Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension

Abstract

In this paper, we consider the Cauchy problem for the generalized Davey-Stewartson systemWe first use a variational approach to give a dichotomy of blow-up and scattering for the solution of mass supercritical equation with the initial data satisfying J(u 0 ) < J(R), where J stands for the Lagrange functional. The basic strategy is the concentration-compactness arguments from Kenig and Merle [17]. We overcome the main difficulties coming from the lack of scaling invariance and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). Furthermore, we adapt the standard method from [9] to obtain the blow up criterion.

Dynamical collapse of cylindrical symmetric dipolar Bose–Einstein condensates

“…For a positive parameter γ, we define by χ {|x|≤γ R} the rescaled function χ(x/γ R) (hence χ {|x|≤γ R} is bounded, positive and supported in the cylinder of radius γ R). The proof of the next lemmas is inspired by [25].…”

Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates

“…For a positive parameter γ, we define by χ {|x|≤γR} the rescaled function χ(x/γR) (hence χ {|x|≤γR} is bounded, positive and supported in the cylinder of radius γR). The proof of the next lemmas is inspired by [33].…”

Asymptotic dynamic for Dipolar Quantum Gases below the ground state energy threshold

“…R 2 1 f. After these considerations, we report the point-wise estimates contained in [22]. f be a smooth function, and η a compactly supported function on B(0, 1) with |η| ≤ 1.…”

“…Then it remains to properly estimate the first integral in the right-hand side of (6.8), namely B 0 . Following the strategy in [22] we introduce the functionχ R = R 2 χ x R − |x| 2 and it straightforwardly yields the equality (6.10)…”

“…B defined in (6.3). Before starting with the analysis we recall some preliminary tools introduced by Lu and Wu in [22], where the authors study the Davey-Stewartson equation. The nonlocal nonlinearity in that case is given by a convolution with a kernel having symbol…”