Invariant Manifolds and Dispersive Hamiltonian Evolution Equations

10.4171/095 ◽  
2011 ◽  
Author(s):  
Kenji Nakanishi ◽  
Wilhelm Schlag
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Hamiltonian Evolution

Random invariant manifolds for ill-posed stochastic evolution equations

2019 ◽  
Vol 20 (02) ◽  
pp. 2050013
Author(s):  
Alexandra Neamţu
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Linear Part ◽  
Time Behavior ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Stable And Unstable Manifolds ◽  
Long Time ◽  
Ill Posed ◽  
Unstable Manifolds

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a [Formula: see text]-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.

scholarly journals SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

2006 ◽  
Vol 03 (02) ◽  
pp. 247-267 ◽  
Author(s):  
JOHN K. HUNTER
Keyword(s):  
Rayleigh Waves ◽  
Evolution Equations ◽  
Hyperbolic Surface ◽  
Smooth Solutions ◽  
Scale Invariant ◽  
Weakly Nonlinear ◽  
Hamiltonian Evolution ◽  
Short Time ◽  
Nonlinear Scale ◽  
Time Existence

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

Operations involving momentum variables in non-Hamiltonian evolution equations

1988 ◽  
Vol 101 (3) ◽  
pp. 333-355 ◽  
Author(s):  
F. Benatti ◽  
G. C. Ghirardi ◽  
A. Rimini ◽  
T. Weber
Keyword(s):  
Evolution Equations ◽  
Hamiltonian Evolution

NONCOMMUTATIVE KdV HIERARCHY

2007 ◽  
Vol 19 (07) ◽  
pp. 677-724 ◽  
Author(s):  
FRANÇOIS TREVES
Keyword(s):  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Equation ◽  
Completely Integrable Systems ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Kdv Hierarchy ◽  
Noncommutative Version ◽  
Constants Of Motion ◽  
Hamiltonian Evolution

The noncommutative version of the Korteweg–de Vries equation studied here is shown to admit infinitely many constants of motion and to give rise to a hierarchy of higher-order Hamiltonian evolution equations, each one the noncommutative version of the commutative KdV equation of the same order. The noncommutative KdV polynomials span, topologically, a maximal Abelian subalgebra of the Lie algebra of noncommutative Bäcklund transformations. Two classes of examples of "completely integrable" systems of evolution equations to which the theory applies are described in the last two sections.

scholarly journals Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions

2021 ◽  
Vol 8 (21) ◽  
pp. 252-266
Author(s):  
Maximilian Engel ◽  
Felix Hummel ◽  
Christian Kuehn
Keyword(s):  
Invariant Manifolds ◽  
Evolution Equations ◽  
Galerkin Approximation ◽  
Infinite Dimensions ◽  
Galerkin Approach ◽  
Infinite Dimensional ◽  
Slow Manifolds ◽  
Finite Dimensional ◽  
Abstract Evolution Equation ◽  
Scales Of Banach Spaces

In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.

scholarly journals GROUPOID SYMMETRY AND CONSTRAINTS IN GENERAL RELATIVITY

2013 ◽  
Vol 15 (01) ◽  
pp. 1250061 ◽  
Author(s):  
CHRISTIAN BLOHMANN ◽  
MARCO CEZAR BARBOSA FERNANDES ◽  
ALAN WEINSTEIN
Keyword(s):  
General Relativity ◽  
Initial Data ◽  
Cotangent Bundle ◽  
Evolution Equations ◽  
Gauge Theories ◽  
Riemannian Metrics ◽  
Einstein Equations ◽  
Cauchy Hypersurface ◽  
Lagrangian Field Theory ◽  
Hamiltonian Evolution

When the vacuum Einstein equations are cast in the form of Hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold [Formula: see text] of Riemannian metrics on a Cauchy hypersurface Σ. As in every Lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any Hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection. In an appendix, we develop some aspects of diffeology, the basic framework for our treatment of function spaces.

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