Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system

Abstract

In this paper, the existence and stability results for a two-parameter family of vector solitary-wave solutions (i.e both components are nonzero) of the nonlinear Schrödinger system iut + uxx + (a|u| 2 + b|v| 2)u = 0, ivt + vxx + (b|u| 2 + c|v| 2)v = 0, where u, v are complex-valued functions of (x, t) ∈ R 2 , and a, b, c ∈ R are established. The results extend our earlier ones as well as those of Ohta, Cipolatti and Zumpichiatti and de Figueiredo and Lopes. As opposed to other methods used before to establish existence and stability where the two constraints of the minimization problems are related to each other, our approach here characterizes solitary-wave solutions as minimizers of an energy functional subject to two independent constraints. The set of minimizers is shown to be stable; and depending on the interplay between the parameters a, b and c, further information about the structures of this set are given.

Existence and orbital stability of standing waves for nonlinear Schrödinger systems

“…Moreover in most of them there is either exactly one constraint [6] or the two constraints cannot be chosen independently [21,22,24]. Concerning (1.4) the more complete results seem to be due to [23]. In [23] the precompactness of minimizing sequences is obtained assuming N = 1.…”

“…Concerning (1.4) the more complete results seem to be due to [23]. In [23] the precompactness of minimizing sequences is obtained assuming N = 1. To exclude the dichotomy the authors crucially applied [1, Lemma 2.10] which depends in turn on original ideas introduced in [5], see also [12].…”

Multiple normalized solutions for a competing system of Schrödinger equations

“…With regard to this, we mention that while for L 2subcritical problems many results are available (see e.g. [13,14,21,22] for equations and [7,11,15] for systems), the L 2 -critical or supercritical ones are much less understood, and we refer to [3,12] for equations and to [4,5,6] for systems. Let us focus now on system (1.1) in the symmetric case a 1 = a 2 and µ 1 = µ 2 .…”

Multiple positive normalized solutions for nonlinear Schrödinger systems

“…It is also well known, under the assumption that the Cauchy problem is locally well posed, that if such sequences are, up to translation, compact then the set of global minimizers is orbitally stable. In that direction there had been a good amount of works, directly on (1.1)- (1.2) or on related problems, see in particular [8,14,32,33], and the more complete result was recently obtained in [19]. On the contrary, when either p i > 2 + 4 N for some i = 1, 2 or r 1 + r 2 > 2 + 4 N , the functional J becomes unbounded from below on S(a 1 , a 2 ).…”