## Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions.(English)Zbl 1031.35137

Authors’ abstract: We consider a linear Schrödinger equation with a nonlinear perturbation in $$\mathbb{R}^3$$. Assume that the linear Hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition. We prove that if the initial data is sufficiently small and is near a nonlinear ground state, then the solution approaches to certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schrödinger equation and its asymptotic profile can have two different types of decay. The resonance-dominated solution decay as $$t^{-1/2}$$ or the dispersion-dominated solutions decay at least like $$t^{-3/2}$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35P25 Scattering theory for PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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