On the initial value problem for the Davey-Stewartson systems

Abstract

In the theory of water waves, the 2D generalisation of the usual cubic ID Schrodinger equation turns out to be a family of systems: the Davey-Stewartson systems. For special values of the parameters characterising these systems, one obtains systems of the inverse scattering type. Our work addresses the very general case and our methods belong to the more standard theory of nonlinear partial differential equations. Well-posedness of the Cauchy problem and also finite-time blow-up are studied.Rhumb. Dam le contexte des ondes aquatiques de surface, la ghbralisation bidimensionnelle de l'bquation de Schrodinger cubique monodimensionnelle s'avere etre une famille de systemes: les systdmes de Davey-Stewartson. Pour des valeurs distingubs des parametres, ceux-ci sont justiciables des techniques de la diffusion inverse. Notre travail conceme ces systemes dans leur complete gbnbalitb et nos mbthodes relivent de la thborie usuelle des bquations aux d6riv6es partielles non linbaires. Le caractire bien pose du probleme de Cauchy mais aussi des questions libs a I'explosion en temps fini sont btudibs.

Stability and instability of standing waves for nonlinear Schrödinger equations

“…The unique local existence of H 1 solution to the Cauchy problem of (4.29) has already been established (see [22]). …”

“…where 1 < p < 2 * − 1, n = 2, or 3 and E 1 is the singular integral operator with symbol σ 1 (ξ) = ξ For ω > 0, Cipolatti [12] showed the existence of ground states for (4.30) by studying the following variational problem: Equation (4.29) describes the evolution of weakly nonlinear waves that travel predominantly in one direction (see [14,22]). The unique local existence of H 1 solution to the Cauchy problem of (4.29) has already been established (see [22]).…”

Estimates for the Scattering Map Associated with a Two-Dimensional First-Order System

“…The statement (12) is immediate from (8) and (13) follows from the definitions of p j and s j . The third, (14), follows since (13) implies s j = 4p j /(3p j − 4) and 1/r…”

“…Our contribution is the observation that the maps Q and S are continuous in L 2 which immediately implies the following result. See Ghidaghlia and Saut [8] and Linares and Ponce [10] for additional results on the Davey-Stewartson systems.…”

Multidimensional localized solitons

“…In particular the DSI equation with boundaries compatible with solitons can be embedded into the KPI equation [37]. These results open the way to the search of physical applications other than hydrodynamics, studied in [38,39, and references quoted therein]. In this respect a more discouraging analysis is made in [40].…”