics. The key to the study of such an interaction is the scale separation, whereby the length and time scales of the waves are much shorter than those of the mean flow. The wave-mean flow interaction has been extensively studied for the cases when the mean flow is prescribed externally---as a stationary or time-dependent current (a ``potential barrier'').

In this talk, I will describe a new type of the wave-mean flow interaction whereby a short-scale ``wave projectile''---a soliton or a linear wave packet---is incident on the evolving large-scale nonlinear dispersive hydrodynamic state: a rarefaction wave or a dispersive shock wave (DSW). Modulation equations are derived for the coupling between the soliton (wavepacket) and the mean flow in the nonlinear dispersive hydrodynamic state. These equations admit particular classes of solutions that describe the transmission or trapping of the soliton (wavepacket) by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified in both cases that determine the transmission, trapping conditions and show that solitons (wavepackets) incident upon smooth expansion waves or compressive, rapidly oscillating DSWs exhibit so-called hydrodynamic reciprocity. The latter is confirmed in a laboratory fluids experiment for soliton-hydrodynamic state interactions.

The developed theory is general and can be applied to integrable and non-integrable nonlinear dispersive wave equations in various physical contexts including nonlinear optics and cold atom physics. As concrete examples we consider the Korteweg-de Vries, the defocusing nonlinear Schroedinger and the viscous fluid conduit equations. The talk is based on the recent papers [1] - [3].

[1] M. D. Maiden, D. V. Anderson, N. A. Franco, G.A. El, and M.A. Hoefer, Solitonic dispersive hydrodynamics: theory and observation, Phys. Rev. Lett., **120** (2018) 144101.

[2] P. Sprenger, M.A. Hoefer, and G.A. El, Hydrodynamic optical soliton tunnelling, Phys. Rev. E **97** (2018) 032218

[3] T. Congy, G.A. El and M.A. Hoefer, Interaction of linear modulated waves with unsteady dispersive hydrodynamic states, arXiv:1812.06593