# A Centre-Stable Manifold for the Focussing Cubic NLS in $${\mathbb{R}}^{1+3}$$

@article{Beceanu2007ACM,
title={A Centre-Stable Manifold for the Focussing Cubic NLS in \$\$\{\mathbb\{R\}\}^\{1+3\}\$\$},
author={Marius Beceanu},
journal={Communications in Mathematical Physics},
year={2007},
volume={280},
pages={145-205}
}
• M. Beceanu
• Published 26 January 2007
• Mathematics
• Communications in Mathematical Physics
Consider the focussing cubic nonlinear Schrödinger equation in $${\mathbb{R}}^3$$ :$$i\psi_t+\Delta\psi = -|\psi|^2 \psi. \quad (0.1)$$It admits special solutions of the form eitαϕ, where $$\phi \in {\mathcal{S}}({\mathbb{R}}^3)$$ is a positive (ϕ > 0) solution of$$-\Delta \phi + \alpha\phi = \phi^3. \quad (0.2)$$The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional…
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## References

SHOWING 1-10 OF 39 REFERENCES
Time decay for solutions of Schrödinger equations with rough and time-dependent potentials
• Mathematics
• 2001
In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions 0.1$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
• Mathematics
• 2003
AbstractWe consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u0.For u0$\in$H1, local existence in time of solutions on an interval
On the Formation of Singularities in Solutions of the Critical Nonlinear Schrödinger 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
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
• Mathematics
• 2006
We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave
A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr
• Mathematics, Physics
• 1999
A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold the- orems to nonlinear Schr- odinger type equations. The theo- rem is applied to the operator that
Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation
• Mathematics
• 2006
We make a detailed numerical study of the spectrum of two Schrödinger operators L± arising from the linearization of the supercritical nonlinear Schrödinger equation (NLS) about the standing wave, in
Uniqueness of positive solutions of Δu−u+up=0 in Rn
We establish the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n≧1, with the Neumann
Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue
Let H=−Δ+V(x) be a three dimensional Schrödinger operator. We study the time decay in Lp spaces of scattering solutions e−itHPcu, where Pc is the orthogonal projection onto the continuous spectral
Tools for Pde: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials
This book develops three related tools that are useful in the analysis of partial differential equations, arising from the classical study of singular integral operators: pseudodifferential
Invariant Manifolds for Semilinear Partial Differential Equations
• Mathematics
• 1989
When studying the behaviour of a dynamical system in the neighbourhood of an equilibrium point the first step is to construct the stable, unstable and centre manifolds. These are manifolds that are