Guide Properties of Infinite Dimensional Hamiltonian Systems

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Walter Craig McMaster University.

A paper by VE Zakharov gives a formulation of the equations for water waves as a Hamiltonian dynamical system, and shows that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context.

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Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed to the speaker by T. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions. We transform the water wave problem with nonzero surface tension to third order Birkhoff normal form, and in the case of zero surface tension in deep water, to fourth order Birkhoff normal form. The result includes a discussion of the dynamics of the normal form, and a quantification of the function space mapping properties of these transformations.

Properties of Infinite Dimensional Hamiltonian Systems - P.R. Chernoff, J.E. Marsden - Google книги

Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem. PDF abstract Size: 40 kb. In my talk I will describe numerical continuation and bifurcation techniques in a boundary value setting used to follow Lyapunov families of periodic orbits. These arise from the polygonal system of n bodies in a rotating frame of reference.

When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. PDF abstract Size: 37 kb. I shall discuss the existence and stability of quasi-periodic invariant tori for classes of evolution PDEs, both in Hamiltonian and reversible setting,trying to give an idea of the general strategy and the main difficulties in problems of this kind.

I will also discuss the related problem of unstable solutions. Yannick Sire John Hopkins University. I will describe recent results in collaboration with R. In particular, our methods uses very little of symplectic geometry and does not use transformation theory. It applies to ill-posed equations in the Hadamard sense and we will give applications to the so-called Boussinesq equation by constructing periodic solutions for it. Slim Ibrahim University of Victoria.

We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates BEC. The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations.

Bibliographic Information

Whitham-Boussinesq model for variable depth topography. Results on normal and trapped modes for non trivial geometries.

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PDF abstract Size: 42 kb. The water-wave problem describes the evolution of an incompressible ideal, irrotational fluid with a free surface under the influence of gravity.

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A significant development in water-wave theory was the discovery by Zakharov in that the problem has a Hamiltonian structure and later W. Craig and C. Sulem introduced the Dirichlet-Neumann operator explicitly on the Hamiltonian. In this talk I will present a joint work with Prof. Panayotis Panayotaros and Prof.

Integrable system

Standard notions of differential geometry derivative, differential, tangent space Integrability in finite dimensions Frobenius, Liouville, more general notions , basic properties, examples. Important examples of integrable systems in mathematics and physics: Spherical pendulum, rigid body, top Lagrange, Euler, Kovalevskaya , coupled spin oscillators, coupled angular momenta Local behaviour regular points : Theorem of Arnold-Liouville, transformation to action-angle coordinates.

Local behaviour singular points : local normal form of Eliasson-Miranda-Zung for nondegenerate singular points in terms of hyperbolic, elliptic, and focus-focus components. Toric systems: Symplectic classification by Delzant by means of polygons. Semitoric systems: properties and interaction with the topology and geometry of the underlying manifold.