back

PROGRAM - NONLINEAR WAVES and DISPERSIVE PDEs @GSSI L’Aquila

June 16-20, 2014

HOME PROGRAM REGISTRATION HOW TO REACH L’AQUILA AND GSSI ACCOMMODATION

 

The School consists in two short courses and seminar-style lectures given by experts in the field.

Informal discussion sessions on open problems will be organized.

SHORT COURSES

 

Gigliola Staffilani, Massachussets Institute of Technology
Title: Dispersive Equations: deterministic and random approach.

Abstract. In these lectures I will summarize well-posedness results for the NLS equation for both subcritical and critical. I will then introduce the concept of Gibbs measures and how its invariance, when available, can be used to extend local solutions to global. I will end by recalling other problems in which the concept of randomization has been used to prove well-posedness almost surely for certain initial value problems where the deterministic approach is either not valid or difficult to prove.

 

Vladimir Georgiev, University of Pisa
Title: Stability of steady states and solitary waves for dispersive and diffusive equations.

Abstract. The program of the course is the following:

1) Stationary points for nonlinear ordinary differential equations, steady states (ground states) and solitary waves for nonlinear evolution partial differential equations.
2) Existence, positivity and uniqueness of ground states.
3) Nonlinear diffusion equations in bounded domains and convergence of the solution to periodic cycles.
4) Dispersive equations and decay estimates for unbounded domains.
Stability of solitary waves for the Schroedinger or wave type equations.

 

SPECIAL LECTURES

 

Scipio Cuccagna, University of Trieste
Title: On the asymptotic stability of black solitons for the one-dimensional Gross-Pitaevskii equation

Abstract. We will discuss an application of the Nonlinear Steepest Descent method of Deift and Zhou to explore the so called asymptotic stability of black solitons. We recall that there is a discussion of the asymptotic stability for dark solitons by Bethuel-Gravejat and Smets, with completely different techniques but which however does not apply to black solitons. Our work was inspired by the use of techniques of integrable systems to prove orbital stability of black solitons due to Gerard and Zhang.

 

Damiano Foschi, University of Ferrara
Title: Local wellposedness of semilinear Schrödinger equations under minimal smoothness assumptions for the nonlinearity

Abstract can be found here

 

Christof Sparber, University of Illinois at Chicago
Title: Dispersive blow-up in Schrödinger equations

Abstract. We revisit the possibility of finite-time, dispersive blow up for nonlinear equations of Schrödinger type. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. We extend the results existing in the literature in several ways. In one direction, the theory is broadened to include the Davey-Stewartson and Gross-Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel’s formula is obtained.