Analytic smoothing effect for a system of Schrödinger equations with three wave interaction

Abstract

We consider the global Cauchy problem for a system of Schrödinger equations with quadratic interaction. Two types of analytic smoothing effect for the solutions are formulated in the small data setting under the mass resonance condition. One is the usual analytic smoothing effect in space variables in terms of the generator of Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to Galilei generators for sufficiently small data with exponential decay at infinity in space ℝn with n ≥ 3. The other is analytic smoothing effect in space-time variables in terms of generator of pseudo-conformal and Galilei transforms. We prove the existence and uniqueness of global solutions which are analytic with respect to pseudo-conformal and Galilei generators for sufficiently small data with exponential decay in ℝ4. We also discuss the associated Lagrange structure.

Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six

“…The mass-resonance assumption has also been appeared in various works involving two and three-component Schrödinger systems (see for instance [38], [42], [39], [19], [20], [21] and [22] and references therein). When (RC) does not hold a more careful analysis must be performed and we do not know if solutions in Σ with negative energy, for instance, blow-up or not.…”

On the dynamics of a quadratic Schrödinger system in dimension $n = 5$

“…From the mathematical point of view, the study of nonlinear Schrödinger systems with quadratic interaction has been increasing in recent years. To cite a few, we refer the reader to [3], [4], [7], [8], [9], [12], [14], [15], [20], and references therein. An almost complete study of system (1.1) in L 2 (R n ) and H 1 (R n ) was undertaken in [9] (see also [7], [8]).…”

Strong instability of standing waves for a system NLS with quadratic interaction

“…The Cauchy problem for (1.1) in L 2 , H 1 and in the weighted L 2 space x −1 L 2 = F(H 1 ) under mass resonance condition have been studied in [13]. The space-time analytic smoothing effect has been studied in [7][8][9]. The sharp threshold for scattering and blow-up for (1.1) under the mass resonance condition in dimension d = 5 has been studied in [6].…”