This is joint work with Xin Zhou.

The speaker will consider, in particular, different aspects of the
solution of the Cauchy problem for the perturbed defocusing NLS equation,
(1) iq_t + q_xx - 2(|q|^2)q -(\epsilon)W(|q|^2)q = 0
q(x,0)=q_0(x)--> 0 as |x|--> \infty.

Here (\epsilon)>0, W(s) is non-negative and W(s) behaves like s^k as s --> 0 for some (sufficiently large) exponent l.

For fixed k>7/4, and \epsilon sufficiently small, the authors
(i) describe the long-time behavior of solutions of (1)
(ii) show that on an invariant, open, connected set in phase space, equation (1) is completely integrable in the sense of Liouville
(iii)show that the solution of (1) is universal in the following sense: one uses W to set the macroscopic scales for the solution, but once the scale is set, the solution of (1) looks the same independent of W.

The main technical tool in proving (i)(ii)(iii) is to use the Zakaharov-Shabat scattering map for NLS to transform the problem to normal form in the manner of Kaup and Newell, and then to analyze the normal form using Riemann-Hilbert/steepest-descent-type methods.

Celestial mechanics is a main parent" of the modern theory of
dynamical systems. Poincare proved non-integrability of the three body
problem when he discovered the homoclinic picture. Alexeev explained the
existence of the oscillatory motions (a planet approaches infinity
always returning to a bounded domain) in Sitnikov model (one of the
restricted versions of the three body problem) using methods of
hyperbolic dynamics.
We show that the structures related to the most recent works in the
smooth dynamical systems (e.g. conservative Henon family, lateral
thickness of a Cantor set, persistent tangencies, splitting of
separatrices) also appear in the three body problem. After we get some
new results in smooth dynamics (parameterized version of conservative
Newhouse phenomena, relation between lateral thicknesses and Hausdorff
dimension of a Cantor set, etc), we prove that in many cases the set of
oscillatory motions has a full Hausdorff dimension.
This is a joint work with V.Kaloshin.