Stable solitary waves with prescribed $L^2$-mass for the cubic Schr\"odinger system with trapping potentials

By Benedetta Noris, Hugo Tavares, Gianmaria Verzini. 2014

94 citations recorded by the source corpus.

For the cubic Schr\"odinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through of a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/111286b2f02f7b782c21744fcf5c587b0fdd0828

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Citations in the corpus

Normalized solutions for a coupled Schrödinger system (2019, background): “In [27,28] the masses a2, b2 have to be small.”
Uniqueness (2021, unknown): “…of view, there has been intense activity in the last ten years, both regarding existence results (check for example the introduction of [77] or [10,66] and their references) as well as the regularity and asymptotic study of solutions as the competition increases, namely, β ij → + ∞ for i (cid:8)…”
Normalized solutions of mass supercritical Schrödinger equations with potential (2020, background): “We refer to [9–15] for autonomous systems, and to [16,17] for systems with trapping potentials or in bounded domains.”
Normalized solutions to mass supercritical Schrödinger equations with negative potential (2021, unknown): “potentials as in (Pρ) satisfying lim |x|→+∞ −V (x) = +∞ (or even −V ≡ +∞ outside some bounded Ω), have been considered in [28, 29, 33, 30] (see also [11] for the case of partial confinement).”
Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case (2021, background): “, lim |x|→+∞ V (x) = +∞, we remark that this coercive condition can provide enough compactness to cause the existence of solutions which are local minimizers of J on Sa also in the mass-supercritical case, at least when a is sufficiently small, see [3, 18].”
Multiple positive normalized solutions for nonlinear Schrödinger systems (2017, background): “Let us also mention the recent papers [28, 29, 34] dealing with situations where the equation is set on a bounded domain or when a trapping potential is acting.”