On the existence of standing waves for a davey–stewartson system

Abstract

We consider the standing waves for the Davey-Stewartson systemin R* and R~. By reducing this system to a single nonlinear equation of Schradinger type, we study the existence, the regularity and asyrnptotics of ground states.

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

“…For each fixed β ∈ (0, ) the function s → c(s), s ≥ 0 has a unique global minimum at s = ω > 0 (the formula β = f (ω)/(2ωf (ω) − ω 2 f (ω)) defines a bijection between the values of β ∈ (0, 1 3 ) and ω ∈ (0, ∞)); we denote the minimum value of c 2 by Λ (so that Λ = 2ω/(2ωf (ω) − ω 2 f (ω))). Note for later use that 6) with equality precisely when s = ±ω. Bifurcations of nonlinear solitary waves are expected whenever the linear group and phase speeds are equal, so that c (s) = 0 (see Dias & Kharif [8, §3]).…”

“…This functional has a nontrivial critical point (Cipolatti [6], Papanicolaou et al Let us now return to the water-wave problem and in particular the task of finding a nontrivial critical point of the functional…”

“…Fully localised solitary wave solutions to the Davey-Stewartson equation have a variational characterisation, and the direct methods of the calculus of variations have been used to show that it indeed has such a solution (see Cipolatti [6] and Papanicolaou et al [16, §5]). In this paper we seek critical points of the functional E − cI.…”

Stability and instability of standing waves for nonlinear Schrödinger equations

“…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]).…”

“…There were some papers concerned with the stability and instability of standing waves for (4.29) (see [13,41,42,43,44]). …”