Soliton dynamics for CNLS systems with potentials

Asymptotic Analysis (Impact Factor: 0.53). 10/2008; 66(2). DOI: 10.3233/ASY-2009-0959
Source: arXiv


The semiclassical limit of a weakly coupled nonlinear focusing Schrödinger system in presence of a nonconstant potential is studied. The initial data is of the x−˜x i form (u1, u2) with ui = ri ε e ε x· ˜ ξ, where (r1, r2) is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For ε sufficiently small, the solution (φ1, φ2) will been shown to have, locally in time, the x−x(t)) i form (r1 ε e ε x·ξ(t) ( x−x(t)) i, r2 ε e ε x·ξ(t)), where (x(t), ξ(t)) is the solution of the Hamiltonian system ˙x(t) = ξ(t), ξ(t) ˙ = −∇V (x(t)) with x(0) = ˜x and ξ(0) = ξ.

Download full-text


Available from: Marco Squassina, Jan 04, 2013
  • Source
    • ",m. Following the blueprint of [25, Lemma 6.1], we get the assertion (see also [21]). 2 "
    [Show abstract] [Hide abstract]
    ABSTRACT: The soliton dynamics for a general class of nonlinear focusing Schrödinger problems in presence of non-constant external (local and nonlocal) potentials is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation.
    Journal of Mathematical Analysis and Applications 05/2010; 365:776-796. DOI:10.1016/j.jmaa.2009.11.045 · 1.12 Impact Factor
  • Source
    • "More precisely, when one considers (1.1) when the Plank's constant explicitly appears in the equations, and studies the evolution, in the semi-classical limit ( → 0), of the solution of (1.1) starting from a -scaling of a soliton, once the action of external forces appears. We refer the reader to [3] [9] [10] for the scalar case and to [15] for systems, where the authors have recently showed, in semi-classical regime, how the soliton dynamics can be derived from Theorem 1.2. "
    [Show abstract] [Hide abstract]
    ABSTRACT: We study the spectral structure of the complex linearized operator for a class of nonlinear Schr\"odinger systems, obtaining as byproduct some interesting properties of non-degenerate ground state of the associated elliptic system, such as being isolated and orbitally stable. Comment: 18 pages, 1 figure
    Communications on Pure and Applied Analysis 05/2009; 9(4). DOI:10.3934/cpaa.2010.9.867 · 0.84 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Orbital stability property for weakly coupled nonlinear Schr\"odinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable single Schr\"odinger weakly coupled system, even if they are not ground states. Comment: 21 pages, original article
    Advanced Nonlinear Studies 09/2008; 10(3). · 0.92 Impact Factor
Show more