Local dynamics and invariant manifolds of traveling wave manifolds of Hamiltonian PDEs

Some Hamiltonian PDEs which are invariant under spatial translations possess traveling wave solutions which form finite dimensional invariant manifolds parametrized by their spatial locations. Extensive studies have been carried out for their stability analysis. In this talks we shall focus on local dynamics and invariant manifolds of the traveling wave manifolds for the Gross-Pitaevskii equation in $R^3$ and the gKdV equation as our main PDE models, while our approach works for a general class of problems. Noting that the symplectic operators of some of these models happen to be unbounded in the energy space, violating a commonly assumed assumption for the study of the linearized systems at these traveling waves, we could carry out linearized analysis in a general framework we developed recently. Nonlinearly our main results are the existence of local invariant manifolds of unstable traveling waving manifolds and the implications on the local dynamics. In addition to applying certain space-time estimates, we use a bundle coordinate system to handle an issue of a seemingly regularity loss caused by the spatial translation parametrization.

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