Frédéric Dias

Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation.

Numerical simulations of the onset phase of continuous wave supercontinuum generation from modulation instability show that the structure of the field as it develops can be interpreted in terms of the properties of Akhmediev Breathers. Numerical and analytical results are compared with experimental measurements of spectral broadening in photonic crystal fiber using nanosecond pulses.

Instabilities, breathers and rogue waves in optics

Curious wave phenomena that occur in optical fibres due to the interplay of instability and nonlinear effects are reviewed.

A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom

An accurate three-dimensional numerical model, applicable to strongly non-linear waves, is proposed. The model solves fully non-linear potential flow equations with a free surface using a higher-order three-dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second-order explicit Taylor series expansions with adaptive time steps. The model is applicable to non-linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16-node cubic ‘sliding’ quadrilateral elements, providing local inter-element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well-posed in all cases of mixed boundary conditions. Higher-order tangential derivatives, required for the time updating, are calculated in a local curvilinear co-ordinate system, using 25-node ‘sliding’ fourth-order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio-temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two-dimensional solution proposed earlier. Finally, three-dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright

Observation of Kuznetsov-Ma soliton dynamics in optical fibre

The nonlinear Schrödinger equation (NLSE) is a central model of nonlinear science, applying to hydrodynamics, plasma physics, molecular biology and optics. The NLSE admits only few elementary analytic solutions, but one in particular describing a localized soliton on a finite background is of intense current interest in the context of understanding the physics of extreme waves. However, although the first solution of this type was the Kuznetzov-Ma (KM) soliton derived in 1977, there have in fact been no quantitative experiments confirming its validity. We report here novel experiments in optical fibre that confirm the KM soliton theory, completing an important series of experiments that have now observed a complete family of soliton on background solutions to the NLSE. Our results also show that KM dynamics appear more universally than for the specific conditions originally considered, and can be interpreted as an analytic description of Fermi-Pasta-Ulam recurrence in NLSE propagation.

Gravity-capillary solitary waves in water of infinite depth and related free-surface flows

Two-dimensional free-surface flows due to a pressure distribution moving at a constant velocity U at the surface of a fluid of infinite depth are considered. Both gravity g and surface tension T are included in the dynamic boundary condition. The velocity U is assumed to be smaller than (4 gT /ρ) ¼ , so that there are no waves in the far field. Here ρ is the density of the fluid. The problem is solved numerically by a boundary integral equation technique. It is shown that for some values of U , four different flows are possible. Three of these flows are interpreted as perturbations of solitary waves in water of infinite depth. It is found that both elevation and depression solitary waves are possible in water of infinite depth. The numerical results for depression waves confirm and extend the solutions previously computed by Longuet-Higgins (1989).

Real-time full bandwidth measurement of spectral noise in supercontinuum generation

The ability to measure real-time fluctuations of ultrashort pulses propagating in optical fiber has provided significant insights into fundamental dynamical effects such as modulation instability and the formation of frequency-shifting rogue wave solitons. We report here a detailed study of real-time fluctuations across the full bandwidth of a fiber supercontinuum which directly reveals the significant variation in measured noise statistics across the spectrum, and which allows us to study correlations between widely separated spectral components. For two different propagation distances corresponding to the onset phase of spectral broadening and the fully-developed supercontinuum, we measure real time noise across the supercontinuum bandwidth, and we quantify the supercontinuum noise using statistical higher-order moments and a frequency-dependent intensity correlation map. We identify correlated spectral regions within the supercontinuum associated with simultaneous sideband generation, as well as signatures of pump depletion and soliton-like pump dynamics. Experimental results are in excellent agreement with simulations.

Comparison between three-dimensional linear and nonlinear tsunami generation models

The modeling of tsunami generation is an essential phase in understanding tsunamis. For tsunamis generated by underwater earthquakes, it involves the modeling of the sea bottom motion as well as the resulting motion of the water above. A comparison between various models for three-dimensional water motion, ranging from linear theory to fully nonlinear theory, is performed. It is found that for most events the linear theory is sufficient. However, in some cases, more-sophisticated theories are needed. Moreover, it is shown that the passive approach in which the seafloor deformation is simply translated to the ocean surface is not always equivalent to the active approach in which the bottom motion is taken into account, even if the deformation is supposed to be instantaneous.

Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions

Abstract Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier–Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoullis equation), but also to the kinematic boundary condition. The nonlinear Schrodinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.

Water-Waves as a Spatial Dynamical System

Abstract The mathematical study of travelling waves, in the context of two-dimensional potential flows in one or several layers of perfect fluid(s) and in the presence of free surface and interfaces, can be formulated as an ill-posed evolution problem, where the horizontal space variable plays the role of “time”. In the finite depth case, the study of near equilibria waves reduces to a low-dimensional reversible ordinary differential equation. In most cases, it appears that the problem is a perturbation of an integrable system, where all types of solutions are known. We describe the method of study and review typical results. In addition, we study the infinite depth limit, which is indeed a case of physical interest. In such a case, the above reduction technique fails because the linearized operator possesses an essential spectrum filling the whole real axis, and new adapted tools are necessary. We also discuss the latest results on the existence of travelling waves in stratified fluids and on three-dimensional travelling waves, in the same spirit of reversible dynamical systems. Finally, we review the recent results on the classical two-dimensional standing wave problem.

Linear pulse structure and signalling in a film flow on an inclined plane

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k -roots of D ( k , omega ) = 0, where k is a wavenumber, omega is a frequency and D ( k , omega ) is the dispersion relation function. The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability—as predicted by some model equations—is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem. The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V . The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.