Spatial dispersive shock waves generated in supersonic flow of Bose–Einstein condensate past slender body

Supersonic flow of Bose–Einstein condensate past macroscopic obstacles is studied theoretically. It is shown that in the case of large obstacles the Cherenkov cone transforms into a stationary spatial shock wave which consists of a number of spatial dark solitons. Analytical theory is developed for the case of obstacles having a form of a slender body. This theory explains qualitatively the properties of such shocks observed in recent experiments on nonlinear dynamics of condensates of dilute alkali gases

An Integrable Model For Undular Bores On Shallow Water

On the basis of the integrable Kaup-Boussinesq version of the shallow water equations, an analytical theory of undular bores is constructed. The problem of the decay of an initial discontinuity is considered

Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation

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 analytical theory of breather and soliton gases by considering a special, thermodynamic-type limit of the wave-number–frequency relations for multiphase (finite-gap) solutions of the focusing nonlinear Schrödinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator, and it yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the “background” Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, noninteracting breathers (solitons) to a special limiting state, which we term a breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathers (solitons). For a nonhomogeneous breather gas, we derive a full set of kinetic equations describing the slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating the efficacy of the developed general theory with broad implications for nonlinear optics, superfluids, and oceanography. In particular, our work provides the theoretical underpinning for the recently observed remarkable connection of the soliton gas dynamics with the long-term evolution of spontaneous modulational instability

Macroscopic dynamics of incoherent soliton ensembles: soliton-gas kinetics and direct numerical modeling

We undertake a detailed comparison of the results of direct numerical simulations of the integrable soliton gas dynamics with the analytical predictions inferred from the exact solutions of the relevant kinetic equation for solitons. We use the KdV soliton gas as a simplest analytically accessible model yielding major insight into the general properties of soliton gases in integrable systems. Two model problems are considered: (i) the propagation of a `trial' soliton through a one-component `cold' soliton gas consisting of randomly distributed solitons of approximately the same amplitude; and (ii) collision of two cold soliton gases of different amplitudes (soliton gas shock tube problem) leading to the formation of an incoherend dispersive shock wave. In both cases excellent agreement is observed between the analytical predictions of the soliton gas kinetics and the direct numerical simulations. Our results confirm relevance of the kinetic equation for solitons as a quantitatively accurate model for macroscopic non-equilibrium dynamics of incoherent soliton ensembles.Comment: 20 pages, 8 figures, 34 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Generation of oblique dark solitons in supersonic flow of Bose-Einstein condensate past an obstacle

Nonlinear and dispersive properties of Bose-Einstein condensate (BEC) provide a possibility of formation of various nonlinear structures such as vortices and bright and dark solitons (see, e.g., [1]). Yet another type of nonlinear wave patterns has been observed in a series of experiments on the BEC flow past macroscopic obstacles [2]. In [3] these structures have been associated with spatial dispersive shock waves. Spatial dispersive shock waves represent dispersive analogs of the the well-known viscous spatial shocks (oblique jumps of compression) occurring in supersonic flows of compressible fluids past obstacles. In a viscous fluid, the shock can be represented as a narrow region within which strong dissipation processes take place and the thermodynamic parameters of the flow undergo sharp change. On the contrary, if viscosity is negligibly small compared with dispersion effects, the shock discontinuity resolves into an expanding in space oscillatory structure which transforms gradually, as the distance from the obstacle increases, into a \fan" of stationary solitons. If the obstacle is small enough, then such a \fan" reduces to a single spatial dark soliton [4]. Here we shall present the theory of these new structures in BEC

Generation of undular bores in the shelves of slowly-varying solitary waves

We study the long-time evolution of the trailing shelves that form behind solitary waves moving through an inhomogeneous medium, within the framework of the variable-coefficient Korteweg-de Vries equation. We show that the nonlinear evolution of the shelf leads typically to the generation of an undular bore and an expansion fan, which form apart but start to overlap and nonlinearly interact after a certain time interval. The interaction zone expands with time and asymptotically as time goes to infinity occupies the whole perturbed region. Its oscillatory structure strongly depends on the sign of the inhomogeneity gradient of the variable background medium. We describe the nonlinear evolution of the shelves in terms of exact solutions to the KdV-Whitham equations with natural boundary conditions for the Riemann invariants. These analytic solutions, in particular, describe the generation of small "secondary" solitary waves in the trailing shelves, a process observed earlier in various numerical simulations

Unified Approach to KdV Modulations

We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial value problem for the zero dispersion KdV as the steepest descent for the scalar Riemann-Hilbert problem, developed by Deift, Venakides, and Zhou, 1997, and on the method of generating differentials for the KdV-Whitham hierarchy proposed by El, 1996. By assuming the hyperbolicity of the zero-dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the $x, t$ - plane. The resulting system effectively solves the zero dispersion KdV with an arbitrary initial data.Comment: 27 pages, Latex, 5 Postscript figures, to be submitted to Comm. Pure. Appl. Mat

Optical Random Riemann Waves in Integrable Turbulence

We examine integrable turbulence (IT) in the framework of the defocusing cubic one-dimensional nonlinear Schr\"{o}dinger 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 show that the development of IT can be divided into two distinct stages, the initial, pre-breaking stage being described by a system of interacting random Riemann waves. We explain the low-tailed statistics of the wave intensity in IT and show that the Riemann invariants of the asymptotic nonlinear geometric optics system represent the observable quantities that provide new insight into statistical features of the initial stage of the IT development by exhibiting stationary probability density functions