**EMS Zurich Lectures in Advanced Mathematics**

Volume: 14;
2011;
258 pp;
Softcover

MSC: Primary 35; 37;
**Print ISBN: 978-3-03719-095-1
Product Code: EMSZLEC/14**

List Price: $52.00

AMS Member Price: $41.60

# Invariant Manifolds and Dispersive Hamiltonian Evolution Equations

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*Kenji Nakanishi; Wilhelm Schlag*

A publication of the European Mathematical Society

The notion of an invariant manifold arises naturally in the
asymptotic stability analysis of stationary or standing wave solutions
of unstable dispersive Hamiltonian evolution equations such as the
focusing semilinear Klein–Gordon and Schrödinger
equations. This is due to the fact that the linearized operators about
such special solutions typically exhibit negative eigenvalues (a
single one for the ground state), which lead to exponential
instability of the linearized flow and allows for ideas from
hyperbolic dynamics to enter.

One of the main results proved here for energy subcritical equations is
that the center-stable manifold associated with the ground state appears
as a hyper-surface which separates a region of finite-time blowup in
forward time from one which exhibits global existence and scattering to
zero in forward time. The authors' entire analysis takes place in the energy
topology, and the conserved energy can exceed the ground state energy
only by a small amount.

This monograph is based on recent research by the authors. The proofs
rely on an interplay between the variational structure of the ground
states and the nonlinear hyperbolic dynamics near these
states. A key element in the proof is a virial-type
argument excluding almost homoclinic orbits originating near the ground
states, and returning to them, possibly after a long excursion.

These lectures are suitable for graduate students and researchers in
partial differential equations and mathematical physics. For the cubic
Klein–Gordon equation in three dimensions all details are provided,
including the derivation of Strichartz estimates for the free equation
and the concentration-compactness argument leading to scattering due to
Kenig and Merle.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and researchers interested in partial differential equations and mathematical physics.