jean-Claude Saut

Université Paris-Sud 11 | Paris 11 · Département de Mathématiques

We present a detailed numerical study of the transverse stability of line solitons of two-dimensional, generalized Zakharov-Kusnetsov equations with various power nonlinearities. In the $L^{2}$-subcritical case, in accordance with a theorem due to Yamazaki we find a critical velocity, below which the line soliton is stable. For higher velocities, t...

We consider in this paper modified fractional Korteweg–de Vries and related equations (modified Burgers–Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low freq...

This introductory chapter contains some well-known facts and a few more recent results. It is mainly addressed to non-specialists of either nonlinear dispersive PDEs or Inverse Scattering techniques and vocabulary. We first recall some basic elementary notions on (dispersive) wave propagation. Then, taking as a guideline the Korteweg–de Vries equat...

This chapter focuses on two asymptotic models for internalwaves, the Benjamin–Ono (BO) and Intermediate LongWave (ILW) equations, which are integrable by inverse scattering techniques (IST). After briefly recalling their (rigorous) derivations, we will review old and recent results on the Cauchy problem, comparing those obtained by IST and PDE tech...

This chapter aims to survey the known results on the Kadomtsevs–Petviashvili equations and their variants from the point of viewof modeling, PDEs and IST.Numerical simulations will illustrate the results and provide useful hints for open problems and conjectures.

We have tried in this book to develop in some detail rigorous results on the Cauchy problem for relevant dispersive integrable equations (and some of their variants) by methods of inverse scattering or PDEs (in the large).

To appear in Applied Mathematical Sciences (Springer)

We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial ze...

This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and fractional cubic nonlinear Schrödinger (fNLS) equations, with the dispersions |D|α∂x and |D|α+1, respectively. We prove the global existence of small solutions for both the Cauchy problems to the modified fKdV and fNLS equations, with a modified scattering which has a...

Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical stratification of the density. We show the local wel...

We prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev–Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev–Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of i...

In this chapter we first recall the derivation of Davey–Stewartson systems in the context of water waves. Actually the Davey–Stewartson systems are singular limits of more general, “universal” systems, the Benney–Roskes/Zakharov–Rubenchik systems that describe the interactions of short and long waves. Then we survey the rigorous results obtained by...

Wave phenomena are omnipresent in nature, the most familiar everyday example probably being waves on lakes or the sea. Despite this fact, the mathematical description ofwaves is very involved and in many cases still incomplete, see for instance [13] for an overview of the mathematical theory of water waves.

This paper concerns the modified fractional Korteweg-de Vries (modified fKdV) and nonlinear Schr\"{o}dinger (modified fNLS) equations, with the dispersions |D|^{\alpha}\partial_x and |D|^{\alpha+1}, respectively. We prove the global existence of small solutions for both the Cauchy problems to the modified fKdV and fNLS equations, with a modified sc...

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering resul...

We consider in this paper modified fractional Korteweg-de Vries and related equations (modified Burgers-Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low freq...

We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial ze...

We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial ze...

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering resul...

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schr\"{o}dinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering r...

We prove wave breaking (shock formation) for some Whitham-type equations which include the Burgers-Hilbert equation, the fractional Korteweg-de Vries equation, and the classical Whitham equation. The result seems to be new for the Burgers-Hilbert equation. In the other cases we provide simpler proofs than the known ones.

We prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev-Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev-Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of i...

We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

We establish the long time existence of solutions for the “Boussinesq-Full dispersion” systems modeling the propagation of internal waves in a two-layer system. For the two-dimensional Hamiltonian case b=d>0,a≤0,c<0, we study the global existence of small solutions of the corresponding system.

Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical strati-fication of the density. We show the local we...

This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their (rigorous) derivations we will review old and recent results on the Cauchy problem, comparing those obtained by IST and...

We show that the limit infimum, as time $\,t\,$ goes to infinity, of any uniformly bounded in time $H^{3/2+}\cap L^1$ solution to the Intermediate Long Wave equation converge to zero locally in an increasing-in-time region of space of order $\,t/\log(t)$. Also, for solutions with a mild $L^1$-norm growth in time is established that its limit infimu...

This paper is a continuation of our previous study [13] on the long time behavior of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the quadratic nonlinearity in two dimensions. More precisely, for a given asymptotic profile u+, we construct a soluti...

We establish the long time existence of solutions for the "Boussinesq-Full dispersion" systems model-ing the propagation of internal waves in a two-layer system. For the two-dimensional Hamiltonian case b = d > 0, a ≤ 0, c < 0, we study the global existence of small solutions of the corresponding system.

We establish the long time existence of solutions for the "Boussinesq-Full Dispersion" systems modeling the propagation of internal waves in a two-layer system. For the two-dimensional Hamiltonian case we prove the global existence of small solutions of the corresponding systems.

This paper is concerned with the one-dimensional version of a specific member of the (abcd) family of Boussinesq systems having the higher possible dispersion. We will establish two different long time existence results for the solutions of the Cauchy problem. The proofs involve normal form transformations suitably modified away from the zero set o...

This paper is concerned with the one-dimensional version of a specific member of the (abcd) family of Boussinesq systems having the higher possible dispersion. We will establish two different long time existence results for the solutions of the Cauchy problem. The first result concerns the system (1.4) without a small parameter. If the initial data...

recall that for a scalar function a, one defines curl o: = (gx~,-g~), while when :g = (u1, u2) is a vector-valued function, curl u = 8 8 u 2-8 8 u 1). Finally n denotes the-X] X2 unit outward normal to r. System (0.1) has been recently studied by Barcilon, Constantin and Titi [3]. They proved the existence of a weak solution of (0.1) by considering...

This paper is a continuation of our previous study on the long time behavior of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the quadratic nonlinearity in two dimensions. More precisely, for a given asymptotic profile $u_{+}$, we construct a solut...

This paper is a continuation of our previous study [13] on the long time behavior of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the quadratic nonlinearity in two dimensions. More precisely, for a given asymptotic profile u+, we construct a soluti...

This volume contains lectures and invited papers from the Focus Program on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" held at the Fields Institute from July 31-August 18, 2017. The conference brought together researchers in completely integrable systems and PDE with the goal of advancing the understanding of qualit...

This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their (rigorous) derivations we will review old and recent results on the Cauchy problem, comparing those obtained by IST and...

This survey article is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briey their (rigorous) derivations we will review old and recent results on the Cauchy problem, comparing those obtained by IST and P...

This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that the asymptotic expansion leads surprisingly to a kind of Poincaré-Dulac normal form of the Navier-Stokes equations. We will also discuss some related results and a few open is...

The aim of this paper is to establish the existence of solitary wave solutions for two classes of two-layers systems modeling the propagation of internal waves. More precisely we will consider the Boussinesq-Full dispersion system and the Intermediate Long Wave (ILW) system together with its Benjamin-Ono (BO) limit.

The aim of this paper is to establish the existence of solitary wave solutions for two classes of two-layers systems modeling the propagation of internal waves. More precisely we will consider the Boussinesq-Full dispersion system and the Intermediate Long Wave (ILW) system together with its Benjamin-Ono (BO) limit.

We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system (known as the Benney-Roskes system in water waves theory), which models the interaction of short and long waves in many physical situations. Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Ca...

This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that that the asymptotic expansion leads surprisingly to a Poincar\' e-Dulac normal form of the Navier-Stokes equations. We will also discuss some related results and a few open is...

Dedicated to Claude-Michel Brauner on the occasion of his 70th birthday. Abstract. This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that the asymptotic expansion leads surprisingly to a Poincaré-Dulac normal form of the Navier-St...

We consider the asymptotic behavior in time of solutions to the nonlinear Schr"odinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatte...

The aim of this paper is to study, via theoretical analysis and numerical simulations, the dynamics of Whitham and related equations. In particular we establish rigorous bounds between solutions of the Whitham and KdV equations and provide some insights into the dynamics of the Whitham equation in different regimes, some of them being outside the r...

The aim of this paper is to prove various ill-posedness and well-posedness results on the Cauchy problem associated to a class of fractional Kadomtsev-Petviashvili (KP) equations including the KP version of the Benjamin-Ono and Intermediate Long Wave equations.

This bibliography is by no means an exhaustive one and it contains essentially the references directly linked to the Lectures. LECTURE 1. Derivation of surface Water Waves models • The original Lagrange Mémoire (in French) is [98]. A nice historical survey of Fluid Mechanics between the Bernoullis and Prandtl, with plenty of details on water waves...

In this paper, we investigate the properties of solitonic structures arising in quadratic media. First, we recall the derivation of systems governing the interaction process for waves propagating in such media and we check the local and global well-posedness of the corresponding Cauchy problem. Then, we look for stationary states in the context of...

This is a continuation of a previous work of two of the Authors. In particular we consider systems with surface tension, one of them leading to new difficulties.

Emphasize a paper where Lagrange derived the so-called water wave system and derive from it in an appropriate limit the linear wave equation.

We consider the Cauchy problem for the Full Dispersion Davey-Stewartson systems derived in [23] for the modeling of surface water waves in the modulation regime and we investigate some of their mathematical properties, emphasizing in particular the differences with the classical Davey-Stewartson systems.

The aim of this paper is to provide a proof of the (conditional) orbital stability of solitary waves solutions to the fractional Korteweg- de Vries equation (fKdV) and to the fractional Benjamin-Bona-Mahony (fBBM) equation in the $L^2$ subcritical case. We also discuss instability and its possible scenarios.

We survey and compare, mainly in the two-dimensional case, various results obtained by IST and PDE techniques for integrable equations. We also comment on what can be predicted from integrable equations on non integrable ones.

We provide a numerical study of various issues pertaining to the dynamics of the Davey-Stewartson systems of the DS II type. In particular we investigate whether or not the properties (blow-up, radiation,...) displayed by the focusing and defocusing DS II integrable systems persist in the non integrable case.

On rappelle brìèvement les étapes de la vie intellectuelle de Hans Reichenbach et on passe en revue ses principaux apports à l'interprétation fréquentiste de la probabilité en insistant sur sa solution au probìème de l'induction.

We provide a detailed numerical study of various issues pertaining to the dynamics of the Burgers equation perturbed by a weak dispersive term: blow-up in finite time versus global existence, nature of the blow-up, existence for "long" times, and the decomposition of the initial data into solitary waves plus radiation. We numerically construct soli...

We consider in this paper the Full Dispersion Kadomtsev-Petviashvili Equation (FDKP) introduced in [19] in order to overcome some shortcomings of the classical KP equation. We investigate its mathematical properties, emphasizing the differences with the Kadomtsev-Petviashvili equation and their relevance to the approximation of water waves. We also...

The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the possibility of dispersive blow up for nonlinear Schroedinger equations already appears in [9]. In the present work, t...

ou mécanique des fluides? breves-de-maths.fr/brooke-benjamin-musique-ou-mecanique-des-fluides/ Le jeune Brooke Benjamin (1929-1995) a longtemps hésité entre la musique et la science. Musicien précoce et très doué, aussi bien au piano qu'au violon, il composa un quintette avec piano à l'âge de 16 ans, dirigea orchestres et choeurs… Il choisit finale...

The aim of this paper is to show how a weakly dispersive perturbation of the inviscid Burgers equation improve (enlarge) the space of resolution of the local Cauchy problem. More generally we will review several problems arising from weak dispersive perturbations of nonlinear hyperbolic equations or systems.

Motivated by the study of boundary control problems for the Zakharov-Kuznetsov equation, we study in this article the initial and boundary value problem for the ZK (short for Zakharov-Kuznetsov) equation posed in a limited domain Ω = (0, 1)x × (−π/2, π/2)d, d = 1, 2. This article is related to Saut and Temam [“An initial boundary-value problem for...

By a nonlinear change of variables from the original one, we derive a "small steepness full dispersion" system for surface water waves which is consistent with the water wave system. This system is symmetrizable and we prove that the Cauchy problem is well-posed on large time of order 1/epsilon where epsilon is the steepness coefficient, implying (...

This paper is a continuation of a previous work by two of the Authors on long time existence for Boussinesq systems modeling the propagation of long, weakly nonlinear water waves. We provide proofs on examples not considered previously in particular we prove a long time well-posedness result for a delicate "strongly dispersive" Boussinesq system.

We consider in this paper the rigorous justification of the Zakharov-Kuznetsov equation from the Euler-Poisson system for uniformly magnetized plasmas. We first provide a proof of the local well-posedness of the Cauchy problem for the aforementioned system in dimensions two and three. Then we prove that the long-wave small-amplitude limit is descri...

We consider the Cauchy problem for dispersion managed nonlinear Schroedinger equations, where the dispersion map is assumed to be periodic and piecewise constant in time. We establish local and global well-posedness results and the possibility of finite time blow-up. In addition, we shall study the scaling limit of fast dispersion management and es...

This document reproduces essentially the slides of the three lectures I gave at the University of Buenos-Aires in March 2012. I would like to thank Constanza Sanchez de la Vega and Diego Rial for the invitation and hospitality and to the audience for its attention. The references are not mentionned in the text and are to be found in the bibliograph...

We establish the well-posedness on time scales of order 1/ϵ of the Cauchy problem for a general class of Boussinesq systems modeling long-wave, small-amplitude gravity surface waves. Here ϵ is the small-parameter that measures the dispersive and nonlinear effects. Our proof relies on the use of the Nash-Moser theorem applied to a suitable transform...

This paper surveys various aspects of the hydrodynamic formulation of the nonlinear Schrodinger equation obtained via the Madelung transform in connexion to models of quantum hydrodynamics and to compressible fluids of the Korteweg type.

We study the incompressible Navier–Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper [C. Foias, J.-C. Saut, Linearization and normal form of the Navier–Stokes equations with potential forces, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 1–47], produces a Poin...

We consider in this paper the well-posedness for the Cauchy problem associated to two-dimensional dispersive systems of Boussinesq type which model weakly nonlinear long wave surface waves. We emphasize the case of the {\it strongly dispersive} ones with focus on the "KdV-KdV" system which possesses the strongest dispersive properties and which is...