Mathematical Sciences Research Institute

Home » Workshop » Schedules » Local dynamics and invariant manifolds of traveling wave manifolds of Hamiltonian PDEs

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

Hamiltonian systems, from topology to applications through analysis I October 08, 2018 - October 12, 2018

October 12, 2018 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Chongchun Zeng (Georgia Institute of Technology)
Location: MSRI: Simons Auditorium
  • invariant manifolds

  • Hamiltonian PDEs

Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



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.

Asset no preview Notes 2.8 MB application/pdf Download
Video/Audio Files


H.264 Video 20-Zeng.mp4 185 MB video/mp4 Download
Buy the DVD

If none of the options work for you, you can always buy the DVD of this lecture. The videos are sold at cost for $20USD (shipping included). Please Click Here to send an email to MSRI to purchase the DVD.

See more of our Streaming videos on our main VMath Videos page.