Soliton Gas Research Articles

Two-dimensional bright solitons in dipolar Bose-Einstein condensates with tilted dipoles

The effect of dipolar orientation with respect to the soliton plane on the physics of two-dimensional bright solitons in dipolar Bose-Einstein condensates is discussed. Previous studies on such a soliton involved dipoles either perpendicular or parallel to the condensate-plane. The tilting angle constitutes an additional tuning parameter, which help us to control the in-plane anisotropy of the soliton as well as provides access to previously disregarded regimes of interaction parameters for soliton stability. In addition, it can be used to drive the condensate into phonon instability without changing its interaction parameters or trap geometry. The phonon-instability in a homogeneous 2D condensate of tilted dipoles always features a transient stripe pattern, which eventually breaks into a metastable soliton gas. Finally, we demonstrate how a dipolar BEC in a shallow trap can eventually be turned into a self-trapped matter wave by an adiabatic approach, involving the tuning of tilting angle.

Elastic and inelastic collisions of interfacial solitons and integrability of a two-layer fluid system subject to horizontal vibrations

We study interfacial waves in a system of two horizontal layers of immiscible inviscid fluids involved into horizontal vibrational motion. We analyze the linear and nonlinear stability properties of the solitons in the system and consider two-soliton collision scenarios. We describe the events of explosive formation of sharp peaks on the interface, which may presumably lead to the layer rapture, and find that beyond the vicinity of these peaks the system dynamics can be represented as a kinetics of a soliton gas.

Soliton turbulence in shallow water ocean surface waves.

We analyze shallow water wind waves in Currituck Sound, North Carolina and experimentally confirm, for the first time, the presence of soliton turbulence in ocean waves. Soliton turbulence is an exotic form of nonlinear wave motion where low frequency energy may also be viewed as a dense soliton gas, described theoretically by the soliton limit of the Korteweg-deVries equation, a completely integrable soliton system: Hence the phrase "soliton turbulence" is synonymous with "integrable soliton turbulence." For periodic-quasiperiodic boundary conditions the ergodic solutions of Korteweg-deVries are exactly solvable by finite gap theory (FGT), the basis of our data analysis. We find that large amplitude measured wave trains near the energetic peak of a storm have low frequency power spectra that behave as ∼ω-1. We use the linear Fourier transform to estimate this power law from the power spectrum and to filter densely packed soliton wave trains from the data. We apply FGT to determine the soliton spectrum and find that the low frequency ∼ω-1 region is soliton dominated. The solitons have random FGT phases, a soliton random phase approximation, which supports our interpretation of the data as soliton turbulence. From the probability density of the solitons we are able to demonstrate that the solitons are dense in time and highly non-Gaussian.

Exciton-polariton patterns in coherently pumped semiconductor microcavities

We theoretically study spatially periodic exciton-polariton patterns in a semiconductor microcavity coherently driven by an optical pump. Patterns grow spontaneously in a uniformly pumped cavity above a certain threshold (Turing instability). By means of a perturbation analysis in the vicinity of this bifurcation point, we determine the conditions for pattern formation and predict their properties. The shape of those patterns depends strongly on the detuning of the pump frequency from the system's resonances. For instance, near the bottom of the lower branch of the polariton dispersion relation, a hexagonal pattern is the most favorable periodic solution similar to pattern formation in optical cavities endowed with a defocusing Kerr nonlinearity. Towards the excitonic resonance, the influence of the upper polariton branch becomes important giving rise to a diversity of other solutions, such as roll, labyrinthine, and honeycomb patterns. Our numerical simulations reveal patterns with point and line defects and the formation of disordered patterns of a soliton gas.

Motion of solitons of the complex Ginzburg-Landau equation: The effect of an external frequency-shifted source

The effect of an external continuous wave (cw) on the motion of solitons of the complex Ginzburg-Landau (CGL) equation is investigated. It is shown that soliton motion, which is prevented by a strong braking due to finite gain bandwitdh, can be induced by the cw. Then the effect of the cw on the collective behavior of a large number of dissipative solitons is considered. Starting from a soliton crystal, we demonstrate that the soliton motion is induced by the cw, so that either the soliton gas, liquid, or crystal can be obtained depending on the intensity and the frequency of the cw.

Generalized hydrodynamic reductions of the kinetic equation for a soliton gas

We derive generalized multiflow hydrodynamic reductions of the nonlocal kinetic equation for a soliton gas and investigate their structure. These reductions not only provide further insight into the properties of the new kinetic equation but also could prove to be representatives of a novel class of integrable systems of hydrodynamic type beyond the conventional semi-Hamiltonian framework.

Possibility of an Akhmediev breather decaying into solitons

We consider the process by which a constant or wide-pulse solution of the nonlinear Schr\"odinger equation can become converted into an ensemble of solitons by modulational instability. This process is generally believed to be one important step in the generation of supercontinuum in optical fibers. Starting from the Akhmediev breather solution we study the conversion by two methods: One is pulse shape-oriented and uses the soliton relation between width and peak power. The other is eigenvalue-oriented and uses results from scattering theory. It becomes clear that an evolution according to the unmodified nonlinear Schr\"odinger equation, even in the presence of noise, will not lead to the transformation into solitons. If one takes the Raman effect into account, however, a conversion to a soliton gas takes place.

Coalescence cascade of dissipative solitons in parametrically driven systems

Parametrically driven spatially extended systems exhibit uniform oscillations which are modulationally unstable. The resulting periodic state evolves to the creation of a gas of dissipative solitons. Driven by the interaction of dissipative solitons, the multisoliton state undergoes a cascade of coalescence processes, where the average soliton separation distance obeys a temporal self-similar law. Starting from the soliton pair interaction law, we have derived analytically and characterized the law of this multisoliton coarsening process. A comparison of numerical results obtained with different models such as the parametrically driven damped nonlinear Schrödinger equation, a vertically driven chain of pendula, and a parametrically forced magnetic wire, shows remarkable agreement. Both phenomena, the pair interaction law and the coarsening process, are also observed experimentally in a quasi-one-dimensional layer of Newtonian fluid which is oscillated vertically.

Formation of matter-wave soliton molecules

We propose an experiment for forming matter-wave soliton molecules. In the proposed setup, we show that if two solitons are initially prepared in phase and with a sufficiently small separation and relative velocity, a bound pair will always form. This is verified by direct numerical simulation of the Gross–Pitaevskii equation and by the derivation of the exact interaction energy of two solitons, which takes the form of a Morse potential. This interaction potential depends not only on the separation but also on the relative phase of the solitons and is essential for an analytical treatment of a host of other problems, such as the soliton gas and the Toda lattice of solitons.

Time-reversal focusing of an expanding soliton gas in disordered replicas

We investigate the properties of time reversibility of a soliton gas, originating from a dispersive regularization of a shock wave, as it propagates in a strongly disordered environment. An original approach combining information measures and spin glass theory shows that time reversal focusing occurs for different replicas of the disorder in forward and backward propagation, provided the disorder varies on a length scale much shorter than the width of the soliton constituents. The analysis is performed by starting from a new class of reflectionless potentials, which describe the most general form of an expanding soliton gas of the defocusing nonlinear Schroedinger equation.

Universal soliton pattern formations in passively mode-locked fiber lasers

We investigate multiple-soliton pattern formations in a figure-of-eight passively mode-locked fiber laser. Operation in the anomalous dispersion regime with a double-clad fiber amplifier allows generation of up to several hundreds of solitons per round trip. We report the observation of remarkable soliton distributions: soliton gas, soliton liquid, soliton polycrystal, and soliton crystal, thus indicating the universality of such complexes.

Nonlinear random wave field in shallow water: variable Korteweg-de Vries framework

Abstract. The transformation of a random wave field in shallow water of variable depth is analyzed within the framework of the variable-coefficient Korteweg-de Vries equation. The characteristic wave height varies with depth according to Green's law, and this follows rigorously from the theoretical model. The skewness and kurtosis are computed, and it is shown that they increase when the depth decreases, and simultaneously the wave state deviates from the Gaussian. The probability of large-amplitude (rogue) waves increases within the transition zone. The characteristics of this process depend on the wave steepness, which is characterized in terms of the Ursell parameter. The results obtained show that the number of rogue waves may deviate significantly from the value expected for a flat bottom of a given depth. If the random wave field is represented as a soliton gas, the probabilities of soliton amplitudes increase to a high-amplitude range and the number of large-amplitude (rogue) solitons increases when the water shallows.

Mid-infrared extension of supercontinuum in chalcogenide suspended core fibre through soliton gas pumping

Reported is an experimental demonstration of the mid-infrared extension of supercontinuum in a 1.9 µm suspended core chalcogenide fibre, the zero dispersion wavelength of which is shifted to 1.9 µm, i.e. into the transparency window of standard silica fibres. Based on the well-known long-pulse regime of supercontinuum generation in silica fibre, a low-cost optical quasi-CW source at 1.53 µm has been converted and shifted into a large number of interacting femtosecond solitons localised up to 2.1 µm. This optical source, called a soliton gas, allows the pumping of a 50 cm long chalcogenide microstructured fibre near its zero dispersion wavelength, thus leading to an efficient extension of a mid-infrared supercontinuum beyond 2.4 µm.

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of N-component ‘cold-gas’ hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the ‘cold-gas’ component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.

Phonon Instability with Respect to Soliton Formation in Two-Dimensional Dipolar Bose-Einstein Condensates

The partially attractive character of the dipole-dipole interaction leads to phonon instability in dipolar Bose-Einstein condensates, which is followed by collapse in 3D geometries. We show that in 2D, the nature of the post-instability dynamics is fundamentally different, due to the stabilization of 2D solitons. As a result, a transient gas of attractive solitons is formed, and collapse may be avoided. In the presence of an harmonic trap, the post-instability dynamics is characterized by a transient pattern formation followed by the creation of stable 2D solitons. This dynamics should be observable in ongoing experiments, allowing for the creation of stable 2D solitons for the first time ever in quantum gases.

Kinetic Equation for a Dense Soliton Gas

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schrödinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.

Stationary solitary three-dimensional waves on a vertically flowing fluid film

Three-dimensional waves appear at the final stage of the evolving wave regime of a fluid-film flow [1]. At this stage, for values of the Reynolds number of the film flow that are sufficiently low (Re < 100), the wave regime manifests itself as a soliton gas. The film surface is covered by numerous horseshoe-shaped waves, which propagate over a thin residual layer, randomly interacting with each other and explicitly demonstrat-

Soliton gas in space-charge-dominated beams

Based on the Vlasov–Maxwell equations describing self-consistent nonlinear beam dynamics and collective processes, the evolution of an intense sheet beam propagating through a periodic focusing field has been studied. It has been shown that in the case of a beam with uniform phase-space density, the Vlasov–Maxwell equations can be replaced exactly by the hydrodynamic equations with a triple adiabatic pressure law coupled with the Maxwell equations. We further demonstrate that starting from the system of hydrodynamic and Maxwell equations, a set of coupled nonlinear Schrödinger equations for the slowly varying amplitudes of density waves can be derived. In the case where a parametric resonance between a certain mode of density waves and the external focusing occurs, the slow evolution of the resonant amplitudes in the cold-beam limit is shown to be governed by a system of coupled Gross–Pitaevskii equations. Properties of the nonlinear Schrödinger equation as well as properties of the Gross–Pitaevskii equation are discussed, together with soliton and condensate formation in intense particle beams.

The thermodynamic limit of the Whitham equations

The infinite-genus limit of the KdV–Whitham equations is derived. The limit involves special scaling for the associated spectral surface such that the integrated density of states remains finite as N→∞ (the thermodynamic type limit). The limiting integro-differential system describes slow evolution of the density of states and can be regarded as the kinetic equation for soliton gas.

Soliton dynamics in a random Toda chain

This paper addresses the soliton dynamics in a Toda lattice with a randomly distributed chain of masses. Applying the inverse scattering transform, we derive effective equations for the decay of the soliton amplitude that take into account radiative losses. It is shown that the soliton energy decays as approximately N(-3/2) for small-amplitude solitons and approximately exp(-N) for large-amplitude solitons. The decay rate does not depend on the incoming energy for large-amplitude soliton. An important feature is the generation of a soliton gas consisting of a large collection of small solitons (a number of the order of epsilon (-2) of solitons with momenta of the order of epsilon (2), where epsilon is the strength of fluctuations). The soliton gas plays an important role in that the changes in the conservation equations cannot be correctly understood if the soliton production is neglected. The role of the correlation length of fluctuations on the soliton decay is discussed. It is shown that in the presence of long-range correlation the Toda soliton is not backscattered, but progressively converted into forward-going radiation. The analytical predictions are confirmed by full numerical simulations.