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Many of the principal equations of mathematical physics can be formulated as dynamical systems in an infinite
dimensional phase space, very often in the form of a Hamiltonian system with infinity mant degrees of
freedom. Such systems of partial differential equations (PDEs) possess at least one, and commonly many
conserved integrals of motion, corresponding to energy conservation, momentum conservation and similar
physical notions. Principal examples of equations of this form include the nonlinear Schroedinger equation
(NLS), the nonlinear wave equation (NLW) and the Korteweg deVries equation (KdV). More complex systems of
similar form include Einstein's equations of general relativity and the Euler equations of fluid dynamics.
The range of physical scales modeled by partial differential equations is enormous, from sub-quantum
mechanical to cosmological dimensions.
Our own interests in Hamiltonian PDE stem from our research in modeling physical phenomena which are energetically
conservative, and from the idea that detailed analytic techniques of Hamiltonian mechanics can be usefull
extended to PDE in the context of extensions to an infinite dimensional phase space. Examples of research
results include versions of KAM theory for partial differential equations, Birkhoff normal forms analysis for
PDE, and stability results related to the theory of Krein and Moser, and to long time Nekhoroshev stability
which gives upper bounds on diffusion of the action variables in many situations in perturbation theory.
There are numerous physical applications which involve Hamiltonian PDE. In particular, the NLS equation
appears in descriptions of the envelope dynamics of nearly monochromatic wave propagation in nonlinear
media, an application which arises in physical contexts as diverse as laser propagation in optical devices,
free surface water waves in the ocean, and in waves in plasmas. Nonlinear dispersive equations such as
the KdV equation are used to model tsunami waves and large interfacial waves in the pycnocline (internal
layer) of tropical seas. Nonlinear wave equations arise in a variety of models in continuum mechanics
ranging from Euler's equations of fluid motion to deformation waves in elastic media. The analysis of
phenomena such as self-focusing, propagation of ultra-short pulses, singularity formation, and other
results of nonlinear interactions are very important in these contexts. |
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Hamiltonian Partial Differential Equations
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