Scipio Cuccagna

ON ASYMPTOTIC STABILITY OF GROUND STATES OF NLS

We prove in dimension n = 3 an asymptotic stability result for ground states of the Nonlinear Schrodinger Equation which contain one internal mode.

On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations

AbstractWe consider nonlinear Schrödinger equations

On stability of standing waves of nonlinear Dirac equations

iu_t +\Delta u +\beta (|u|^2)u=0\, ,\, \text{for} (t,x)\in \mathbb{R}\times \mathbb{R}^d,

Bifurcations from the endpoints of the essential spectrum in the linearized nonlinear Schrödinger problem

where d ≥ 3 and β is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t → ∞ assuming the so called the Fermi Golden Rule (FGR) hypothesis. We improve the “sign condition” required in a recent paper by Gang Zhou and I.M.Sigal.

On asymptotic stability in 3D of kinks for the ⁴ model

In this paper we study small amplitude solutions of nonlinear Klein Gordon equations with a potential. Under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity, we prove that all small energy solutions are asymptotically free. In cases where the linear system has at most one bound state the result was already proved by Soffer and Weinstein: we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states. The proof is based on a combination of Birkhoff normal form techniques and dispersive estimates.

On asymptotic stability of moving ground states of the nonlinear Schrödinger equation

We consider the stability problem for standing waves of nonlinear Dirac models. Under a suitable definition of linear stability, and under some restriction on the spectrum, we prove at the same time orbital and asymptotic stability. We are not able to get the full result proved in [24] for the nonlinear Schrödinger equation, because of the strong indefiniteness of the energy.

Dispersion for Schrödinger Equation with Periodic Potential in 1D

We use the inverse scattering transform, the auto-Bäcklund transformation, and the steepest descent method of Deift and Zhou to obtain the asymptotic stability of the solitons in the cubic NLS (nonlinear Schrödinger) equation.

NONLINEAR INSTABILITY OF A CRITICAL TRAVELING WAVE IN THE GENERALIZED KORTEWEG-DE VRIES EQUATION ∗

We study bifurcations of eigenvalues from the endpoints of the essential spectrum in the linearized nonlinear Schrodinger problem in three dimensions. We show that a resonance and an eigenvalue of positive energy at the endpoint may bifurcate only to a real eigenvalue of positive energy, while an eigenvalue of negative energy at the endpoint may also bifurcate to complex eigenvalues.