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This study investigates the evolution of dispersive shock wave (DSW) solutions within the focusing and defocusing cylindrical modified Korteweg-de Vries (cmKdV(f)) and (cmKdV(d)) equations under Riemann-type initial conditions. Using Whitham modulation theory, we derive and numerically solve the Whitham systems, enabling a comparison between these asymptotic solutions and direct numerical simulations of the cmKdV equations. The results provide a detailed classification of wave structures in both focusing and defocusing cases of the cmKdV equations. This research offers new insights into the behavior and classification of nonlinear periodic waves in the cmKdV equations. [ABSTRACT FROM AUTHOR]

The Riemann problem for the discrete conservation law 2u̇n+un+12−un−12=0$2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi‐continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations. [ABSTRACT FROM AUTHOR]

The Serre–Green–Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre–Green–Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that the modified system effectively approximates the classical Serre–Green–Naghdi equations over sufficiently extended time intervals and admits dispersive–diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre–Green–Naghdi equations and the effects of diffusion in the generation and propagation of undular bores. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a survey and a detailed numerical study on a remarkable Boussinesq system describing weakly nonlinear, long surface water waves. In the one‐dimensional case, this system can be viewed as a dispersive perturbation of the hyperbolic Saint‐Venant (shallow water) system. The asymptotic stability of the solitary waves is numerically established. Blow‐up of solutions for initial data not satisfying the noncavitation condition as well as the appearance of dispersive shock waves are studied. [ABSTRACT FROM AUTHOR]

In this paper dispersive hydrodynamics associated with the non-Hermitian nonlinear Schrödinger (NLS) equation with generic complex external potential is studied. In particular, a set of dispersive hydrodynamic equations are obtained. They differ from their classical counterparts (without an external potential), by the presence of additional source terms that alter the density and momentum equations. When restricted to a class of Wadati-type complex potentials, the resulting hydrodynamic system conserves a modified momentum and admits constant intensity/density solutions. This motivates the construction and study of an initial value problem (IVP) comprised of a centred (or non-centred) step-like initial condition that connects two constant intensity/density states. Interestingly, this IVP is shown to be related to a Riemann problem posed for the hydrodynamic system in an appropriate traveling reference frame. The study of such IVPs allows one to interpret the underlying non-Hermitian Riemann problem in terms of an 'optical flow' over an obstacle. A broad class of non-Hermitian potentials that lead to modulationally stable constant intensity states are identified. They are subsequently used to numerically solve the associated Riemann problem for various initial conditions. Due to the lack of translation symmetry, the resulting long-time dynamics show a dependence on the location of the step relative to the potential. This is in sharp contrast to the NLS case without potential, where the dynamics are independent of the step location. This fact leads to the formation of diverse nonlinear wave patterns that are otherwise absent. In particular, various gain-loss generated near-field features are present, which in turn drive the optical flow in the far-field which could be comprised of various rich nonlinear wave structures, including DSW-DSW, DSW-rarefaction, and soliton-DSW interactions. [ABSTRACT FROM AUTHOR]

We study in this paper linear and weakly nonlinear waves within the framework of a Hall-magnetohydrodynamic model. An optimal ordering, which allows the Hall effect to be seen in the leading order equations, is used to discuss the propagation of such waves; an evolution equation is obtained where the nonlinearity and Hall effect enter through the parameters that influence the wave propagation significantly. The interplay between nonlinearity and Hall effect leads to the emergence of a dispersive shock wave, which appears as the solution to the initial value problem associated with the evolution equation. The present study reveals a number of interesting flow characteristics which are not seen in the theory of ideal magnetohydrodynamics. • The presence of the Hall effect triggers the formation of a dispersive shock wave in Hall-MHD. • An initial profile with higher amplitude leads to an early appearance of a DSW. • An increase in the Hall parameter enhances the dispersive effect. • An increase in the nonlinearity parameter leads to a more intense oscillatory behavior and results in an early appearance of DSW. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the defocusing self-induced transparency system which can describe propagation characters of short pulses in Erbium doped fibers with a two-level medium. The single-phase periodic solution and Whitham modulation equations will be derived via finite-gap integration method and Whitham modulation theory. In addition, the oscillatory structure of dispersive shock waves will be constructed and analyzed. [ABSTRACT FROM AUTHOR]

The flow of a density stratified fluid over obstacles has been intensively explored from a natural and scientific point of view. This flow has been successfully governed by using the forced Korteweg–de Vries-Burgers equation that generated solitons in a viscous flow. This is done by adding the viscous term beyond the Korteweg–de Vries approximation. It is based on the conservation laws of the Korteweg–de Vries-Burgers equation for mass and energy, and assumes that the upstream wavetrains are composed of solitary waves. Our results show that the influence of viscosity plays a key role in determining the upstream solitary wave amplitude of the bore. A good comparison is obtained between the numerical and analytical solutions. • Uniform solitary wave theory solutions show that the undular bores are generated upstream and downstream of the forcing in a viscous flow. • Predictions prove that the effect of viscosity plays a key role in determining the properties of waves. [ABSTRACT FROM AUTHOR]

We consider the modulationally stable version of the Kaup–Boussinesq system which models propagation of nonlinear waves in various physical situations. It is shown that the Whitham modulation equations for this model have a new type of solutions which describe trigonometric shock waves. In the Gurevich–Pitaevskii problem of evolution of an initial discontinuity, these solutions correspond to a nonzero wave excitation on one of the sides of the discontinuity. As a result, the trigonometric shock wave propagates along a rarefaction wave and we consider the problem of the analytical description of such an evolution. Our analytical results are confirmed by numerical calculations. [ABSTRACT FROM AUTHOR]

We present a numerical approach to study solutions to the dispersionless Kadomtsev–Petviashvili (dKP) equation on R × T . The dependence on the coordinate x is considered on the compactified real line, and the dependence on the coordinate y is assumed to be periodic. Critical behavior, the formation of a shock in the solutions, is of special interest. The latter permits the numerical study of Dubrovin's universality conjecture on the break-up of solutions to the Kadomtsev–Petviashvili equation. Examples from a previous paper on dKP solutions studied numerically on T 2 are addressed, and the influence of the periodicity or not in the x-coordinate on the break-up is studied. [ABSTRACT FROM AUTHOR]

We study the long-time asymptotic behavior of the solution q(x, t), x ∈ ℝ, t ∈ ℝ+, of the modified Korteweg--de Vries equation (MKdV) qt +6q²qx +qxxx = 0 with step-like initial datum q(x, 0) →{c- for x → -∞; c+ for x → +∞; with c- > c+ ≥ 0. For the step initial data q(x, 0) = c- for x ≤ 0; c+ for x ≥ 0; the solution develops an oscillatory region called the dispersive shock wave region that connects the two constant regions c+ and c-. We show that the dispersive shock wave is described by a modulated periodic traveling wave solution of the MKdV equation where the modulation parameters evolve according to a Whitham modulation equation. The oscillatory region is expanding within a cone in the (x, t) plane defined as -6c²- + 12c²+ < x/t < 4c²- + 2c²+; with t » 1. For step-like initial data we show that the solution decomposes for long times into three main regions: (1) a region where solitons and breathers travel with positive velocities on a constant background c+; (2) an expanding oscillatory region (that generically contains breathers); (3) a region of breathers traveling with negative velocities on the constant background c-. When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the step initial data. The phase shift depends on the solitons, the breathers, and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers, and radiation. [ABSTRACT FROM AUTHOR]

The solar wind in the inner heliosheath beyond the termination shock (TS) is a nonequilibrium collisionless plasma consisting of thermal solar wind ions, suprathermal pickup ions, and electrons. In such multi‐ion plasma, two fast magnetosonic wave modes exist, the low‐frequency fast mode and the high‐frequency fast mode. Both fast modes are dispersive on fluid and ion scales, which results in nonlinear dispersive shock waves. We present high‐resolution three‐fluid simulations of the TS and the inner heliosheath up to a few astronomical units (AU) downstream of the TS. We show that downstream propagating nonlinear fast magnetosonic waves grow until they steepen into shocklets, overturn, and start to propagate backward in the frame of the downstream propagating wave. The counterpropagating nonlinear waves result in 2‐D fast magnetosonic turbulence, which is driven by the ion‐ion hybrid resonance instability. Energy is transferred from small scales to large scales in the inverse cascade range, and enstrophy is transferred from large scales to small scales in the direct cascade range. We validate our three‐fluid simulations with in situ high‐resolution Voyager 2 magnetic field observations in the inner heliosheath. Our simulations reproduce the observed magnetic turbulence spectrum with a spectral slope of −5/3 in frequency domain. However, the fluid‐scale turbulence spectrum is not a Kolmogorov spectrum in wave number domain because Taylor's hypothesis breaks down in the inner heliosheath. The magnetic structure functions of the simulated and observed turbulence follow the Kolmogorov‐Kraichnan scaling, which implies self‐similarity. Key Points: Nonlinear dispersive fast magnetosonic waves produce 2‐D compressible turbulence downstream of the termination shockTaylor's hypothesis breaks down in the subfast magnetosonic solar wind in the inner heliosheathThe magnetic turbulence spectrum observed by Voyager 2 in the inner heliosheath is reproduced by self‐consistent three‐fluid MHD simulation [ABSTRACT FROM AUTHOR]

The evolution of high-intensity light pulses in nonlinear single-mode optical waveguides, the dynamics of light in which is described by the nonlinear Schrödinger equation with a Raman term taking into account stimulated Raman self-scattering of light, is investigated. It is demonstrated that dispersive shock waves the behavior of which is much more diverse than in the case of ordinary nonlinear Schrödinger equation with a Kerr nonlinearity are formed in the process of evolution of pulses of substantially high intensity. The Whitham equations describing slow evolution of the dispersive shock waves are derived under the assumption of the Raman term being small. It is demonstrated that the dispersive shock waves can asymptotically assume a stationary profile when the Raman effect is taken into account. Analytical theory is corroborated by numerical calculations. [ABSTRACT FROM AUTHOR]

Dispersive shock waves (DSWs), also termed undular bores in fluid mechanics, governed by the non-local Whitham equation are studied in order to investigate short wavelength effects that lead to peaked and cusped waves within the DSW. This is done by combining the weak nonlinearity of the Korteweg- de Vries equation with full linear dispersion relations. The dispersion relations considered are those for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120° peaked Stokes wave of highest amplitude. A dispersive shock fitting method is used to find the leading (solitary wave) and trailing (linear wave) edges of the DSW. This method is found to produce results in excellent agreement with numerical solutions up until the lead solitary wave of the DSW reaches its highest amplitude. Numerical solutions show that the DSWs for the water wave and Whitham peaking kernels become modulationally unstable and evolve into multi-phase wavetrains after a critical amplitude which is just below the DSW of maximum amplitude. [ABSTRACT FROM AUTHOR]

We demonstrate a possibility to make rogue waves (RWs) in the form of the Peregrine soliton (PS) and Kuznetsov–Ma breathers (KMBs) effectively stable objects, with the help of properly defined dispersion or nonlinearity management applied to the continuous-wave (CW) background supporting the RWs. In particular, it is found that either management scheme, if applied along the longitudinal coordinate, making the underlying nonlinear Schrödinger equation (NLSE) self-defocusing in the course of disappearance of the PS, indeed stabilizes the global solution with respect to the modulational instability of the background. In the process, additional excitations are generated, namely, dispersive shock waves and, in some cases, also a pair of slowly separating dark solitons. Further, the nonlinearity-management format, which makes the NLSE defocusing outside of a finite domain in the transverse direction, enables the stabilization of the KMBs, in the form of confined oscillating states. On the other hand, a nonlinearity-management format applied periodically along the propagation direction, creates expanding patterns featuring multiplication of KMBs through their cascading fission. [ABSTRACT FROM AUTHOR]

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit. [ABSTRACT FROM AUTHOR]

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 ow over obstacles. The convective instability results in an effective stabilization of the DSW. [ABSTRACT FROM AUTHOR]

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory. [ABSTRACT FROM AUTHOR]

The propagation of an optical dispersive shock wave, generated from a jump discontinuity in light intensity, in a defocusing colloidal medium is analysed. The equations governing nonlinear light propagation in a colloidal medium consist of a nonlinear Schrödinger equation for the beam and an algebraic equation for the medium response. In the limit of low light intensity, these equations reduce to a perturbed higher order nonlinear Schrödinger equation. Solutions for the leading and trailing edges of the colloidal dispersive shock wave are found using modulation theory. This is done for both the perturbed nonlinear Schrödinger equation and the full colloid equations for arbitrary light intensity. These results are compared with numerical solutions of the colloid equations. [ABSTRACT FROM AUTHOR]

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose--Ein stein condensates. As in the classical theory of hyperbolic equations where a nonconvex flux leads to nonclassical solution structures, a nonconvex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth order Korteweg--de Vries (KdV) equation, also known as the Kawahara equation, a classical model for shallow water waves, is shown to be a universal model of Eulerian hydrodynamics with higher order dispersive effects. Utilizing asymptotic methods an d numerical computations, this work classifies the long-time behavior of solutions for step-like initial data. For convex dispersion, the result is a dispersive shock wave (DSW), qualitatively and quantitatively bearing close resemblance to the KdV DSW. For nonconvex dispersion, three distinct dynamic regimes are observed. For small amplitude jumps, a perturbed KdV DSW with positive polarity and orientation is generated, accompanied by small amplitude radiation from an embedded solitary wave leading edge, termed a radiating DSW. For moderate jumps, a crossover regime is obser ved with waves propagating forward and backward from the sharp transition region. For jumps excee ding a critical threshold, a new type of DSW is observed that we term a traveling DSW (TDSW). The TDSW consists of a traveling wave that connects a partial, nonmonotonic, negative solitary wave at the trailing edge to an interior nonlinear periodic wave. Its speed, a generalized Rankine--Hugoniot jump condition, is determined by the far-field structure of the traveling wave. The TDSW is resolved at the leading edge by a harmonic wavepacket moving with the linear group velocity. The nonclassical TDSW exhibits features common to both dissipative and dispersive shock waves. [ABSTRACT FROM AUTHOR]

A dispersive shock wave (DSW), with a circular geometry, is studied in a colloidal medium. The colloidal particle interaction is based on the repulsive theoretical hard sphere model, where a series in the particle density, or packing fraction is used for the compressibility. Experimental results show that the particle interactions are temperature dependent and can be either repulsive or attractive, so the second term in the compressibility series is modified to allow for temperature dependent effects, using a power-law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump in electric field amplitude is resolved via a DSW, which forms before the onset of modulational instability. A semi-analytical solution for the amplitude of the solitary waves in a DSW of large radius, is derived based on a combination of conservation laws and geometrical considerations. The effect of temperature and background packing fraction on the evolution of the DSW and the amplitude of the solitary waves is discussed and the semi-analytical solutions are found to be very accurate, when compared with numerical solutions. [ABSTRACT FROM AUTHOR]

Dispersive shock waves (DSWs) in the Kadomtsev–Petviashvili (KP) equation and two dimensional Benjamin–Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time ( 2 + 1 ) dimensions to finding DSW solutions of ( 1 + 1 ) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg–de Vries (cKdV) and cylindrical Benjamin–Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the ( 2 + 1 ) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced ( 1 + 1 ) dimensional equations. [ABSTRACT FROM AUTHOR]

Surface and interfacial weakly-nonlinear ring waves in a two-layer fluid are modelled numerically, within the framework of the recently derived 2 + 1 -dimensional cKdV-type equation. In a case study, we consider concentric waves from a localised initial condition and waves in a 2D version of the dam-break problem, as well as discussing the effect of a piecewise-constant shear flow. The modelling shows, in particular, the formation of 2D dispersive shock waves and oscillatory wave trains. [ABSTRACT FROM AUTHOR]

The Serre–Green–Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is known to describe accurately the wave motion at the surface of an incompressible inviscid fluid in the case when the fluid flow is irrotational and two-dimensional. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre–Green–Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre–Green–Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling. One consequence of the present analysis is that the energy loss appearing in the shallow-water theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre–Green–Naghdi equations using a finite-element discretization coupled with an adaptive Runge–Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre–Green–Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling. [ABSTRACT FROM AUTHOR]

Considering the nonlinear Schrödinger (NLS) equation as a representative model, we report a unified presentation of different forms of incoherent shock waves that emerge in the long-range interaction regime of a turbulent optical wave system. These incoherent singularities can develop either in the temporal domain through a highly noninstantaneous nonlinear response, or in the spatial domain through a highly nonlocal nonlinearity. In the temporal domain, genuine dispersive shock waves (DSW) develop in the spectral dynamics of the random waves, despite the fact that the causality condition inherent to the response function breaks the Hamiltonian structure of the NLS equation. Such spectral incoherent DSWs are described in detail by a family of singular integro-differential kinetic equations, e.g. Benjamin–Ono equation, which are derived from a nonequilibrium kinetic formulation based on the weak Langmuir turbulence equation. In the spatial domain, the system is shown to exhibit a large scale global collective behavior, so that it is the fluctuating field as a whole that develops a singularity, which is inherently an incoherent object made of random waves. Despite the Hamiltonian structure of the NLS equation, the regularization of such a collective incoherent shock does not require the formation of a DSW — the regularization is shown to occur by means of a different process of coherence degradation at the shock point. We show that the collective incoherent shock is responsible for an original mechanism of spontaneous nucleation of a phase-space hole in the spectrogram dynamics. The robustness of such a phase-space hole is interpreted in the light of incoherent dark soliton states, whose different exact solutions are derived in the framework of the long-range Vlasov formalism. [ABSTRACT FROM AUTHOR]

We investigate surface gravity waves in a shallow water tank, in the limit of long wavelengths. We report the observation of non-stationary dispersive shock waves rapidly expanding over a 90 m flume. They are excited by means of a wave maker that allows us to launch a controlled smooth (single well) depression with respect to the unperturbed surface of the still water, a case that contains no solitons. The dynamics of the shock waves are observed at different levels of nonlinearity equivalent to a different relative smallness of the dispersive effect. The observed undulatory behavior is found to be in good agreement with the dynamics described in terms of a Korteweg–de Vries equation with evolution in space, though in the most nonlinear cases the description turns out to be improved over the quasi linear trailing edge of the shock by modeling the evolution in terms of the integro-differential (nonlocal) Whitham equation. [ABSTRACT FROM AUTHOR]

Dispersive shock waves (DSWs), which connect states of different amplitude via a modulated wave train, form generically in nonlinear dispersive media subjected to abrupt changes in state. The primary tool for the analytical study of DSWs is Whitham's modulation theory. While this framework has been successfully employed in many space-continuous settings to describe DSWs, the Whitham modulation equations are cumbersome in most spatially discrete systems. In this article, we illustrate the relevance of the reduction of the DSW dynamics to a planar ODE in a broad class of lattice examples. Solutions of this low-dimensional ODE accurately describe the orbits of the DSW in self-similar coordinates and the local averages in a manner consistent with the modulation equations. We use data-driven and quasi-continuum approaches within the context of a discrete system of conservation laws to demonstrate how the underlying low dimensional structure of DSWs can be identified and analyzed. The connection of these results to Whitham modulation theory is also discussed. • A planar ODE is found to describe a lattice DSW well. • The planar ODE is identified using data driven and quasi-continuum approaches. • The planar ODE description is consistent with modulation theory. • The presented methods could be applied to a broad class of lattices. [ABSTRACT FROM AUTHOR]

The formation of a dispersive shock wave in a colloidal medium, due to an initial jump in the light intensity, is studied. The compressibility of the colloidal particles is modeled using a series in the particle density, or packing fraction, where the virial coefficients depend on the particle interaction model. Both the theoretical hard disk and sphere repulsive models, and a model with temperature dependent compressibility, are considered. Experimental results for the second virial coefficient show that it is temperature dependent and that the particle interactions can be either repulsive or attractive. These effects are modeled using a power-law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump is resolved via a dispersive shock wave which forms before the onset of modulational instability. A semi-analytical solution is developed for the one-dimensional line bore case which predicts the amplitude of the solitary waves which form in the dispersive shock wave. The solitary wave amplitude versus jump height diagrams can exhibit three different kinds of behaviors. A unique solution, an S-shaped solution curve and multiple solution branches where the upper branch has separated from the lower branches. A bifurcation from the low to the high power branch can occur for many parameter choices as the amplitude of the initial jump increases. The effect of temperature on the evolution of the bore, the amplitude of the solitary waves and the bifurcation patterns are all discussed and the semi-analytical solutions are found to be very accurate. [ABSTRACT FROM AUTHOR]

Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Numerical simulations of the NLSE (or GPE) are presented demonstrating emission of short pulses of the matter (light) density formed in the course of tunneling in wave-guided light and/or trapped BEC. The phenomenon is observed under various conditions, for nonlinearities of different signs, zero nonlinearity included. We study, both numerically and analytically, pulsations of matter (light) remaining within the trap and use the results in order to induce emission of sequential pulses by properly narrowing the trap. This allows us to propose a mechanism for a realization of Atom Pulse Laser. [Copyright &y& Elsevier]

Abstract: Collisions and interactions of dispersive shock waves in defocusing (repulsive) nonlinear Schrödinger type systems are investigated analytically and numerically. Two canonical cases are considered. In one case, two counterpropagating dispersive shock waves experience a head-on collision, interact and eventually exit the interaction region with larger amplitudes and altered speeds. In the other case, a fast dispersive shock overtakes a slower one, giving rise to an interaction. Eventually the two merge into a single dispersive shock wave. In both cases, the interaction region is described by a modulated, quasi-periodic two-phase solution of the nonlinear Schrödinger equation. The boundaries between the background density, dispersive shock waves and their interaction region are calculated by solving the Whitham modulation equations. These asymptotic results are in excellent agreement with full numerical simulations. It is further shown that the interactions of two dispersive shock waves have some qualitative similarities to the interactions of two classical shock waves. [Copyright &y& Elsevier]

Abstract: We show the relevance of the dispersive analogue of the shock waves in the FPU dynamics. In particular we give strict numerical evidence that metastable states emerging from low frequency initial excitations are indeed constituted by dispersive shock waves travelling through the chain. Relevant characteristics of the metastable states, such as their frequency extension and their time scale of formation, are correctly obtained within this framework, using the underlying continuum model, the KdV equation. [Copyright &y& Elsevier]