Research Seminar: Geometric Analysis by Jingxuan Zhang II

Speaker:  Jingxuan Zhang 

Title: Recent works on effective dynamics

Abstract: In the study of nonlinear evolutions, the method of adiabatic approximation is an essential tool to reduce an infinite dimensional dynamical system to a simpler, possibly finite dimensional one. In these talks, we formulate a generic scheme of adiabatic approximation that is valid for an abstract dissipative, conservative or dispersive evolution under mild regularity assumptions. The key prerequisite for the scheme is the existence of what we call approximate solitons. These are some low energy but not necessarily stationary configurations, which we use to approximate nearby evolutions with systematically minimized errors.

As an application, we consider the concentration property of solutions to the dispersive Ginzburg-Landau (or Gross-Pitaevskii) equation in three dimensions.  On a spatial domain, it has long been conjectured that such a solution concentrates near some curve evolving according to the binormal curvature flow, and conversely, that a curve moving this way can be realized in a suitable sense by some solution to the dispersive Ginzburg-Landau equation. Some partial results are known with rather strong symmetry assumptions. Our main theorems here provide affirmative answer to both conjectures under certain small curvature assumption.