Frontiers in Nonlinear Waves
in honor of Vladimir Zakharov's 70th birthday
March 26–29, 2010
University of Arizona, Tucson, AZ, USA

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Conference Program

Mar 26 Mar 27 Mar 28 Mar 29 Speakers

Sunday March 28, 15:15–15:40

Dynamics of Solutions to the Focusing Cubic NLS Equation in 2 and 3 Dimensions
Svetlana Roudenko
School of Mathematical and Statistical Sciences, Arizona State University , Tempe, AZ
We consider the cubic NLS equation iut + Δu + |u|2u = 0 in 2 and 3 spatial dimensions. The 3d version appears as a limiting model equation in plasma physics which was first described by Zakharov in his fundamental paper of 1972. Solutions to the focusing NLS may blow up in finite time and existence of such blow up solutions has been known since the works of Zakharov and Petrishev–Talanov–Vlasov. However, the precise dynamics of blow-up solutions is known only in a few cases and we discuss the situation for the 3d cubic NLS equation. For the 2d cubic NLS, we study the space-time (Strichartz L4{tx}) norm of small L2 solutions: we show that the maximum of this norm is attained, and give a precise estimate of it as the L2 norm tends to 0. In particular, this maximum is greater than the corresponding maximum for the linear equation, which was computed by Foschi and Hundertmark–Zharnitsky.