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Control of a Korteweg-de Vries equation: A tutorial

The Cauchy problem for the Gross-Pitaevskii equation , Ann. Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation , J. Pures Appl. Stability theory of solitary waves in the presence of symmetry I , J. Stability and instability of traveling solitonic bubbles , Adv.

ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

Differential Equations , Tome 7 no. Stability of two soliton collision for nonintegrable gKdV equations , Commun. Inelastic interaction of nearly equal solitons for the quartic gKdV equation , Invent. Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations , Commun. PhD thesis In preparation. Thus, this type of regularity propagates with infinite speed to its left as time evolves. The numerical approach allows high precision numerical studies of solutions on the whole real line.

In addition it is shown that the Peregrine breather being discussed as a model for rogue waves can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied. It is especially useful to obtain results for space-dependent problems, and beyond power-law nonlinearities. This will be illustrated in the case of the asymptotically linear NLS.

Oct 14, Solitary waves for Boussinesq type systems The aim of this talk is to exhibit specific properties of Boussinesq type models. After recalling the usual asymptotic method leading to BT models, we will present a new asymptotic model and present a local Cauchy theory. We then provide an effective method to compute solitary waves for Boussinesq type models.

Wolfgang Pauli Institute (WPI) Vienna

We will conclude by discussing shoaling properties of such models. This is a joint work with S. Oct 14, Global existence and scattering for KP II in three space dimensions The Kadomtsev-Petviasvhili II equation describes wave propagating in one direction with weak transverse effect. I will explain the proof of global existence and scattering for three space dimensions.

Korteweg-De Vries and Nonlinear Schrodinger Equations : Qualitative Theory

This is joint work with Junfeng Li. We show the existence of solutions behaving at large time as a couple of scalar solitary waves. Finally, we present some numerical simulations to understand more the qualitative behavior of the solitary waves. Oct 14, Internal waves in continuously stratified media Many things are known about the propagation of waves at the interface of two fluids of different densities, for which dispersion plays an important role it plays a stabilizing role controlling Kelvin-Helmholtz instabilities and balances the long time effects of the nonlinearities.

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When a flow is continuously stratified, the notion of wave is less clear, as well as the nature of dispersive effects. We show that they are encoded in a Sturm Liouville problem and are therefore of 'nonlocal type'; we also derive simpler, local, asymptotic models. This is a joint work with JC Saut and B.

Korteweg-de Vries and Nonlinear Schroedinger Equations: Qualitative Theory

Oct 14, Kelvin-Helmholtz instabilities in shallow water Kelvin-Helmholtz instabilities arise when a sufficiently strong shear velocity lies at the interface between two layers of immiscible fluids. The typical wavelength of the unstable modes are very small, which goes against the natural shallow-water assumption in oceanography. As a matter of fact, the usual shallow-water asymptotic models fail to correctly reproduce the formation of KH instabilities. With this in mind, our aim is to motivate and study a new class of shallow-water models with improved dispersion behavior.

This is a joint work with Samer Israwi and Raafat Talhouk. Oct 14, Long time control of large topography effects for the water waves equations We explain how we can get a large time of existence for the Water-Waves equation with large topography variations. We explain the method on the simplier example of the Shallow-Water equation and then present its implementation for the WW equations itselves.

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  • Peter E. Zhidkov (Author of Korteweg-de Vries and Nonlinear Schrodinger Equations)?
  • Oct 14, Solitary water waves in three dimensions I will discuss some existence results for solitary waves with surface tension on a three-dimensional layer of water of finite depth. The waves are fully localized in the sense that they converge to the undisturbed state of the water in every horizontal direction.

    The existence proofs are of variational nature and different methods are used depending on whether the surface tension is weak or strong. In the case of strong surface tension, the existence proof also gives some information about the stability of the waves. The solutions are to leading order described by the Kadomtsev-Petviashvili I equation for strong surface tension or the Davey-Stewartson equation for weak surface tension. These model equations play an important role in the theory. This is joint work with B. Buffoni, M. Groves and S. Oct 14, On the inviscid limit for a 2D incompressible fluid "In this talk, we will present some results of inviscid limit of the 2D Navier-stokes system with data in spaces with BMO flavor.

    The issue of uniform in viscosity estimates for these equations will be also considered. It is a joint work with F. Bernicot and T. This class includes the Whitham equation, the linear terms of which match the dispersion relation for gravity water waves on finite depth. In this talk we present recent results for this equation and its generalisations, including periodic bifurcation results, existence of solitary waves via minimisation, and well-posedness local. Oct 14, Travelling waves for a non-local Korteweg—de Vries—Burgers equation We study travelling wave solutions of a Korteweg—de Vries—Burgers equation with a non-local diffusion term.

    This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation which is an extension of classical boundary layer theory. The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem.

    We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors.

    In contrast, travelling waves of the non-local KdV—Burgers equation are not in general monotone, as is the case for the corresponding classical KdV—Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV—Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. Nonlinear dispersive wave equations provide excellent examples of infinite dimensional dynamical systems which possess diverse and fascinating phenomena including solitary waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive turbulence and the propagation of spatial-temporal chaos.

    Nonlinear dispersive waves occur throughout physical and natural systems whenever dissipation is weak. Important applications include nonlinear optics and long distance communication devices such as transoceanic optical fibers, waves in the atmosphere and the ocean, and turbulence in plasmas. Examples of nonlinear dispersive partial differential equations include the Korteweg de Vries equation, nonlinear Klein Gordon equations, nonlinear Schroedinger equations, and many others.