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Zawartość zarchiwizowana w dniu 2024-05-07

Chaotic dynamics of multi - dimensional hamiltonian systems and applications to partial differential equations of physical significance


Research objectives and content
The main purpose of this project is to investigate analytically and numerically the chaotic properties of multi-dimensional dynamical systems. In particular, we will study near-integrable discretizations of certain partial differential equations (pdes) of physical significance like the Nonlinear Schrodinger equation describing pulse transmission in optical fibers and the sine-Gordon equation monitoring the flux of supercontacting current in Josephson junctions.
Besides the relevance of our study to the continuum limit of the corresponding pdes, we will also analyze our systems as
multi-degree-of-freedom Hamiltonian lattice models. Thus we expect to obtain results concerning the existence and stability of localized oscillatory excitations called breathers, as well as investigate energy transport between interacting breathers.
Our basic analytical tools will be the application of invariant manifold theory, Mel'nikov analysis and 'horseshoe' dynamics to establish the occurrence of homoclinic chaos. We also intend to apply various other methods, like the implicit function theorem in the so-called anti-continuum limit which have recently proved very useful in the investigation of such breather solutions. Guided by our analytical results we also plan to carry out extensive numerical computations to study the stability of our localized oscillatory states and global behavior of these systems for very long times.
Training content (objective, benefit and expected impact)
My stay at Cambridge is expected to be of great benefit as it well enable me to work with outstanding researchers in the field of Hamiltonian Dynamics. Furthermore, my background in higher-dimensional Mel'nikov analysis, invariant manifold theory and pdes should be beneficial for the success of the proposed collaboration. It is expected that important results will be obtained towards a better understanding of the several open problems concerning chaos in multi-dimensional systems. Links with industry / industrial relevance (22)

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