Existence of Pseudo-Conformally Invariant Solutions to the Davey–Stewartson System

Abstract

We prove that the pseudo-conformal group of the Schrödinger equation acts on solutions of the Davey-Stewartson system and that there exists an infinite family of solutions which are invariant under the action of that group. We also exhibit two different time behaviors for these invariant solutions of the Davey-Stewartson system, and we study the stability of some of these solutions, and prove that initial data close to them give rise to global solutions asymptotically behaving like an invariant solution.

Bose–Einstein condensates in optical lattices: Mathematical analysis and analytical approximate formulae

“…Proof: We proceed as in [14]. Since ϕ ∈ H 1 (R), we have that ϕ(ξ) is a continuous function satisfying…”

“…To simplify the notation, we still write ψ n for this subsequence. Since the embedding X ⊂ L p (R) (see [14,15]) is compact for all 2 ≤ p < +∞, we have…”

Variable coefficient Davey-Stewartson system with a Kac-Moody-Virasoro symmetry algebra

“…Note that this formula holds both integrable and nonintegrable cases. This special invariance (the so-called pseudo-conformal) was used for constructing exact blow-up solutions for the DS [9,11] and the nonlinear Schrödinger equations [12,13]. Putting T = t, α 0 = β 0 = 0 we obtain the gauge symmetry group (arbitrary time dependent phase change in ψ and a shift in w)…”

On the blow-up phenomenon for a generalized Davey-Stewartson system

“…In other words, established a necessary condition for the blow-up that the initial mass is larger than a critical values ( u 0 L 2 (R 2 ) N c ). Moreover, the sharpness of this criterion follows from the existence of the pseudo-conformal symmetry (see Cipolatti & Kavian, 2001;Ozawa, 1992). If u(t, x) solves (1.8), then so does [C u](t, x), which is defined by…”

“…which appears in non-linear optics. There is a large volume of the literature on the blow-up solutions and the standing wave of NLS-like equations including the DS system, and for details see, for example, Cazenave (2003), Cipolatti (1992), Cipolatti & Kavian (2001), Gan & Zhang (2008), Ghidaglia & Saut (1990), Guo & Wang (1999), Glassey (1977), Hmidi & Keraani (2005), Li et al (2011), Merle (1993), Merle & Raphaël (2005), Merle & Tsutsumi (1990), Ogawa & Tsutsumi (1990), Ohta (1994), Papanicolaou et al (1994), Sulem & Sulem (1999) and Weinstein (1983Weinstein ( , 1986).…”