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.

1990年9月考入南开大学主修数学专业，1997年获得Brigham Young University数学专业博士学位，1997年8月—2000年在美国纽约大学做博士后，2000—2005年作为助理教授在弗吉尼亚大学工作，2005年8月至今在佐治亚理工学院从事教学研究工作。他的主要研究方向是应用动力系统以及非线性偏微分方程组。