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We consider two fluids in a 2-dimensional region: The lower fluid occupies an infinitely depth region, while the upper fluid occupies a region with a fixed upper boundary. We study the dynamics of the interface between the two fluids (interface problem) in the limit in which the interface has a space periodic profile, is close to horizontal, and has a "long wave profile". We use a Hamiltonian normal form approach to show that up to corrections of second order, the equations are approximated by two decoupled Benjamin-Ono equations. [ABSTRACT FROM AUTHOR]

In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincaré normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighborhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincaré normal form and of the transformation to it, through a rapid convergence algorithm. [ABSTRACT FROM AUTHOR]

We prove reducibility of a transport equation on the d-dimensional torus T d with a time quasiperiodic unbounded perturbation. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. Furthermore, the unperturbed problem has eigenvalues whose differences are dense on the real axis. [ABSTRACT FROM AUTHOR]

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state. Such result is valid under a CFL condition of the form $$\tau h^{-2}\le C$$ where $$\tau $$ and $$h$$ denote the time and space step size respectively. Furthermore we prove that if the initial datum is symmetric and close to the continuous ground state $$\eta $$ then the associated numerical solution remains close to the orbit of $$\eta ,\Gamma =\cup _\alpha \{e^{i\alpha }\eta \}$$, for very long times. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Scho¨dinger equation −iut=−Δu+V*u+g(u,u¯) with periodic boundary conditions on [−π,π]d, d⩾1; g is analytic and g(0,0)=Dg(0,0)=0; V is a potential in L2. Under a nonresonance condition which is fulfilled for most Vs we prove that, for any integer M there exists a canonical transformation that puts the Hamiltonian in Birkhoff normal form up to a reminder of order M. The canonical tranformation is well defined in a neighbourhood of the origin of any Sobolev space of sufficiently high order. From the dynamical point of view this means in particular that if the initial data is smaller than ϵ, the solution remains smaller than 2ϵ for all times t smaller than ϵ−(M−1). Moreover, for the same times, the solution is close to an infinite dimensional torus. To cite this article: D. Bambusi, B. Gre´bert, C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]