Dispersive hydrodynamics has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media, encompassing both dynamic and stochastic aspects of wave propagation. See the Special Issue “Dispersive Hydrodynamics” of
Physica D **33** (2016) dedicated to the memory of G.B. Whitham.

Within the general framework of dispersive hydrodynamics two major projects are being developed by **Prof. Gennady El** and his collaborators. This research is supported by EPSRC (Grant No. EP/R00515X/1) and Dstl (Contract No DSTLX-1000116851).

Shock waves are ubiquitous nonlinear phenomena observed in nature. Thunder, the crack of a whip, and the boom heard from a jet plane surpassing the speed of sound are familiar occurrences in human experience and all result from the generation of viscous shock waves in air. Dispersive shock waves (DSWs), the subject of this project, are of a very different type, lacking dissipation and realized as expanding oscillatory disturbances in a dispersive medium. The generation of DSWs represents a universal mechanism to resolve hydrodynamic singularities in dispersive media. Physical manifestations of DSWs include undular bores on shallow water and in the atmosphere (the Morning Glory), nonlinear diffraction patterns in optics, and matter waves in ultracold atoms. Any approximately conservative, nonlinear, hydrodynamic medium exhibiting weak dispersion can develop DSWs.

This project explores DSWs in systems described by nonlinear PDEs such as the Korteweg – de Vries (KdV) and nonlinear Schrodinger (NLS) equations, as well as their integrable and non-integrable extensions and counterparts with the emphasis on non-convex dispersive hydrodynamic systems. The mathematics employed involve a synthesis of exact and asymptotic methods from soliton theory, hyperbolic conservation laws and Whitham modulation theory. The analytical developments are supported by careful numerical simulations and, when possible, comparisons with available experimental and observational data.

**References**

[2] G.A. El, M.A. Hoefer and M. Shearer, Dispersive and diffusive-dispersive shock waves for non-convex conservation laws, SIAM Review

[3] 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.

Turbulence is one of the most recognisable, and at the same time, one of the most intriguing forms of nonlinear motion that is commonly observed in everyday phenomena such as wind blasts or fast flowing rivers. Despite its widespread occurrence, the mathematical description of turbulence remains one of the most challenging problems of modern science. Physical mechanisms giving rise to turbulent motion can be very different but typically they involve some sort of dissipation, e.g. viscosity.

This project explores a very different kind of turbulence that does not involve any dissipation but is concerned with dynamics and statistics of random nonlinear waves that are modelled by integrable PDEs, hence the term “integrable turbulence” [1]. One of the refined forms of integrable turbulence is soliton gas [2]. Integrable turbulence in modulationally unstable systems modelled by the focusing NLS equation and its multi-component generalisations represents a promising theoretical framework to understand the spontaneous formation of rogue waves, particularly in fiber optics [3]. The mathematics employed in the project involve the Riemann-Hilbert problem approach to the semi-classical inverse scattering combined with finite-gap theory and detailed numerical simulations. The theoretical finding of the project will inform fibre optics experimental research conducted at the PhLAM laboratory at the University of Lille (France).

**References**

[2] G.A. El and A.M. Kamchatnov, Kinetic equation for a dense soliton gas, Phys. Rev. Lett.

[3]P. Suret, G. El, M. Onorato, and S. Randoux, Rogue Waves in Integrable Turbulence: Semi-Classical Theory and Fast Measurements, in: