Dispersive Hydrodynamics Research Articles

Dispersive hydrodynamics in non-Hermitian nonlinear Schrödinger equation with complex external potential

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.

Generalized hydrodynamics of the KdV soliton gas

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 equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport.

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.

Soliton gas in integrable dispersive hydrodynamics

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 dynamics are modelled by a thermodynamic type limit of modulated finite-gap spectral solutions of the Korteweg–de Vries and the focusing nonlinear Schrödinger (NLS) equations. For the focusing NLS equation the notions of soliton condensate and breather gas are introduced that are related to the phenomena of spontaneous modulational instability and the rogue wave formation. The integrability properties of the kinetic equation for soliton gas are discussed and some physically relevant solutions are presented and compared with direct numerical simulations of dispersive hydrodynamic systems.

Magnetic Guiding with Permanent Magnets: Concept, Realization and Applications to Nanoparticles and Cells.

The idea of remote magnetic guiding is developed from the underlying physics of a concept that allows for bijective force generation over the inner volume of magnet systems. This concept can equally be implemented by electro- or permanent magnets. Here, permanent magnets are in the focus because they offer many advantages. The equations of magnetic fields and forces as well as velocities are derived in detail and physical limits are discussed. The special hydrodynamics of nanoparticle dispersions under these circumstances is reviewed and related to technical constraints. The possibility of 3D guiding and magnetic imaging techniques are discussed. Finally, the first results in guiding macroscopic objects, superparamagnetic nanoparticles, and cells with incorporated nanoparticles are presented. The constructed magnet systems allow for orientation, movement, and acceleration of magnetic objects and, in principle, can be scaled up to human size.

Soliton gas in bidirectional dispersive hydrodynamics.

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 overtaking collisions of solitons take place. We distinguish between two qualitatively different types of bidirectional soliton gases: isotropic gases, in which the position shifts accompanying the head-on and overtaking soliton collisions have the same sign, and anisotropic gases, in which the position shifts for head-on and overtaking collisions have opposite signs. We construct kinetic equations for both types of bidirectional soliton gases and solve the respective shock-tube problems for the collision of two "monochromatic" soliton beams consisting of solitons of approximately the same amplitude and velocity. The corresponding weak solutions of the kinetic equations consisting of differing uniform states separated by contact discontinuities for the mean flow are constructed. Concrete examples of bidirectional Eulerian soliton gases for the defocusing nonlinear Schrödinger (NLS) equation and the resonant NLS equation are considered. The kinetic equation of the resonant NLS soliton gas is shown to be equivalent to that of the shallow-water bidirectional soliton gas described by the Kaup-Boussinesq equations. The analytical results for shock-tube Riemann problems for bidirectional soliton gases are shown to be in excellent agreement with direct numerical simulations.

Stability in turbulence: The interplay between shocks and vorticity in a superfluid with higher-order dispersion

Higher-order dispersion can lead to intriguing dynamics that are becoming a focus of modern hydrodynamics research. Such systems occur naturally, for example in shallow water waves and nonlinear optics, for which several types of novel dispersive shocks structures have been identified. Here we introduce ultracold atoms as a tunable quantum simulations platform for higher-order systems. Degenerate quantum gases are well controlled model systems for the experimental study of dispersive hydrodynamics in superfluids and have been used to investigate phenomena such as vortices, solitons, dispersive shock waves and quantum turbulence. With the advent of Raman-induced spin-orbit coupling, the dispersion of a dilute gas Bose-Einstein condensate can be modified in a flexible way, allowing for detailed investigations of higher-order dispersion dynamics. Here we present a combined experimental and theoretical study of shock structures generated in such a system. The breaking of Galilean invariance by the spin-orbit coupling allows two different types of shock structures to emerge simultaneously in a single system. Numerical simulations suggest that the behavior of these shock structures is affected by interactions with vortices in a manner reminiscent of emerging viscous hydrodynamics due to an underlying quantum turbulence in the system. This result suggests that spin-orbit coupling can be used as a powerful means to tun the effective viscosity in cold-atom experiments serving as quantum simulators of turbulent hydrodynamics, with applications from condensed matter and optics to quantum simulations of neutron stars.

Hydrodynamic X Waves.

Stationary wave groups exist in a range of nonlinear dispersive media, including optics, Bose-Einstein condensates, plasma, and hydrodynamics. We report experimental observations of nonlinear surface gravity X waves, i.e., X-shaped wave envelopes that propagate over long distances with constant form. These can be described by the 2D+1 nonlinear Schrödinger equation, which predicts a balance between dispersion and diffraction when the envelope (the arms of the X) travel at ±arctan(1/sqrt[2])≈±35.26° to the carrier wave. Our findings may help improve understanding the lifetime of extremes in directional seas and motivate further studies in other nonlinear dispersive media.

Controlling dispersive hydrodynamic wavebreaking in a viscous fluid conduit

The driven, cylindrical, free interface between two miscible, Stokes fluids with high viscosity contrast have been shown to exhibit dispersive hydrodynamics. A hallmark feature of dispersive hydrodynamic media is the dispersive resolution of wavebreaking that results in a dispersive shock wave. In the context of the viscous fluid conduit system, the present work introduces a simple, practical method to precisely control the location, time, and spatial profile of wavebreaking in dispersive hydrodynamic systems with only boundary control. The method is based on tracking the dispersionless characteristics backward from the desired wavebreaking profile to the boundary. In addition to the generation of approximately step-like Riemann and box problems, the method is generalized to other, approximately piecewise-linear dispersive hydrodynamic profiles including the triangle wave and N-wave. A definition of dispersive hydrodynamic wavebreaking is used to obtain quantitative agreement between the predicted location and time of wavebreaking, viscous fluid conduit experiment, and direct numerical simulations for a range of flow conditions. Observed space-time characteristics also agree with triangle and N-wave predictions. The characteristic boundary control method introduced here enables the experimental investigation of a variety of wavebreaking profiles and is expected to be useful in other dispersive hydrodynamic media. As an application of this approach, soliton fission from a large, box-like disturbance is observed both experimentally and numerically, motivating future analytical treatment.

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)1/n (x<0, positive pulse) or -x1/n (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. In the positive pulse case, the dispersive shock wave forms at the leading edge of the wave structure and in the negative pulse case, at its rear edge. The dynamics of dispersive shock waves is described by the Whitham modulation equations. For power law initial profiles, this dynamics is self-similar and the solution of the Whitham equations is obtained in a closed form for arbitrary n>1.

Solitonic Dispersive Hydrodynamics: Theory and Observation.

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 lacking. Here, a general soliton-mean field theory is introduced and used to describe the propagation of solitons in macroscopic hydrodynamic flows. Two universal adiabatic invariants of motion are identified that predict trapping or transmission of solitons by hydrodynamic states. The result of solitons incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves is the same, an effect termed hydrodynamic reciprocity. Experiments on viscous fluid conduits quantitatively confirm the soliton-mean field theory with broader implications for nonlinear optics, superfluids, geophysical fluids, and other dispersive hydrodynamic media.

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.

Evolution of initial discontinuities in the DNLS equation theory

We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive hydrodynamics problem, solutions of the Whitham modulation equations are mapped to parameters of a modulated wave by two-valued functions what makes situation much richer than that for a convex case of the NLS equation type. In particular, new types of simple-wave-like structures, such as trigonometric shocks and combined shocks, appear as building elements of the whole wave pattern observable in the long-time evolution of pulses after the wave-breaking point. The developed theory can find applications to propagation of light pulses in fibers and to the theory of Alfvén dispersive shock waves.

Shock Waves in Dispersive Hydrodynamics with Nonconvex Dispersion

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose-Einstein condensates. As in the classical theory of hyperbolic equations where a non-convex flux leads to non-classical solution structures, a non-convex 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 and 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 non-convex dispersion, three distinct dynamic regimes are observed. For small 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 or RDSW. For moderate jumps, a crossover regime is observed with waves propagating forward and backward from the sharp transition region. For jumps exceeding a critical threshold, a new type of DSW is observed we term a translating DSW or TDSW. The TDSW consists of a traveling wave that connects a partial, non-monotonic, 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 non-classical TDSW exhibits features common to both dissipative and dispersive shock waves.

Dispersive hydrodynamics of nonlinear polarization waves in two-component Bose-Einstein condensates

We study one dimensional mixtures of two-component Bose-Einstein condensates in the limit where the intra-species and inter-species interaction constants are very close. Near the mixing-demixing transition the polarization and the density dynamics decouple. We study the nonlinear polarization waves, show that they obey a universal (i.e., parameter free) dynamical description, identify a new type of algebraic soliton, explicitly write simple wave solutions, and study the Gurevich-Pitaevskii problem in this context.

Observation of Dispersive Shock Waves, Solitons, and Their Interactions in Viscous Fluid Conduits

Dispersive shock waves and solitons are fundamental nonlinear excitations in dispersive media, but dispersive shock wave studies to date have been severely constrained. Here, we report on a novel dispersive hydrodynamic test bed: the effectively frictionless dynamics of interfacial waves between two high viscosity contrast, miscible, low Reynolds number Stokes fluids. This scenario is realized by injecting from below a lighter, viscous fluid into a column filled with high viscosity fluid. The injected fluid forms a deformable pipe whose diameter is proportional to the injection rate, enabling precise control over the generation of symmetric interfacial waves. Buoyancy drives nonlinear interfacial self-steepening, while normal stresses give rise to the dispersion of interfacial waves. Extremely slow mass diffusion and mass conservation imply that the interfacial waves are effectively dissipationless. This enables high fidelity observations of large amplitude dispersive shock waves in this spatially extended system, found to agree quantitatively with a nonlinear wave averaging theory. Furthermore, several highly coherent phenomena are investigated including dispersive shock wave backflow, the refraction or absorption of solitons by dispersive shock waves, and the multiphase merging of two dispersive shock waves. The complex, coherent, nonlinear mixing of dispersive shock waves and solitons observed here are universal features of dissipationless, dispersive hydrodynamic flows.

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.

A new ripplon branch in He II

We analyse the dispersion relation of ripplons, on the surface of superfluid helium, using the dispersive hydrodynamics approach and find a new ripplon branch. We obtain an analytical equation for the dispersion relation and analytic expressions for the limiting cases. The probabilities of decay of unstable ripplons above the roton gap into rotons are derived. A numerical solution for the ripplon dispersion curve is obtained. A new ripplon branch is found at energies just below the instability point of the bulk spectrum, and is investigated; its stability is discussed.

Hypersonic flow past slender bodies in dispersive hydrodynamics

The problem of two-dimensional steady hypersonic flow past a slender body is formulated for dispersive media. It is shown that for the hypersonic flow, the original 2 + 0 boundary-value problem is asymptotically equivalent to the 1 + 1 piston problem for the fully nonlinear flow in the same physical system, which allows one to take advantage of the analytic methods developed for one-dimensional systems. This type of equivalence, well known in ideal Euler gas dynamics, has not been established for dispersive hydrodynamics so far. Two examples pertaining to collisionless plasma dynamics are considered.

Hydrodynamics of magnetic and dielectric colloidal dispersions

We discuss the hydrodynamic behavior of colloidal suspensions, the particles of which possess electric or magnetic moments. Particular attention is given to the case where an external electric or magnetic field acts on a system in which the polarization does not relax instantaneously, so that reorientational Brownian motion is coupled to both the field and to hydrodynamic degrees of freedom. Magnetosonic and magnetoviscous effects are derived, with emphasis on anisotropy with respect to the external field.