Law of spreading of the crest of a breaking wave

In a wide range of conditions, ocean waves break. This can be seen as the manifestation of a singularity in the dynamics of the fluid surface, moving under the effect of the fluid motion underneath. We show that, at the onset of breaking, the wave crest expands in the spanwise direction as the square root of time. This is first derived from a theoretical analysis and then compared with experimental findings. The agreement is excellent.

Optical solitons as quantum objects

The intensity of classical bright solitons propagating in linearly coupled identical fibers can be distributed either in a stable symmetric state at strong coupling or in a stable asymmetric state if the coupling is small enough. In the first case, if the initial state is not the equilibrium state, the intensity may switch periodically from fiber to fiber, while in the second case the asymmetrical state remains forever, with most of its energy in either fiber. The latter situation makes a state of propagation with two exactly reciprocal realizations. In the quantum case, such a situation does not exist as an eigenstate because of the quantum tunneling between the two fibers. Such a tunneling is a purely quantum phenomenon without counterpart in the classical theory. We estimate the rate of tunneling by quantizing a simplified dynamics derived from the original Lagrangian equations with test functions. This tunneling could be within reach of the experiments, particularly if the quantum coherence of the soliton can be maintained over a sufficient amount of time.

Coexistence of ordinary elasticity and superfluidity in a model of a defect-free supersolid

The mechanical behavior of a supersolid is studied in the framework of a fully explicit model derived from the Gross-Pitaevskii equation without assuming any defect or vacancy. A set of coupled nonlinear partial differential equations plus boundary conditions is derived. The conditions of mechanical equilibrium are studied under external constraints such as steady rotation and external stress. Our model explains the experimentally observed paradoxical behavior: a nonclassical rotational inertia fraction in the limit of small rotation speed but a solidlike elastic response to small stress or an external force field.

Condensation of classical nonlinear waves

We study the formation of a large-scale coherent structure (a condensate) in classical wave equations by considering the defocusing nonlinear Schrödinger equation as a representative model. We formulate a thermodynamic description of the classical condensation process by using a wave turbulence theory with ultraviolet cutoff. In three dimensions the equilibrium state undergoes a phase transition for sufficiently low energy density, while no transition occurs in two dimensions, in complete analogy with standard Bose-Einstein condensation in quantum systems. On the basis of a modified wave turbulence theory, we show that the nonlinear interaction makes the transition to condensation subcritical. The theory is in quantitative agreement with the numerical integration of the nonlinear Schrödinger equation.

Casimir-like force arising from quantum fluctuations in a slowly moving dilute Bose-Einstein condensate

We calculate a force due to zero-temperature quantum fluctuations on a stationary object in a moving superfluid flow. We model the object by a localized potential varying only in the flow direction and model the flow by a three-dimensional weakly interacting Bose-Einstein condensate at zero temperature. We show that this force exists for any arbitrarily small flow velocity and discuss the implications for the stability of superfluid flow.

Diffusion and reaction-diffusion in fast cellular flows

Cellular structures, as the rolls generated by Rayleigh-Bénard instability, have always been an important topic in nonlinear science. The diffusion of a passive scalar in a given steady cellular flow becomes an interesting question in the limit of a large Péclet number, often realistic. The main result there is that the effective diffusion is somewhere in between the molecular diffusion and the "turbulent" diffusion. A new added twist to this is the reaction-diffusion case, where the front speed is the laminar propagation velocity (without flow) times the Péclet number to the power 1/4. I refine this last result and give the behavior of the prefactor in the Zel'dovich limit of a narrow reaction zone.

Dissipative flow and vortex shedding in the Painlevé boundary layer of a Bose-Einstein condensate

This paper addresses the drag force and formation of vortices in the boundary layer of a Bose-Einstein condensate stirred by a laser beam following the experiments of Phys. Rev. Lett. 83, 2502 (1999)]. We make our analysis in the frame moving at constant speed where the beam is fixed. We find that there is always a drag around the laser beam. We also analyze the mechanism of vortex nucleation. At low velocity, there are no vortices and the drag has its origin in a wakelike phenomenon: This is a particularity of trapped systems since the density gets small in an extended region. The shedding of vortices starts only at a threshold velocity and is responsible for a large increase in drag. This critical velocity for vortex nucleation is lower than the critical velocity computed for the corresponding 2D problem at the center of the cloud.

Example of a chaotic crystal: the labyrinth

Labyrinthine structures often appear as the final steady state of pattern forming systems. Being disordered, they exhibit the same kind of short range positional order as the Newell-Pomeau turbulent crystal. Labyrinths can be seen as a limit case of the texture of disordered rolls with a coherence length of the same order as the wavelength. In the various two-dimensional model equations we looked at, labyrinths and parallel rolls are steady states for the same parameters, their occurrence depending on the initial conditions. Comparing the stability of these two structures, we find that in variational models their energy is very close, rolls always being more stable than labyrinths. For the nonvariational model we propose a numerical experiment which displays a well defined bifurcation from parallel rolls to labyrinths as the more stable state.

Sliding drops in the diffuse interface model coupled to hydrodynamics

Using a film thickness evolution equation derived recently combining long-wave approximation and diffuse interface theory [L. M. Pismen and Y. Pomeau, Phys. Rev. E 62, 2480 (2000)] we study one-dimensional surface profiles for a thin film on an inclined plane. We discuss stationary flat film and periodic solutions including their linear stability. Flat sliding drops are identified as universal profiles, whose main properties do not depend on mean film thickness. The flat drops are analyzed in detail, especially how their velocity, advancing and receding dynamic contact angles and plateau thicknesses depend on the inclination of the plane. A study of nonuniversal drops shows the existence of a dynamical wetting transition with hysteresis between droplike solutions and a flat film with small amplitude nonlinear waves.

Film rupture in the diffuse interface model coupled to hydrodynamics

The process of dewetting of a thin liquid film is usually described using a long-wave approximation yielding a single evolution equation for the film thickness. This equation incorporates an additional pressure term-the disjoining pressure-accounting for the molecular forces. Recently a disjoining pressure was derived coupling hydrodynamics to the diffuse interface model [L. M. Pismen and Y. Pomeau, Phys. Rev. E 62, 2480 (2000)]. Using the resulting evolution equation as a generic example for the evolution of unstable thin films, we examine the thickness ranges for linear instability and metastability for flat films, the families of stationary periodic and localized solutions, and their linear stability. The results are compared to simulations of the nonlinear time evolution. From this we conclude that, within the linearly unstable thickness range, there exists a well defined subrange where finite perturbations are crucial for the time evolution and the resulting structures. In the remainder of the linearly unstable thickness range the resulting structures are controlled by the fastest flat film mode assumed up to now for the entire linearly unstable thickness range. Finally, the implications for other forms of disjoining pressure in dewetting and for spinodal decomposition are discussed.

Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics

The hydrodynamic phase field model is applied to the problem of film spreading on a solid surface. The disjoining potential, responsible for modification of the fluid properties near a three-phase contact line, is computed from the solvability conditions of the density field equation with appropriate boundary conditions imposed on the solid support. The equations describing the motion of a spreading film are derived in the lubrication approximation (in the limit of small contact angles). In the case of quasiequilibrium spreading, it is shown that the correct sharp-interface limit is obtained, and sample solutions are obtained by numerical integration. It is further shown that evaporation or condensation may strongly affect the dynamics near the contact line, and that it is necessary to account for kinetic retardation of the interphase transport to build up a consistent theory.