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,
$$iq_t + q_xx - 2(|q|^2)q -\epsilon W(|q|^2)q = 0$$
(1)
$$ q(x,0)=q_0(x)\rightarrow 0 as |x|\rightarrow \infty $$
Here $\epsilon >0, W(s)$ is non-negative and W(s) behaves like $s^k$ as $s \rightarrow 0$ for some (sufficiently large) exponent $k$. 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.