| Language |
English
|
| From |
2011/08/31 15:00
|
| To |
2011/08/31 16:00
|
| Place | Room 251, Building No.3, Faculty of Science, Kyoto University |
| Seminar Name | NLPDE Seminar |
| Title |
Explicit Formula for the Solution of the Cubic Szeg? Equation on the Real Line and its Applications¡¡ |
| Field |
Analysis
|
| Speakers | Oana Pocovnicu |
| Affiliation | Universite Paris-Sud, Orsay |
| Abstract | In this talk we consider the cubic Szeg? equation: i u_t = Pi_+
(|u|^2u) on the real line,
where Pi_+ is the Szeg? projector onto non-negative frequencies.
This equation was introduced as a model of a non-dispersive Hamiltonian
equation.
Like 1-d cubic NLS and KdV, it is known to be completely integrable in the
sense that it possesses a Lax pair structure.
As a consequence, it has an infinite sequence of conservation laws, that
can all be controlled by the H^{1/2}-norm.
First, we present an explicit formula for the solutions. Then, as an
application, we prove soliton resolution in H^s, s>0,
for "generic" data. Namely, we show that solutions with generic rational
functions as initial data can be decomposed in H^s,
for all s>0, into a sum of solitons plus a remainder, as time tends to
infinity. As for non-generic rational function data,
we construct an example for which soliton resolution holds only in H^s,
0< s < 1/2, while the high Sobolev norms grow
to infinity over time. In the last part of the talk we show that the Szeg?
equation is the first approximation
of a non-linear wave equation with well-prepared initial condition. As a
consequence,
we give an example of solution of the non-linear wave equation
for which the high Sobolev norms grow relatively to the initial condition. |
| Link | http://www.math.kyoto-u.ac.jp/~yosihiro/nlpde/nlpde.html |
|