Hamiltonian Partial Differential Equations


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.



  Walter Craig  

Dmitri Pelinovsky


Andrei Biryuk


Philippe Guyenne


Slim Ibrahim


Vladislav Panferov



  • Invariant tori for Hamiltonian PDE (Craig, 2004)
  • Traveling gravity water waves in two and three dimensions, (W. Craig and D. Nicholls, 2002)
  • Problémes de petits diviseurs dans les équations aux dérivées partielles (W. Craig, 2000)