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We consider a linear Schrödinger equation with a nonlinear perturbation in R 3. Assume that the linear Hamiltonian has exactly two bound states and its eigen-values 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… (More)

We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schrödinger equation, this implies asymptotic stability of all plane… (More)

We study asymptotic behaviour at time infinity of solutions close to the non-zero constant equilibrium for the Gross-Pitaevskii equation in two and three spatial dimensions. We construct a class of global solutions with prescribed dispersive asymptotic behavior, which is given in terms of the linearized evolution.

- Tai-Peng Tsai
- 1998

This paper proves that Leray's self-similar solutions of the three-dimensional Navier-Stokes equations must be trivial under very general assumptions , for example, if they satisfy local energy estimates. 1. Introduction In 1934 Leray Le] raised the question of the existence of self-similar solutions of the Navier-Stokes equations. For a long time, the… (More)

We consider the nonlinear Hartree equation describing the dynamics of weakly interacting non-relativistic Bosons. We show that a nonlinear Møller wave operator describing the scattering of a soliton and a wave can be defined. We also consider the dynamics of a soliton in a slowly varying background potential W (εx). We prove that the soliton decomposes into… (More)

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in R 3 with nontrivial swirl. Such solutions are not known to be globally defined, but it is shown in ([1], Partial regularity of suitable weak solutions of the Navier–Stokes equations. they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r… (More)

Nonlinear Schrödinger equations (NLSs) with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves and to the long-time dynamics of solutions of NLSs. We study these spectra in detail, both analytically and… (More)

We consider a nonlinear Schrödinger equation in R 3 with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear excited state. We give a sufficient condition on the initial data so that the solution to… (More)

- Tai-Peng Tsai
- 2002

We consider a nonlinear Schrödinger equation with a bounded local potential in R 3. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data are localized and small in H 1. We prove that exactly three local-in-space behaviors can occur as the time tends to infinity: 1.… (More)