Gennady El: Research

Gennady El's Research Page

Current Research Areas

	Whitham modulation theory
	Dispersive shock waves and undular bores
	Soliton gases in integrable systems
	Nonlinear waves and solitons in Bose-Einstein condensates and optical media
	Interaction of dispersive fluid flows with obstacles

My main research interests are in the general area of nonlinear dispersive waves with an emphasis on multi-scale inviscid fluid flows involving solitons and modulated nonlinear wavetrains. I am particularly interested in the mathematical theory of dispersive shock waves --- the unsteady rapidly oscillating nonlinear wave structures providing regularisation of `breaking' hydrodynamic singularities in weakly dispersive media. These dispersive shock waves have recently become an object of a very active theoretical and experimental research worldwide, notably due to ground-breaking experiments in Bose-Einstein condensates and nonlinear optical media. In a completely different physical context, dispersive shock waves (also often called undular bores) are believed to play an important role in a number of atmosphere and ocean events such as thunderstorm initiation and coastal tsunami propagation. The main mathematical tool in the description of dispersive shock waves/undular bores is the Whitham method of slow modulations.

Along with theoretical aspects of the Whitham modulation theory for integrable and non-integrable nonlinear dispersive equations (CPAM 54 (2001) 1243, Phys Lett A 311 (2003) 374, Chaos 15 (2005) 037103), my current research is concerned with the applied problems of the interaction of inviscid fluid flows with variable topographies (JFM 585 (2007) 213, JFM 640 (2009) 187), the wave radiation in superfluid flows past obstacles (PRL 97 (2006) 180405, PRA 79 (2009) 063608, PRE 80 (2009) 046317) and nonlinear diffraction of light in optical crystals (PRA 76 (2007) 053813, PRA 78 (2008) 013829).

In a different direction, I am involved in the development of an analytical theory of soliton gases (PRL 95 (2005) 204101). A single soliton represents a stable localised nonlinear wave which interacts elasically with other waves. Solitons are described by a special class of nonlinear partial differential equations, called completely integrable equations. These equations have remarkably rich mathematical structure and, at the same time, capture essential features of many important nonlinear wave phenomena in classical and superfluids as well as in solids and plasmas. Owing to their robustness, solitons are of keen interest to engineers and physicists. The classical soliton theory deals mostly with individual solitons and their ordered finite sequences. The consideration of the collective dynamics of random infinite soliton ensembles (the soliton gases) has led to the discovery of a novel class of nonlocal kinetic equations, whose structure and integrability properties have only begun to be understood (J Nonlin Sci 21 (2011) 151). These equations represent non-trivial finite-density generalisations of the kinetic equation for a "rarefied soliton gas" introduced by V. E. Zakharov in 1971 in the context of the Korteweg -- de Vries equation. This study is part of a general programme of the development of the ''integrable turbulence'' theory, initiated by P.D. Lax (1991) and V.E. Zakharov (2009).

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Sample of recent publications (for a complete list of publications click here)

	El, G.A., Kamchatnov, A.M., Pavlov, M.V. and Zykov, S.A., "Kinetic equation for a soliton gas and its hydrodynamic reductions", J Nonlin Sci 21 (2011) 151-191.
	El, G.A., Grimshaw, R.H.J. and Smyth, N.F., "Transcritical shallow-water flow past topography: finite-amplitude theory", J Fluid Mech 640 (2009) 187 - 214.
	El, G.A., Kamchatnov, A.M., Khodorovskii, V.V., Annibale, E.S. and Gammal, A., "Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle", Phys Rev E 80 (2009) Art No 046317.
	Leszczyszyn, A.M., El, G.A., Gladush, Yu.G. and Kamchatnov, A.M. "Transcritical flow of a Bose-Einstein condensate through a penetrable barrier", Phys Rev A 79 (2009) Art No 063608.
	El, G.A., Grimshaw, R.H.J., and Smyth, N.F., Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory, Physica D 237 (2008) 2423-2435.
	Khamis, E.G., Gammal, A., El, G.A., Gladush, Yu.G., Kamchatnov, A.M., Nonlinear diffraction of light beams propagating in photorefractive media with embedded reflecting wire, Phys Rev A 78 (2008) Art No 013829.
	El, G.A., Gammal, A., Khamis, E.G., Kraenkel, R.A., and Kamchatnov, A.M., Theory of optical dispersive shock waves in photorefractive media,Phys. Rev. A 76 (2007) Art No 053813.
	El, G.A., Grimshaw, R.H.J., and Kamchatnov, A.M., Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction, J Fluid Mech 585 (2007) 213 -244.
	El, G.A., Gammal, A., and Kamchatnov, A.M, Oblique dark solitons in supersonic flow of a Bose-Einstein condensate, Phys Rev Lett, 97 (2006) Art No 180405.
	El, G.A., Grimshaw, R.H.J., and Smyth, N.F., Unsteady undular bores in fully nonlinear shallow-water theory, Phys Fluids, 18 (2006) Art No 027104, 17pp.
	El, G.A. and Kamchatnov, A.M., Kinetic equation for a dense soliton gas, Phys Rev Lett, 95 (2005) Art No 204101.
	El, G.A., Resolution of a shock in hyperbolic systems modified by weak dispersion, Chaos, 15 (2005) Art No 037103, 21 pp.
	El, G.A., Grimshaw R.H.J., and Kamchatnov, A.M., Analytic model for a weakly dissipative shallow-water undular bore, Chaos, 15 (2005) Art No 037102.
	El, G.A., The thermodynamic limit of the Whitham equations, Phys Lett A, 311 (2003) 374-383.
	El, G.A. and Grimshaw, R.H.J., Generation of undular bores in the shelves of slowly-varying solitary waves, Chaos, 12 (2002) 1015-1026.
	El, G.A., Krylov A.L., Molchanov, S.A., and Venakides, S., Soliton turbulence as a thermodynamic limit of stochastic soliton lattices, Advances in Nonlinear Mathematics and Science, Physica D, 152/153 (2001) 653-664.
	El, G.A., Grimshaw, R.H.J., and Pavlov, M.V., Integrable shallow-water equations and undular bores, Stud Appl Math, 106 (2001) 157-186.
	El, G.A., Krylov, A.L., and Venakides, S., Unified approach to KdV modulations, Comm Pure Appl Math, 54 (2001) 1243-1270.