Anomalous conduction in one-dimensional particle lattices: Wave-turbulence approach.

One-dimensional particle chains are fundamental models to explain anomalous thermal conduction in low-dimensional solids such as nanotubes and nanowires. In these systems the thermal energy is carried by phonons, i.e., propagating lattice oscillations that interact via nonlinear resonance. The average energy transfer between the phonons can be described by the wave kinetic equation, derived directly from the microscopic dynamics. Here we use the spatially nonhomogeneous wave kinetic equation of the prototypical ß-Fermi-Pasta-Ulam-Tsingou model, to study thermal conduction in one-dimensional particle chains on a mesoscale description. By means of numerical simulations, we study two complementary aspects of thermal conduction: in the presence of thermostats setting different temperatures at the two ends and propagation of a temperature perturbation over an equilibrium background. Our main findings are as follows. (i) The anomalous scaling of the conductivity with the system size, in close agreement with the known results from the microscopic dynamics, is due to a nontrivial interplay between high and low wave numbers. (ii) The high-wave-number phonons relax to local thermodynamic equilibrium transporting energy diffusively, in the manner of Fourier. (iii) The low-wave-number phonons are nearly noninteracting and transfer energy ballistically. These results present perspectives for the applicability of the full nonhomogeneous wave kinetic equation to study thermal propagation.

Three-dimensional imaging of waves and floes in the marginal ice zone during a cyclone.

The marginal ice zone is the dynamic interface between the open ocean and consolidated inner pack ice. Surface gravity waves regulate marginal ice zone extent and properties, and, hence, atmosphere-ocean fluxes and ice advance/retreat. Over the past decade, seminal experimental campaigns have generated much needed measurements of wave evolution in the marginal ice zone, which, notwithstanding the prominent knowledge gaps that remain, are underpinning major advances in understanding the region's role in the climate system. Here, we report three-dimensional imaging of waves from a moving vessel and simultaneous imaging of floe sizes, with the potential to enhance the marginal ice zone database substantially. The images give the direction-frequency wave spectrum, which we combine with concurrent measurements of wind speeds and reanalysis products to reveal the complex multi-component wind-plus-swell nature of a cyclone-driven wave field, and quantify evolution of large-amplitude waves in sea ice.

Observation of a giant nonlinear wave-packet on the surface of the ocean.

In many physical systems such as ocean waves, nonlinear optics, plasma physics etc., extreme events and rare fluctuations of a wave field have been widely observed and discussed. In the field of oceanography and naval architecture, their understanding is fundamental for a correct design of platforms and ships, and for performing safe operations at sea. Here, we report a measurement of an impressive and unique wave packet recorded in the Bay of Biscay in the North-East of the Atlantic Ocean. An analysis of the spatial extension of the packet that includes three large waves reveals that it extents for more than 1 km, with individual crests moving faster than 100 km/h. The central and largest wave in the packet was 27.8 m high in a sea with significant wave height of 11 m. A detailed analysis of the data using the nonlinear Fourier analysis reveals that the wave packet is characterized by a non trivial nonlinear content. This observation opens a new paradigm which requires new understanding of the dynamics of ocean waves and, more in general, of nonlinear and dispersive waves.

"Extraordinary" modulation instability in optics and hydrodynamics.

The classical theory of modulation instability (MI) attributed to Bespalov-Talanov in optics and Benjamin-Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

Fourier amplitude distribution and intermittency in mechanically generated surface gravity waves.

We examine and discuss the spatial evolution of the statistical properties of mechanically generated surface gravity wave fields, initialized with unidirectional spectral energy distributions, uniformly distributed phases, and Rayleigh distributed amplitudes. We demonstrate that nonlinear interactions produce an energy cascade towards high frequency modes with a directional spread and trigger localized intermittent bursts. By analyzing the probability density function of Fourier mode amplitudes in the high frequency range of the wave energy spectrum, we show that a heavy-tailed distribution emerges with distance from the wave generator as a result of these intermittent bursts, departing from the originally imposed Rayleigh distribution, even under relatively weak nonlinear conditions.

Coexistence of Ballistic and Fourier Regimes in the ß Fermi-Pasta-Ulam-Tsingou Lattice.

Commonly, thermal transport properties of one-dimensional systems are found to be anomalous. Here, we perform a numerical and theoretical study of the ß-Fermi-Pasta-Ulam-Tsingou chain, considered a prototypical model for one-dimensional anharmonic crystals, in contact with thermostats at different temperatures. We give evidence that, in steady state conditions, the local wave energy spectrum can be naturally split into modes that are essentially ballistic (noninteracting or scarcely interacting) and kinetic modes (interacting enough to relax to local thermodynamic equilibrium). We show numerically that the well-known divergence of the energy conductivity is related to how the transition region between these two sets of modes shifts in k space with the system size L, due to properties of the collision integral of the system. Moreover, we show that the kinetic modes are responsible for a macroscopic behavior compatible with Fourier's law. Our work sheds light on the long-standing problem of the applicability of standard thermodynamics in one-dimensional nonlinear chains, testbed for understanding the thermal properties of nanotubes and nanowires.

Starting Flow Past an Airfoil and its Acquired Lift in a Superfluid.

We investigate superfluid flow around an airfoil accelerated to a finite velocity from rest. Using simulations of the Gross-Pitaevskii equation we find striking similarities to viscous flows: from production of starting vortices to convergence of airfoil circulation onto a quantized version of the Kutta-Joukowski circulation. We predict the number of quantized vortices nucleated by a given foil via a phenomenological argument. We further find stall-like behavior governed by airfoil speed, not angle of attack, as in classical flows. Finally we analyze the lift and drag acting on the airfoil.

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.

Experimental Evidence of a Hydrodynamic Soliton Gas.

We report on an experimental realization of a bidirectional soliton gas in a 34-m-long wave flume in a shallow water regime. We take advantage of the fission of a sinusoidal wave to continuously inject solitons that propagate along the tank, back and forth. Despite the unavoidable damping, solitons retain their profile adiabatically, while decaying. The outcome is the formation of a stationary state characterized by a dense soliton gas whose statistical properties are well described by a pure integrable dynamics. The basic ingredient in the gas, i.e., the two-soliton interaction, is studied in detail and compared favorably with the analytical solutions of the Kaup-Boussinesq integrable equation. High resolution space-time measurements of the surface elevation in the wave flume provide a unique tool for studying experimentally the whole spectrum of excitations.

Directional soliton and breather beams.

Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the unidirectional nonlinear Schrödinger equation (NLSE). We report the observation of slanted solitons and breathers propagating at an angle with respect to the direction of propagation of the wave field. As the coherence is diagonal, the scale in the crest direction becomes finite; consequently, beam dynamics form. Spatiotemporal measurements of the water surface elevation are obtained by stereo-reconstructing the positions of the floating markers placed on a regular lattice and recorded with two synchronized high-speed cameras. Experimental results, based on the predictions obtained from the (2D + 1) hyperbolic NLSE equation, are in excellent agreement with the theory. Our study proves the existence of such unique and coherent wave packets and has serious implications for practical applications in optical sciences and physical oceanography. Moreover, unstable wave fields in this geometry may explain the formation of directional large-amplitude rogue waves with a finite crest length within a wide range of nonlinear dispersive media, such as Bose-Einstein condensates, solids, plasma, hydrodynamics, and optics.

Predicting ocean rogue waves from point measurements: An experimental study for unidirectional waves.

Rogue waves are strong localizations of the wave field that can develop in different branches of physics and engineering, such as water or electromagnetic waves. Here, we experimentally quantify the prediction potentials of a comprehensive rogue-wave reduced-order precursor tool that has been recently developed to predict extreme events due to spatially localized modulation instability. The laboratory tests have been conducted in two different water wave facilities and they involve unidirectional water waves; in both cases we show that the deterministic and spontaneous emergence of extreme events is well predicted through the reported scheme. Due to the interdisciplinary character of the approach, similar studies may be motivated in other nonlinear dispersive media, such as nonlinear optics, plasma, and solids, governed by similar equations, allowing the early stage of extreme wave detection.

Double Scaling in the Relaxation Time in the ß-Fermi-Pasta-Ulam-Tsingou Model.

We consider the original ß-Fermi-Pasta-Ulam-Tsingou system; numerical simulations and theoretical arguments suggest that, for a finite number of masses, a statistical equilibrium state is reached independently of the initial energy of the system. Using ensemble averages over initial conditions characterized by different Fourier random phases, we numerically estimate the time scale of equipartition and we find that for very small nonlinearity it matches the prediction based on exact wave-wave resonant interaction theory. We derive a simple formula for the nonlinear frequency broadening and show that when the phenomenon of overlap of frequencies takes place, a different scaling for the thermalization time scale is observed. Our result supports the idea that the Chirikov overlap criterion identifies a transition region between two different relaxation time scalings.

Spatiotemporal optical dark X solitary waves.

We introduce spatiotemporal optical dark X solitary waves of the (2+1)D hyperbolic nonlinear Schrödinger equation (NLSE), which rules wave propagation in a self-focusing and normally dispersive medium. These analytical solutions are derived by exploiting the connection between the NLSE and a well-known equation of hydrodynamics, namely the type II Kadomtsev-Petviashvili (KP-II) equation. As a result, families of shallow water X soliton solutions of the KP-II equation are mapped into optical dark X solitary wave solutions of the NLSE. Numerical simulations show that optical dark X solitary waves may propagate for long distances (tens of nonlinear lengths) before they eventually break up, owing to the modulation instability of the continuous wave background. This finding opens a novel path for the excitation and control of X solitary waves in nonlinear optics.

Route to thermalization in the α-Fermi-Pasta-Ulam system.

We study the original α-Fermi-Pasta-Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ(8), where ϵ is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

Ring-type multisoliton dynamics in shallow water.

The propagation, in a shallow water, of nonlinear ring waves in the form of multisolitons is investigated theoretically. This is done by solving both analytically and numerically the cylindrical (also referred to as concentric) Korteweg-de Vries equation (cKdVE). The latter describes the propagation of weakly nonlinear and weakly dispersive ring waves in an incompressible, inviscid, and irrotational fluid. The spatiotemporal evolution is determined for a cylindrically symmetric response to the free fall of an initially given multisoliton ring. Analytically, localized solutions in the form of tilted solitons are found. They can be thought as single- or multiring solitons formed on a conic-modulated water surface, with an oblique asymptote in arbitrary radial direction (tilted boundary condition). Conversely, the ring solitons obtained from numerical solutions are localized single- or multiring structures (standard solitons), whose wings vanish along all radial directions (standard boundary conditions). It is found that the wave dynamics of these standard ring-type localized structures differs substantially from that of the tilted structures. A detailed analysis is performed to determine the main features of both multiring localized structures, particularly their break-up, multiplet formation, overlapping of pulses, overcoming of one pulse by another, "amplitude-width" complementarity, etc., that are typically ascribed to a solitonlike behavior. For all the localized structures investigated, the solitonlike character of the rings is found to be preserved during (almost) entire temporal evolution. Due to their cylindrical character, each ring belonging to one of these multiring localized structures experiences the physiological decay of the peak and the physiological increase of the width, respectively, while propagating ("amplitude-width" complementarity). As in the planar geometry, i.e., planar Korteweg-de Vries equation (pKdVE), we show that, in the case of the tilted analytical solutions, the instantaneous product P=(maximumamplitude)×(width)(2) is rigorously constant during all the soliton spatiotemporal evolution. Nevertheless, in the case of the numerical solutions, we show that this product is not preserved; i.e., the instantaneous physiological variations of both peak and width of each ring do not compensate each other as in the tilted analytical case. In fact, the amplitude decay occurs faster than the width increase, so that P decreases in time. This is more evident in the early times than in the asymptotic ones (where actually cKdVE reduces to pKdVE). This is in contrast to previous investigations on the early-time localized solutions of the cKdVE.

Intermittency in integrable turbulence.

We examine the statistical properties of nonlinear random waves that are ruled by the one-dimensional defocusing and integrable nonlinear Schrödinger equation. Using fast detection techniques in an optical fiber experiment, we observe that the probability density function of light fluctuations is characterized by tails that are lower than those predicted by a Gaussian distribution. Moreover, by applying a bandpass frequency optical filter, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian. These phenomena are very well described by numerical simulations of the one-dimensional nonlinear Schrödinger equation.

Vector rogue waves and baseband modulation instability in the defocusing regime.

We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue-wave formation. The existence of vector rogue waves in the defocusing regime is expected to be a crucial progress in explaining extreme waves in a variety of physical scenarios described by multicomponent systems, from oceanography to optics and plasma physics.

Rogue waves: from nonlinear Schrödinger breather solutions to sea-keeping test.

Under suitable assumptions, the nonlinear dynamics of surface gravity waves can be modeled by the one-dimensional nonlinear Schrödinger equation. Besides traveling wave solutions like solitons, this model admits also breather solutions that are now considered as prototypes of rogue waves in ocean. We propose a novel technique to study the interaction between waves and ships/structures during extreme ocean conditions using such breather solutions. In particular, we discuss a state of the art sea-keeping test in a 90-meter long wave tank by creating a Peregrine breather solution hitting a scaled chemical tanker and we discuss its potential devastating effects on the ship.

Vortex knots in a Bose-Einstein condensate.

We present a method for numerically building a vortex knot state in the superfluid wave function of a Bose-Einstein condensate. We integrate in time the governing Gross-Pitaevskii equation to determine evolution and shape preservation of the two (topologically) simplest vortex knots which can be wrapped over a torus. We find that the velocity of a vortex knot depends on the ratio of poloidal and toroidal radius: for smaller ratio, the knot travels faster. Finally, we show how vortex knots break up into vortex rings.

Triggering rogue waves in opposing currents.

We show that rogue waves can be triggered naturally when a stable wave train enters a region of an opposing current flow. We demonstrate that the maximum amplitude of the rogue wave depends on the ratio between the current velocity U(0) and the wave group velocity c(g). We also reveal that an opposing current can force the development of rogue waves in random wave fields, resulting in a substantial change of the statistical properties of the surface elevation. The present results can be directly adopted in any field of physics in which the focusing nonlinear Schrödinger equation with nonconstant coefficient is applicable. In particular, nonlinear optics laboratory experiments are natural candidates for verifying experimentally our results.