Edinburgh on James Colliander
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Mon, 28 May 2012 00:00:00 +0000

GWP of GrossPitaevskii Equation on R4
https://example.com/post/20120528gwpofgrosspitaevskiiequationonr4/
Mon, 28 May 2012 00:00:00 +0000
https://example.com/post/20120528gwpofgrosspitaevskiiequationonr4/
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 GrossPitaevskii equation.

Addendum to Arnold Memorial Workshop: Khesin on Pinzari Talk
https://example.com/post/20111024addendumtoarnoldmemorialworkshopkhesinonpinzaritalk/
Mon, 24 Oct 2011 00:00:00 +0000
https://example.com/post/20111024addendumtoarnoldmemorialworkshopkhesinonpinzaritalk/
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
https://example.com/post/20110120edinburghmeetingnotes3/
Thu, 20 Jan 2011 00:00:00 +0000
https://example.com/post/20110120edinburghmeetingnotes3/
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, …

Edinburgh Meeting Notes 2
https://example.com/post/20110119edinburghmeetingnotes2/
Wed, 19 Jan 2011 00:00:00 +0000
https://example.com/post/20110119edinburghmeetingnotes2/
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: NavierStokes. $$ \dot{u} + (u\cdot \nabla) u + \nabla p = \epsilon \Delta u + force; ~ \nabla \cdot u = 0.

Edinburgh Meeting Notes 1
https://example.com/post/20110118edinburghmeetingnotes1/
Tue, 18 Jan 2011 00:00:00 +0000
https://example.com/post/20110118edinburghmeetingnotes1/
These are notes from a meeting entitled Advanced Numerical Studies in Nonlinear PDEs in Edinburgh, Scotland.
Walter Craig (McMaster): Water Wave Interactions I’m an analyst but I’m going to talk about numerics and experiments as well as analysis. We will discuss the problem of water waves and then I’ll talk about two specific settings in which the theory has led to good and quite elegant numerics and the numerics have started to answer some questions.