Tai-Peng Tsai

Stability and Asymptotic Stability for Subcritical gKdV Equations

Abstract: We prove in this paper the stability and asymptotic stability in H1 of a decoupled sum of N solitons for the subcritical generalized KdV equations The proof of the stability result is based on energy arguments and monotonicity of the local L2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [16].

Spectra of Linearized Operators for NLS Solitary Waves

Nonlinear Schrodinger 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 numerically.

Relaxation of excited states in nonlinear Schrödinger equations

We consider a nonlinear Schrodinger equation in

Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves

\R^3

STABLE DIRECTIONS FOR EXCITED STATES OF NONLINEAR SCHRÖDINGER EQUATIONS

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 {\it excited} state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schrodinger equation approaches to certain nonlinear {\it ground} state as the time tends to infinity.

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

We study a class of nonlinear Schrodinger equations which admit families of small solitary wave solutions. We consider solutions which are small in the energy space H 1 , and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we show that as t → ∞, the solitary wave component converges to a fixed solitary wave, and the dispersive component converges strongly in H 1 to a solution of the free Schrodinger equation.

Stability in

ABSTRACT We consider nonlinear Schrödinger equations in . Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden rule.

H^1

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ℝ3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ϵ ≤ 1, |v (x, t)| ≤ C * r −1+ϵ |t|−ϵ/2 for − T 0 ≤ t < 0 and 0 < C * < ∞ allowed to be large. We prove that v is regular at time zero.

of the sum of

In this article we consider nonlinear Schrodinger (NLS) equations in R for d = 1, 2, and 3. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let Rk(t, x) be K solitary wave solutions of the equation with different speeds v1, v2, . . . , vK . Provided that the relative speeds of the solitary waves vk − vk−1 are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the Rk(t) is stable for t 0 in some suitable sense in H 1. To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrodinger equation. This property is similar to the L2 monotonicity property that has been proved by Martel and Merle for the generalized Korteweg–de Vries (gKdV) equations (see [12, Lem. 16, proof of Prop. 6]) and that was used to prove the stability of the sum of K solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).

K

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in