Asymptotic integrability of nonlinear wave equations.

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables.

Undular bore theory for the modified Korteweg-de Vries-Burgers equation.

We consider nonlinear wave structures described by the modified Korteweg-de Vries equation, taking into account a small Burgers viscosity for the case of steplike initial conditions. The Whitham modulation equations are derived, which include the small viscosity as a perturbation. It is shown that for a long enough time of evolution, this small perturbation leads to the stabilization of cnoidal bores, and their main characteristics are obtained.

Propagation of generalized Korteweg-de Vries solitons along large-scale waves.

We consider propagation of solitons along large-scale background waves in the generalized Korteweg-de Vries (gKdV) equation theory when the width of the soliton is much smaller than the characteristic size of the background wave. Due to this difference in scales, the soliton's motion does not affect the dispersionless evolution of the background wave. We obtained the Hamilton equations for soliton's motion and derived simple relationships which express the soliton's velocity in terms of a local value of the background wave.

Asymptotic theory of not completely integrable soliton equations.

We develop the theory of transformation of intensive initial nonlinear wave pulses to trains of solitons emerging at asymptotically large time of evolution. Our approach is based on the theory of dispersive shock waves in which the number of nonlinear oscillations in the shock becomes the number of solitons at the asymptotic state. We show that this number of oscillations, which is proportional to the classical action of particles associated with the small-amplitude edges of shocks, is preserved by the dispersionless flow.

Motion of dark solitons in a non-uniform flow of Bose-Einstein condensate.

We study motion of dark solitons in a non-uniform one-dimensional flow of a Bose-Einstein condensate. Our approach is based on Hamiltonian mechanics applied to the particle-like behavior of dark solitons in a slightly non-uniform and slowly changing surrounding. In one-dimensional geometry, the condensate's wave function undergoes the jump-like behavior across the soliton, and this leads to generation of the counterflow in the background condensate.

Contour dynamics of two-dimensional dark solitons.

Equations for contour dynamics of trough-shaped dark solitons are obtained for the general form of the nonlinearity function. Their self-similar solution which describes the nonlinear stage of the bending instability of dark solitons is studied in detail.

Number of solitons produced from a large initial pulse in the generalized NLS dispersive hydrodynamics theory.

We show that the number of solitons produced from an arbitrary initial pulse of the simple wave type can be calculated analytically if its evolution is governed by a generalized nonlinear Schrödinger (NLS) equation provided this number is large enough. The final result generalizes the asymptotic formula derived for completely integrable nonlinear wave equations such as the standard NLS equation with the use of the inverse scattering transform method.

Quantitative Analysis of Shock Wave Dynamics in a Fluid of Light.

We report on the formation of a dispersive shock wave in a nonlinear optical medium. We monitor the evolution of the shock by tuning the incoming beam power. The experimental observations for the position and intensity of the solitonic edge of the shock, as well as the location of the nonlinear oscillations are well described by recent developments of Whitham modulation theory.

Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse.

The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow, which interacts with the edge wave packets or the edge solitons. A conjecture about the existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated.

Formation of dispersive shock waves in a saturable nonlinear medium.

We use the Gurevich-Pitaevskii approach based on the Whitham averaging method for studying the formation of dispersive shock waves in an intense light pulse propagating through a saturable nonlinear medium. Although the Whitham modulation equations cannot be diagonalized in this case, the main characteristics of the dispersive shock can be derived by means of an analysis of the properties of these equations at the boundaries of the shock. Our approach generalizes a previous analysis of steplike initial intensity distributions to a more realistic type of initial light pulse and makes it possible to determine, in a setting of experimental interest, the value of measurable quantities such as the wave-breaking time or the position and light intensity of the shock edges.

Self-similar wave breaking in dispersive Korteweg-de Vries hydrodynamics.

We discuss the problem of breaking of a nonlinear wave in the process of its propagation into a medium at rest. It is supposed that the profile of the wave is described at the breaking moment by the function (-x) (x<0, positive pulse) or -x (x>0, negative pulse) of the coordinate x. Evolution of the wave is governed by the Korteweg-de Vries equation resulting in the formation of a dispersive shock wave.

Dispersive shock wave theory for nonintegrable equations.

We suggest a method for calculation of parameters of dispersive shock waves in the framework of Whitham modulation theory applied to nonintegrable wave equations with a wide class of initial conditions corresponding to propagation of a pulse into a medium at rest. The method is based on universal applicability of Whitham's "number of waves conservation law" as well as on the conjecture of applicability of its soliton counterpart to the above mentioned class of initial conditions which is substantiated by comparison with similar situations in the case of completely integrable wave equations. This allows one to calculate the limiting characteristic velocities of the Whitham modulation equations at the boundary with the smooth part of the pulse whose evolution obeys the dispersionless approximation equations.

Long-time evolution of pulses in the Korteweg-de Vries equation in the absence of solitons reexamined: Whitham method.

We consider the long-time evolution of pulses in the Korteweg-de Vries equation theory for initial distributions which produce no soliton but instead lead to the formation of a dispersive shock wave and of a rarefaction wave. An approach based on Whitham modulation theory makes it possible to obtain an analytic description of the structure and to describe its self-similar behavior near the soliton edge of the shock. The results are compared with numerical simulations.

Simple waves in a two-component Bose-Einstein condensate.

We study the dynamics of so-called simple waves in a two-component Bose-Einstein condensate. The evolution of the condensate is described by Gross-Pitaevskii equations which can be reduced for these simple wave solutions to a system of ordinary differential equations which coincide with those derived by Ovsyannikov for the two-layer fluid dynamics. We solve the Ovsyannikov system for two typical situations of large and small difference between interspecies and intraspecies nonlinear interaction constants.

Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate.

We provide a classification of the possible flows of two-component Bose-Einstein condensates evolving from initially discontinuous profiles. We consider the situation where the dynamics can be reduced to the consideration of a single polarization mode (also denoted as "magnetic excitation") obeying a system of equations equivalent to the Landau-Lifshitz equation for an easy-plane ferromagnet. We present the full set of one-phase periodic solutions.

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion.

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates.

Oblique breathers generated by a flow of two-component Bose-Einstein condensates past a polarized obstacle.

We predict that oblique breathers can be generated by a flow of two-component Bose-Einstein condensates past a polarized obstacle that attracts one component of the condensate and repels the other one. The breather exists if intraspecies interaction constants differ from the interspecies interaction constant, and it corresponds to the nonlinear excitation of the so-called polarization mode with domination of the relative motion of the components. Analytical theory is developed for the case of small-amplitude breathers that is in reasonable agreement with the numerical results.

Two-dimensional dispersive shock waves in dissipative optical media.

We study generation of two-dimensional dispersive shock waves and oblique dark solitons upon interaction of tilted plane waves with negative refractive index defects embedded into defocusing material with linear gain and two-photon absorption. Different evolution regimes are encountered, including the formation of well-localized disturbances for input tilts below critical one, generation of extended shock waves containing multiple intensity oscillations in the "upstream" region, and gradually vanishing oblique dark solitons in the "downstream" region for input tilts exceeding critical one. The generation of stable dispersive shock waves is possible only below certain critical defect strength.

Undular bore theory for the Gardner equation.

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible.

Stationary one-dimensional dispersive shock waves.

We address shock waves generated upon the interaction of tilted plane waves with negative refractive index defects in defocusing media with linear gain and two-photon absorption. We found that, in contrast to conservative media where one-dimensional dispersive shock waves usually exist only as nonstationary objects expanding away from a defect or generating beam, the competition between gain and two-photon absorption in a dissipative medium results in the formation of localized stationary dispersive shock waves, whose transverse extent may considerably exceed that of the refractive index defect. One-dimensional dispersive shock waves are stable if the defect strength does not exceed a certain critical value.

Two-dimensional supersonic nonlinear Schrödinger flow past an extended obstacle.

Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional (2D) defocusing nonlinear Schrödinger (NLS) equation. This problem is of fundamental importance as a dispersive analog of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed is sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x coordinate and the piston motion curve is defined by the spatial body profile.

Stabilization of solitons generated by a supersonic flow of Bose-Einstein condensate past an obstacle.

The stability of dark solitons generated by supersonic flow of a Bose-Einstein condensate past an obstacle is investigated. It is shown that in the reference frame attached to the obstacle a transition occurs at some critical value of the flow velocity from absolute instability of dark solitons to their convective instability. This leads to the decay of disturbances of solitons at a fixed distance from the obstacle and the formation of effectively stable dark solitons.

Whitham method for the Benjamin-Ono-Burgers equation and dispersive shocks.

The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of the damped Benjamin-Ono equation. The structure of the dispersive shock is considered in this method.

Oblique dark solitons in supersonic flow of a Bose-Einstein condensate.

In the framework of the Gross-Pitaevskii mean field approach, it is shown that the supersonic flow of a Bose-Einstein condensate can support a new type of pattern--an oblique dark soliton. The corresponding exact solution of the Gross-Pitaevskii equation is obtained. It is demonstrated by numerical simulations that oblique solitons can be generated by an obstacle inserted into the flow.