Gennady A. El

Gennady A. El
Northumbria University · Department of Mathematics, Physics and Electrical Engineering

Doctor of Philosophy

About

157
Publications
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Introduction
Gennady A. El currently works at the Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne

Publications

Publications (157)
Article
Full-text available
Solitons and breathers are localized solutions of integrable systems that can be viewed as “particles” of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media, these “integrable” gases present a fundamental interest for nonlinear physics. We develop an...
Preprint
Full-text available
We review spectral theory of soliton gases in integrable dispersive hydrodynamic systems. We first present a phenomenological approach based on the consideration of phase shifts in pairwise soliton collisions and leading to the kinetic equation for a non-equilibrium soliton gas. Then a more detailed theory is presented in which soliton gas dynamics...
Article
Full-text available
Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin-Bona-Mahony (BBM) equation are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg-de Vries equ...
Article
Full-text available
The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamic...
Preprint
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We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg-de Vries (KdV) equati...
Article
Full-text available
Peregrine soliton (PS) is widely regarded as a prototype nonlinear structure capturing properties of rogue waves that emerge in the nonlinear propagation of unidirectional wave trains. As an exact breather solution of the one-dimensional focusing nonlinear Schrödinger equation with nonzero boundary conditions, the PS can be viewed as a soliton on f...
Article
Full-text available
We review the spectral theory of soliton gases in integrable dispersive hydrodynamic systems. We first present a phenomenological approach based on the consideration of phase shifts in pairwise soliton collisions and leading to the kinetic equation for a non-equilibrium soliton gas. Then, a more detailed theory is presented in which soliton gas dyn...
Preprint
Full-text available
Peregrine soliton (PS) is widely regarded as a prototype nonlinear structure capturing properties of rogue waves that emerge in the nonlinear propagation of unidirectional wave trains. As an exact breather solution of the one-dimensional focusing nonlinear Schr\"odinger equation with nonzero boundary conditions, the PS can be viewed as a soliton on...
Article
Full-text available
We numerically realize a breather gas for the focusing nonlinear Schrödinger equation. This is done by building a random ensemble of N∼50 breathers via the Darboux transform recursive scheme in high-precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhm...
Article
Full-text available
The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and...
Preprint
Full-text available
The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to nonconvex flux is studied in the framework of the modified Korteweg-de Vries (mKdV) equation, a canonical model for nonlinear internal gravity wave propagation in stratified fluids. The principal feature of the studied interaction is that bot...
Article
Full-text available
It has been shown analytically that Peregrine solitons emerge locally from a universal mechanism in the so-called semiclassical limit of the one-dimensional focusing nonlinear Schrödinger equation. Experimentally, this limit corresponds to the strongly nonlinear regime where the dispersion is much weaker than nonlinearity at initial time. We review...
Preprint
Full-text available
We numerically realize breather gas for the focusing nonlinear Schr\"odinger equation. This is done by building a random ensemble of N $\sim$ 50 breathers via the Darboux transform recursive scheme in high precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding t...
Article
Full-text available
Soliton gases represent large random soliton ensembles in physical systems that exhibit integrable dynamics at the leading order. Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media, their controlled experimental realization has been missing. We report a controlled synthes...
Preprint
Full-text available
Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin-Bona-Mahony (BBM) equation $u_t + uu_x = u_{xxt}$ are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable...
Preprint
Full-text available
The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and...
Preprint
Full-text available
Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at the leading order. Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media their controlled experimental realization has been missing. We report the first controlled...
Article
Full-text available
We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability bu...
Article
Full-text available
Solitary wave fission of a large disturbance in a viscous fluid conduit - Volume 883 - M. D. Maiden, N. A. Franco, E. G. Webb, G. A. El, M. A. Hoefer
Article
We investigate the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The statistical properties of the noise-induced MI, observed previously in numerical simulations and in experiments, have not been explained theoretically. In this Letter, using the inverse scattering transform (IST) formalism,...
Preprint
Full-text available
We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability bu...
Preprint
Solitons and breathers are localized solutions of integrable systems that can be viewed as "particles" of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media these "integrable" gases present fundamental interest for nonlinear physics. We develop spect...
Article
Full-text available
A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical ex...
Article
Full-text available
We examine statistical properties of integrable turbulence in the defocusing and focusing regimes of one-dimensional small-dispersion nonlinear Schrödinger equation (1D-NLSE). Specifically, we study the 1D-NLSE evolution of partially coherent waves having Gaussian statistics at time t=0. Using short time asymptotic expansions and taking advantage o...
Preprint
Full-text available
We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations but have not been explained so far. In the framework of inverse scattering transform (IS...
Preprint
Full-text available
We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations but have not been explained so far. In the framework of inverse scattering transform (IS...
Preprint
Full-text available
The temporal evolution of a large, localized initial disturbance is considered in the context of the viscous fluid conduit system---the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Due to buoyancy induced nonlinear self-steepening balanced by stress induced interfacial dispersion, the disturba...
Preprint
Full-text available
We examine statistical properties of integrable turbulence in the defocusing and focusing regimes of one-dimensional small-dispersion nonlinear Schrodinger equation (1D-NLSE). Specifically, we study the 1D-NLSE evolution of partially coherent waves having Gaussian statistics at time t=0. Using short time asymptotic expansions and taking advantage o...
Preprint
Full-text available
A new type of wave-mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg-de Vries (KdV) equation is considered as a prototypical ex...
Article
Full-text available
The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude‐frequency domains. In this paper, we take advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions in order to develop...
Article
Full-text available
The data recorded in optical fiber and in hydrodynamic experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrödinger equation (...
Preprint
Full-text available
The data recorded in optical fiber [1] and in hydrodynamic [2] experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrodinger eq...
Preprint
Full-text available
The data recorded in optical fiber [1] and in hydrodynamic [2] experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrodinger eq...
Preprint
Full-text available
We report an optical fiber experiment in which we study nonlinear stage of modulational instability of a plane wave in the presence of a localized perturbation. Using a recirculating fiber loop as experimental platform, we show that the initial perturbation evolves into expanding nonlinear oscillatory structure exhibiting some universal characteris...
Preprint
Full-text available
We report an optical fiber experiment in which we study nonlinear stage of modulational instability of a plane wave in the presence of a localized perturbation. Using a recirculating fiber loop as experimental platform, we show that the initial perturbation evolves into expanding nonlinear oscillatory structure exhibiting some universal characteris...
Article
Full-text available
We report an optical fiber experiment in which we study nonlinear stage of modulational instability of a plane wave in the presence of a localized perturbation. Using a recirculating fiber loop as experimental platform, we show that the initial perturbation evolves into expanding nonlinear oscillatory structure exhibiting some universal characteris...
Article
Full-text available
We report an optical fiber experiment in which we study the nonlinear stage of modulational instability of a plane wave in the presence of a localized perturbation. Using a recirculating fiber loop as the experimental platform, we show that the initial perturbation evolves into an expanding nonlinear oscillatory structure exhibiting some universal...
Article
Full-text available
A conceptually new notion of hydrodynamic optical soliton tunneling is introduced in which a dark soliton is incident upon an evolving, broad potential barrier that arises from an appropriate variation of the input signal. The barriers considered include smooth rarefaction waves and highly oscillatory dispersive shock waves. Both the soliton and th...
Article
Full-text available
Ubiquitous nonlinear waves in dispersive media include localized solitons and extended hydrodynamic states such as dispersive shock waves. Despite their physical prominence and the development of thorough theoretical and experimental investigations of each separately, experiments and a unified theory of solitons and dispersive hydrodynamics are lac...
Article
Full-text available
We develop a general approach to the description of dispersive shock waves (DSWs) for a class of nonlinear wave equations with a nonlocal Benjamin-Ono type dispersion term involving the Hilbert transform. Integrability of the governing equation is not a pre-requisite for the application of this method which represents a modification of the DSW fitt...
Article
Full-text available
Stationary expansion shocks have been recently identified as a new type of solution to hyperbolic conservation laws regularized by non-local dispersive terms that naturally arise in shallow-water theory. These expansion shocks were studied in (El, Hoefer, Shearer 2016) for the Benjamin-Bona-Mahony equation using matched asymptotic expansions. In th...
Chapter
Full-text available
Integrable turbulence arises while noisy initial data are launched into a system described by an integrable wave equation. Over the last years the study of the complex dynamics and non-Gaussian statistics in integrable turbulence has become a growing and fundamental field of research. In this Chapter, we show how recent theoretical results obtained...
Article
Full-text available
In dispersive media, hydrodynamic singularities are resolved by coherent wavetrains known as dispersive shock waves (DSWs). Only dynamically expanding, temporal DSWs are possible in one-dimensional media. The additional degree of freedom inherent in two-dimensional media allows for the generation of time-independent DSWs that exhibit spatial expans...
Article
Full-text available
We report experimental confirmation of the universal emergence of the Peregrine soliton predicted to occur during pulse propagation in the semiclassical limit of the focusing nonlinear Schrödinger equation. Using an optical fiber based system, measurements of temporal focusing of high power pulses reveal both intensity and phase signatures of the P...
Article
Full-text available
We present experimental evidence of the universal emergence of the Peregrine soliton predicted in the semi-classical (zero-dispersion) limit of the focusing nonlinear Schrödinger equation [Comm. Pure Appl. Math. 66, 678 (2012)]. Experiments studying higher-order soliton propagation in optical fiber use an optical sampling oscilloscope and frequency...
Article
Full-text available
We examine integrable turbulence (IT) in the framework of the defocusing cubic one-dimensional nonlinear Schrodinger equation. This is done theoretically and experimentally, by realizing an optical fiber experiment in which the defocusing Kerr nonlinearity strongly dominates linear dispersive effects. Using a dispersive-hydrodynamic approach, we s...
Article
Full-text available
Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is shown that all shear flows having monotonic convex velocity profiles are stable. The hydrodynamic approximatio...
Article
Full-text available
Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of...
Article
Full-text available
Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schr\"odinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions...
Article
Full-text available
In dispersive media, hydrodynamic singularities are resolved by coherent wavetrains known as dispersive shock waves (DSWs). Only dynamically expanding, temporal DSWs are possible in one-dimensional media. The additional degree of freedom inherent in two-dimensional media allows for the generation of time-independent DSWs that exhibit spatial expans...
Article
Full-text available
The nonlinear Schrödinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework...
Article
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This Special Issue on Dispersive Hydrodynamics is dedicated to the memory and work of G. B. Whitham who was one of the pioneers in this field of physical applied mathematics. Some of the papers appearing here are related to work reported on at the workshop "Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications" held i...
Article
Full-text available
We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularised shallow water equations that include the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the ex...