Related papers: Long time dynamics for generalized Korteweg-de Vri…
We consider the non linear wave equation (NLW) on the d-dimensional torus with a smooth nonlinearity of order at least two at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up…
The Korteweg-de Vries equation (KdV) and various generalized, most often semi- linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general…
The distance between the solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations is estimated in appropriate Sobolev spaces. This family of equations includes, as…
We consider a family of Schr\"odinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension $d\geq1$. We study the behaviour of high Sobolev norms $H^{s}$, $s\gg1$, of solutions with initial conditions…
We establish the approximate controllability in $L^2$ for the nonlinear Benjamin-Ono equation on torus via two-dimensional control input. Our proof is based on adaptations of geometric control approach introduced by Agrachev and Sarychev.…
We study the variable bottom generalized Korteweg-de Vries (bKdV) equation dt u=-dx(dx^2 u+f(u)-b(t,x)u), where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water.…
We study extreme wave formation for the Korteweg-de Vries equation on the torus with random initial data of average size $\epsilon$. We establish a large deviations principle for the supremum of the solution over arbitrarily long polynomial…
We study the long-time stability of soliton solutions to the Korteweg-deVries equation. We consider solutions $u$ to the KdV with initial data in $H^s$, $0 \leq s < 1$, that are initially close in $H^s$ norm to a soliton. We prove that the…
This article concerns the nonlinear Korteweg-de Vries equation with boundary time-delay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional…
In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. In order to bypass the analytical ill-posedness of the equation in the Sobolev support of the…
We study the Korteweg-de Vries-type equation dt u=-dx(dx^2 u+f(u)-B(t,x)u), where B is a small and bounded, slowly varying function and f is a nonlinearity. Many variable coefficient KdV-type equations can be rescaled into this equation. We…
We consider linear, time-dependent and skew-adjoint perturbations of periodic transport equations on the one-dimensional torus. We describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving…
We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the…
We study the Benjamin-Ono equation, posed on the torus. We prove that an infinite sequence of weighted gaussian measures, constructed in our previous work, are invariant by the flow of the equation. These measures are supported by Sobolev…
We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations on torus. The normal form is complete up to arbitrary finite order. The proof is based on a valid non-resonant condition and a suitable norm of…
It is shown that plane wave solutions to the cubic nonlinear Schr\"odinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend…
We consider the quantum hydrodynamic system on a $d$-dimensional irrational torus with $d=2,3$. We discuss the behaviour, over a "non trivial" time interval, of the $H^s$-Sobolev norms of solutions. More precisely we prove that, for generic…
In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier-Stokes-Korteweg system with Coriolis force, when the Mach, Rossby and…
In this paper, we prove that the Sobolev norm of solutions of the linear wave equation with unbounded perturbations of order one stay bounded for the all time. The main proof is based on the KAM reducibility of the linear wave equation. To…
The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for $H^s$ initial data, $s>-1/2$, and for any $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$…