Dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations: mass transfer in systems with solitons and degenerate neutral modes. (English) Zbl 1175.35136

Gang, Zhou; Weinstein, Michael I.

Anal. PDE 1, No. 3, 267-322 (2008).

The authors prove the asymptotic stability of the ground state of NLS/GP equation when the linearized spectrum has degenerate neutral modes. They show that the solution has three interacting parts: i) a modulating soliton, parametrized by the motion along ground state; ii) oscillatory, spacially localized, neutral modes, which decay with time; iii) a dispersive part, which decays in a local energy norm. The neutral modes and dispersive waves decay via transferring their mass to the soliton manifold or to spatial infinity. Additionally, degenerate neutral modes are coupled and exchange mass among themselves in addtition to with the soliton and radiation.

Reviewer: Igor Andrianov (Köln)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B35	Stability in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K45	Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35C08	Soliton solutions

Keywords:

soliton; nonlinear bound state; nonlinear scattering; asymptotic stability; dispersive PDE