arXiv subjects

G. A. El

Publications and source records attributed to G. A. El.

At least 19 recordsLinked to original sources

arXiv2026

Soliton gas resolution and statistics of random wave fields in semiclassical integrable turbulence

We develop a general analytical framework for determining the probability distribution of random nonlinear wave fields governed by the focusing nonlinear Schrödinger equation (fNLSE) in regimes where typical realizations are dominated by solitons. We formulate the soliton gas resolution conjecture for the long-time evolution of slowly varying ("semiclassical") random initial states and implement a stochastic analogue of the inverse scattering transform by establishing a relationship between the spectral density of states of the underlying bound-state soliton gas and the probability density function (PDF) of the intensity of the resulting turbulent wave field. The derived explicit integral representation for the PDF is shown to be in excellent agreement with direct numerical simulations across several representative regimes of fNLSE integrable turbulence. The results have broad applicability to physical systems including water waves, nonlinear optics, and superfluids.

nlin.PS↗

arXiv2025

An exactly solvable model of wave-mean field interaction in integrable turbulence

The kinetic theory of soliton gases (SG) is used to develop a solvable model for wave-mean field interaction in integrable turbulence. The waves are stochastic soliton ensembles that scatter off a critically dense SG or soliton condensate -- the mean field. The derived two-fluid kinetic-hydrodynamic equations admit exact solutions predicting SG filtering and an induced mean field. The obtained SG statistical moments agree with ensemble averages of numerical simulations. The developed theory readily generalizes, with applications in fluids, nonlinear optics and condensed matter.

nlin.PS↗

arXiv2024

Spontaneous modulational instability of elliptic periodic waves: the soliton condensate model

We use the spectral theory of soliton gas for the one-dimensional focusing nonlinear Schrödinger equation (fNLSE) to describe the statistically stationary and spatially homogeneous integrable turbulence emerging at large times from the evolution of the spontaneous (noise-induced) modulational instability of the elliptic ``dn'' fNLSE solutions. We show that a special, critically dense, soliton gas, namely the genus one bound-state soliton condensate, represents an accurate model of the asymptotic state of the ``elliptic'' integrable turbulence. This is done by first analytically evaluating the relevant spectral density of states which is then used for implementing the soliton condensate numerically via a random N-soliton ensemble with N large. A comparison of the statistical parameters, such as the Fourier spectrum, the probability density function of the wave intensity, and the autocorrelation function of the intensity, of the soliton condensate with the results of direct numerical fNLSE simulations with dn initial data augmented by a small statistically uniform random perturbation (a noise) shows a remarkable agreement. Additionally, we analytically compute the kurtosis of the elliptic integrable turbulence, which enables one to estimate the deviation from Gaussianity. The analytical predictions of the kurtosis values, including the frequency of its temporal oscillations at the intermediate stage of the modulational instability development, are also shown to be in excellent agreement with numerical simulations for the entire range of the elliptic parameter $m$ of the initial dn potential.

nlin.PS↗

arXiv2024

Statistics of extreme events in integrable turbulence

We use the spectral kinetic theory of soliton gas to investigate the likelihood of extreme events in integrable turbulence described by the one-dimensional focusing nonlinear Schrödinger equation (fNLSE). This is done by invoking a stochastic interpretation of the inverse scattering transform for fNLSE and analytically evaluating the kurtosis of the emerging random nonlinear wave field in terms of the spectral density of states of the corresponding soliton gas. We then apply the general result to two fundamental scenarios of the generation of integrable turbulence: (i) the asymptotic development of the spontaneous (noise induced) modulational instability of a plane wave, and (ii) the long-time evolution of strongly nonlinear, partially coherent waves. In both cases, involving the bound state soliton gas dynamics, the analytically obtained values of the kurtosis are in perfect agreement with those inferred from direct numerical simulations of the the fNLSE, providing the long-awaited theoretical explanation of the respective rogue wave statistics. Additionally, the evolution of a particular non-bound state gas is considered providing important insights related to the validity of the so-called virial theorem.

nlin.PS↗

arXiv2024

Riemann problem for polychromatic soliton gases: a testbed for the spectral kinetic theory

We use Riemann problem for soliton gas as a benchmark for a detailed numerical validation of the spectral kinetic theory for the Korteweg-de Vries (KdV) and the focusing nonlinear Schrödinger (fNLS) equations. We construct weak solutions to the kinetic equation for soliton gas describing collision of two dense "polychromatic" soliton gases composed of a finite number of "monochromatic" components, each consisting of solitons with nearly identical spectral parameters of the scattering operator in the Lax pair. The interaction between the gas components plays the key role in the emergent, large-scale hydrodynamic evolution. We then use the solutions of the spectral kinetic equation to evaluate macroscopic physical observables in KdV and fNLS soliton gases and compare them with the respective ensemble averages extracted from the "exact" soliton gas numerical solutions of the KdV and fNLS equations. To numerically synthesise dense polychromatic soliton gases we develop a new method which combines recent advances in the spectral theory of the so-called soliton condensates and the effective algorithms for the numerical realisation of $n$-soliton solutions with large $n$.

nlin.PS↗

arXiv2022

Dispersive hydrodynamics of soliton condensates for the Korteweg-de Vries equation

We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special "condensate" limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the $N$-phase KdV-Whitham modulation equations derived by Flaschka, Forest and McLaughlin (1980) and Lax and Levermore (1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for "diluted" soliton condensates exhibiting rich incoherent behaviours associated with integrable turbulence.

nlin.PS↗

arXiv2020

Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

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, Korteweg-de Vries equation $u_t + uu_x + u_{xxx} =0.$ The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two-phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two-phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self-similar solution of the BBM equation whose limit as $t \to \infty$ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem.

nlin.PS↗

arXiv2019

Solitary wave fission of a large disturbance in a viscous fluid conduit

This paper presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large, localised initial disturbance into a sequence of solitary waves. This problem is of fundamental importance in a range of applications including tsunami and internal ocean wave modelling. This study is performed 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 disturbance evolves into a slowly modulated wavetrain and further, into a sequence of solitary waves. An extension of Whitham modulation theory, termed the solitary wave resolution method, is used to resolve the fission of an initial disturbance into solitary waves. The developed theory predicts the relationship between the initial disturbance's profile, the number of emergent solitary waves, and their amplitude distribution, quantifying an extension of the well-known soliton resolution conjecture from integrable systems to non-integrable systems that often provide a more accurate modelling of physical systems. The theoretical predictions for the fluid conduit system are confirmed both numerically and experimentally. The number of observed solitary waves is consistently within 1-2 waves of the prediction, and the amplitude distribution shows remarkable agreement. Universal properties of solitary wave fission in other fluid dynamics problems are identified.

physics.flu-dyn↗

arXiv2019

Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves

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 example of dynamic wavepacket-mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wave packet by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Phys. Rev. Lett. 120, 144101 (2018) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.

nlin.PS↗

arXiv2018

Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure

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 a universal analytical description of dispersive shock waves (DSWs) generated in Riemann problems for a broad class of integrable and non-integrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW's edges into the DSW's interior, i.e., this work reveals the DSW structure. Our approach also provides a natural framework in which to analyze DSW stability. We consider several representative, physically relevant examples that illustrate the efficacy of the developed general theory. Comparisons with direct numerical simulations show that inclusion of higher order terms in the NLS equation enables a remarkably accurate description of the DSW structure in a broad region that extends from the harmonic, small amplitude edge.

nlin.PS↗

arXiv2018

Dispersive shock waves in systems with nonlocal dispersion of Benjamin-Ono type

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 fitting method previously developed for dispersive-hydrodynamic systems of Korteweg-de Vries (KdV) type (i.e. reducible to the KdV equation in the weakly nonlinear, long wave, unidirectional approximation). The developed method is applied to the Calogero-Sutherland dispersive hydrodynamics for which the classification of all solution types arising from the Riemann step problem is constructed and the key physical parameters (DSW edge speeds, lead soliton amplitude, intermediate shelf level) of all but one solution type are obtained in terms of the initial step data. The analytical results are shown to be in excellent agreement with results of direct numerical simulations.

nlin.PS↗

arXiv2017

Hydrodynamic Optical Soliton Tunneling

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 the barrier satisfy the same one-dimensional defocusing nonlinear Schrödinger (NLS) equation, which admits a convenient dispersive hydrodynamic interpretation. Under the scale separation assumption of Whitham modulation theory the highly nontrivial nonlinear interaction between the soliton and the evolving hydrodynamic barrier is described in terms of simple wave solutions to an appropriate asymptotic reduction of the Whitham-NLS system. One of the Riemann invariants of the reduced modulation system determines the characteristics of a soliton interacting with an initial mean flow that results in soliton tunneling or trapping. Another Riemann invariant yields the soliton tunneling phase shift. Under certain conditions, soliton interaction with hydrodynamic barriers gives rise to new effects that include reversal of the soliton propagation direction and spontaneous soliton cavitation, which further suggest possible methods of dark soliton control in optical fibers.

nlin.PS↗

arXiv2016

Stationary Expansion Shocks for a Regularized Boussinesq System

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 this paper, we extend the analysis of (El, Hoefer, Shearer 2016) to the regularized Boussinesq system by using Riemann invariants of the underlying dispersionless shallow water equations. The extension for a system is non-trivial, requiring a combination of small amplitude, long-wave expansions with high order matched asymptotics. The constructed asymptotic solution is shown to be in excellent agreement with accurate numerical simulations of the Boussinesq system for a range of appropriately smoothed Riemann data.

nlin.PS↗

arXiv2016

Oblique spatial dispersive shock waves in nonlinear Schrödinger flows

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 expansion. Spatial oblique DSWs, dispersive analogs of oblique shocks in classical media, are constructed utilizing Whitham modulation theory for a class of nonlinear Schrödinger boundary value problems. Self-similar, simple wave solutions of the modulation equations yield relations between the DSW's orientation and the upstream/downstream flow fields. Time dependent numerical simulations demonstrate a convective or absolute instability of oblique DSWs in supersonic flow over obstacles. The convective instability results in an effective stabilization of the DSW.

nlin.PS↗

arXiv2016

Dispersive shock waves and modulation theory

There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G.~B.~Whitham's seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham's averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg-de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs.

nlin.PS↗

arXiv2016

Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

nlin.PS↗

arXiv2013

Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg--de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including solibores, rarefaction waves, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

nlin.PS↗

arXiv2013

Refraction of dispersive shock waves

We study a dispersive counterpart of the classical gas dynamics problem of the interaction of a shock wave with a counter-propagating simple rarefaction wave often referred to as the shock wave refraction. The refraction of a one-dimensional dispersive shock wave (DSW) due to its head-on collision with the centred rarefaction wave (RW) is considered in the framework of defocusing nonlinear Schrödinger (NLS) equation. For the integrable cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain main physical parameters of the DSW refraction. The key features of the DSW-RW interaction predicted by our modulation theory analysis are confirmed by direct numerical solutions of the full dispersive problem.

nlin.PS↗