28 mar 2007

Abstract.

Ore 17:30 - 19:00 - Pisa, Dipartimento di Matematica, Sala delle riunioni

The one dimensional cubic Nonlinear Schrödinger equation (NLS) $$ i u_t - u_{xx} \pm u
u
² = 0, \qquad u(0) = u_0. $$ arises as generic asymptotic equation for modulated wave trains. Its has a particularly rich structure. We consider the cubic NLS equation in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time $H^{s}$ bounds in terms of the $H^s$ size of the initial data for $s \geq -\frac16$. This is related to large time properties of solutions to NLS.