GWP of Gross-Pitaevskii Equation on R4

Last week, I had a chance to visit Edinburgh in part to serve as the external examiner on the PhD Thesis (papers) of Tim Candy. Tim is now Dr. Timothy Candy and has an exciting research program to develop as a postdoc at Imperial. It turned out I had lucky timing since my visit overlapped with a visit by Oana Pocovnicu. I had a chance to hear her speak about her recent work on the Gross-Pitaevskii equation.

Addendum to Arnold Memorial Workshop: Khesin on Pinzari Talk

The following message is a guest post by Boris Khesin. Boris summarizes the wonderful talk given by Gabriella Pinzari at the workshop. –Jim Colliander Gabriella Pinzari (30min talk) described her joint result with her advisor Luigi Chierchia on a recently found fix for the famous KAM theorem, or rather for its application to the stability of the Solar system. Namely, the original KAM theorem inArnold’s 1963 paper claimed the persistence of the Liouville tori for perturbations of integrable systems under some nondegeneracy assumption - some determinant must be nonzero.

Edinburgh Meeting Notes 3

Galina Perelman: 2 soliton collision in NLS $$i \partial_t \psi = - \psi_{xx} + F(|\psi|^2) \psi, ~ x \in R$$ where $F(\xi) = -2 \xi + O (\xi^2), ~ \xi \rightarrow 0.$ This family of equations has solitary wave solutions $$e^{i \theta(x,t) \phi (x - b(t), E)}$$ where $\theta(x,t) = \omega t + \gamma + v \frac{x}{2}, ~b(t) =vt + c$ (all reall parameters). The profile $\phi$ is the associated ground state, which is $C^2$, decays exponentially, is even, …
Sergei Kuksin (École Polytechnique): Nonlinear Schrödinger Equation We consider Hamiltonian PDE. This is of course very interesting. In physics, there is a class of pdes which is also of interest: Hamiltonian PDE = small damping + small forcing Why is it so important? This class contains a very important equation: Navier-Stokes.  \dot{u} + (u\cdot \nabla) u + \nabla p = \epsilon \Delta u + force; ~ \nabla \cdot u = 0.