# 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} }

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…

## 18 Citations

The instability of Bourgain-Wang solutions for the L2 critical NLS

- Mathematics
- 2010

We consider the two dimensional $L^2$ critical nonlinear Schr\"odinger equation $i\partial_tu+\Delta u+uu^2=0$. In their pioneering 1997 work, Bourgain and Wang have constructed smooth solutions…

A critical center‐stable manifold for Schrödinger's equation in three dimensions

- Mathematics
- 2012

Consider the focusing $\dot H^{1/2}$‐critical semilinear Schrödinger equation in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ $$i\partial _t \psi + \Delta \psi + |\psi |^2 \psi = 0.$$

Existence and Stability of Schrödinger Solitons on Noncompact Manifolds

- MathematicsSIAM J. Math. Anal.
- 2019

This work proves the existence of a solitary wave by perturbing off the flat Euclidean case, and provides numerical evidence showing that the introduction of a nontrivial geometry destabilizes the solitary wave in a wide variety of cases, regardless of the curvature of the manifold.

Endpoint Strichartz Estimates for Charge Transfer Hamiltonians

- Mathematics
- 2015

We prove the optimal endpoint Strichartz estimates for Schr\"{o}dinger equation with charge transfer potentials and a general source term in $\mathbb{R}^n$ for $n\geq3$. The proof is based on using…

Conditional stability theorem for the one dimensional Klein-Gordon equation

- Mathematics
- 2011

The paper addresses the conditional non-linear stability of the steady state solutions of the one-dimensional Klein-Gordon equation for large time. We explicitly construct the center-stable manifold…

A Dispersive Bound for Three-Dimensional Schrödinger Operators with Zero Energy Eigenvalues

- Mathematics
- 2008

We prove a dispersive estimate for the evolution of Schrödinger operators H = −Δ + V(x) in ℝ3. The potential is allowed to be a complex-valued function belonging to L p (ℝ3) ∩ L q (ℝ3), , so that H…

Decay Estimates for the Supercritical 3-D Schrödinger Equation with Rapidly Decreasing Potential

- Mathematics
- 2012

We establish an almost optimal decay estimate for the 3-D Schrodinger equation with non-negative potential decaying exponentially and nonlinearity of power \( p > 1 + 2/3 = 5/3 \).The key point is…

New estimates for a time-dependent Schrödinger equation

- Mathematics
- 2009

This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold…

On Stability of Standing Waves of Nonlinear Dirac Equations

- Mathematics
- 2011

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…

## 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

- Mathematics, Physics
- 2001

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

- Mathematics
- 1989

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

- Mathematics
- 2005

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

- Mathematics
- 2000

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…