Galina Perelman

Asymptotic Stability of Multi-soliton Solutions for Nonlinear Schrödinger Equations

Abstract We consider the Cauchy problem for the nonlinear Schrödinger equation iψ t = −▵ψ + F(|ψ|2)ψ, in space dimensions d ≥ 3, with initial data close to a sum of N decoupled solitons. Under some suitable assumptions on the spectral structure of the one soliton linearizations we prove that for large time the asymptotics of the solution is given by a sum of solitons with slightly modified parameters and a small dispersive term.

On the Formation of Singularities in Solutions of the Critical Nonlinear Schrodinger Equation

Abstract. For the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity the Cauchy problem with initial data close to a soliton is considered. It is shown that for a certain class of initial perturbations the solution develops a self-similar singularity infinite time T*, the profile being given by the ground state solitary wave and the limiting self-focusing law being of the form¶¶

Nondispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in ℝ

\lambda(t) \sim (ln \mid ln(T^* -t)\mid)^{1/2} (T^* - t)^{-1/2}

Effective Dynamics of Double Solitons for Perturbed mKdV

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Blow Up Dynamics for Equivariant Critical Schrödinger Maps

The following energy critical focusing nonlinear Schrödinger equation in R3 is considered: iψt = −Δψ − |ψ|4ψ; it is proved that, for any ν and α0 sufficiently small, there exist radial finite energy solutions of the form ψ(x, t) = eiα(t)λ1/2(t)W (λ(t)x)+eζ+oḢ1 (1) as t → +∞, where α(t) = α0 ln t, λ(t) = tν , W (x) = (1 + 1 3 |x|2)−1/2 is the ground state, and ζ∗ is arbitrary small in Ḣ1. §