# Invariant Manifolds and Dispersive Hamiltonian Evolution Equations

@inproceedings{Nakanishi2011InvariantMA, title={Invariant Manifolds and Dispersive Hamiltonian Evolution Equations}, author={Kenji Nakanishi and Wilhelm Schlag}, year={2011} }

By means of certain dispersive PDEs (such as the nonlinear Klein-Gordon equation) we will exhibit a new family of phenomena related to the ground state solitons. These solitons are (exponentially) unstable, and one can construct stable, unstable, and center(-stable) manifolds associated with these ground states in the sense of hyperbolic dynamics. In terms of these invariant manifolds one can completely characterize the global dynamics of solutions whose energy exceeds that of the ground states…

## Figures from this paper

## 133 Citations

### Title Invariant Manifolds Around Soliton Manifolds for the Nonlinear Klein‒Gordon Equation

- Mathematics
- 2017

We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein–Gordon equation with a focusing energy subcritical nonlinearity, associated…

### Invariant Manifolds Around Soliton Manifolds for the Nonlinear Klein-Gordon Equation

- MathematicsSIAM J. Math. Anal.
- 2012

The graph transform method is carried out in the presence of modulation parameters corresponding to the symmetries, which requires less spectral information on the linearized operator, and less decay of the nonlinearity.

### On the spectral properties of L± in three dimensions

- Mathematics
- 2011

This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrödinger or Klein–Gordon equations in three dimensions. We demonstrate by…

### THE METHOD OF CONCENTRATION COMPACTNESS AND DISPERSIVE HAMILTONIAN EVOLUTION EQUATIONS

- Mathematics
- 2012

• Small data theory: (f, g) are small, and F is treated as a perturbation. The main questions are local and global well-posedness, the existence of conserved quantities (energy), their relation to…

### Generic and non-generic behavior of solutions to the defocusing energy critical wave equation with potential in the radial case

- Mathematics
- 2015

In this paper, we continue our study [16] on the long time dynamics of radial solutions to defocusing energy critical wave equation with a trapping radial potential in 3 + 1 dimensions. For generic…

### Dynamical behaviors of a system modeling wave bifurcations

- Mathematics
- 2018

. We rigorously show that a class of systems of partial diﬀerential equations (PDEs) modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics…

### Global Dynamics Above The Ground State Energy For The Energy-Critical Klein-Gordon Equation

- Physics, Mathematics
- 2022

. Consider the focusing energy-critical Klein-Gordon equation in dimension d ∈ { , x ) with data ( f 0 ,f 1 ) ∈ H := H 1 × L 2 . We describe the global dynamics of real-valued solutions of which the…

### Attractors of nonlinear Hamiltonian partial differential equations

- Physics, MathematicsRussian Mathematical Surveys
- 2020

This is a survey of the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. Included are results on global attraction to stationary states, to…

### GLOBAL DYNAMICS ABOVE THE GROUND STATE FOR THE CRITICAL KLEIN-GORDON EQUATION

- Mathematics
- 2014

Consider the focusing critical Klein-Gordon equation in dimension N = 3, 5 ∂ttu−∆u+ u = |u| 4 N−2 u u(0, x) := f0(x) ∂tu(0, x) := f1(x) with data (f0, f1) ∈ H1 × L2. We describe the global…

### Dynamical behavior of a system modeling wave bifurcations with higher order viscosity

- Mathematics
- 2014

We rigorously show that a class of systems of partial differential equations modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics on the…

## References

SHOWING 1-3 OF 3 REFERENCES

### Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein–Gordon equation

- Mathematics
- 2011

We present some numerical findings concerning the nature of the blowup versus global existence dichotomy for the focusing cubic nonlinear Klein–Gordon equation in three dimensions for radial data.…

### Saddle points and instability of nonlinear hyperbolic equations

- Mathematics
- 1975

A number of authors have investigated conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time. For certain classes of…

### Nonlinear Wave Equations

- Mathematics
- 1990

Invariance Existence Singularities Solutions of small amplitude Scattering Stability of solitary waves Yang-Mills equations Vlasov-Maxwell equations.