**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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Program and Speakers
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**Conference Program**

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Sunday March 28, 15:15–15:40

Dynamics of Solutions to the Focusing Cubic NLS Equation in 2 and 3 Dimensions
Svetlana RoudenkoSchool of Mathematical and Statistical Sciences, Arizona State University , Tempe, AZ |

We consider the cubic NLS equation i*u*_{t} +
Δ*u* + |*u*|^{2}*u* = 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
*L*^{4}{*t*, *x*}) norm of small
*L*^{2} solutions: we show that the maximum of this norm is
attained, and give a precise estimate of it as the *L*^{2}
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.