**Astérisque**

Volume: 389;
2017;
114 pp;
Softcover

MSC: Primary 35; 47; 37;
**Print ISBN: 978-2-85629-854-1
Product Code: AST/389**

List Price: $52.00

AMS Member Price: $41.60

# The Cubic Szegő Equation and Hankel Operators

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*Patrick Gérard; Sandrine Grellier*

A publication of the Société Mathématique de France

This monograph is devoted to the dynamics on Sobolev spaces
of the cubic Szegő equation on the circle \(\mathbb{S}^1\),
\[i\partial_t u=\Pi(\vert u\vert^{2} u).\]
Here \(\Pi\) denotes the orthogonal projector
from \(L^2(\mathbb{S}^1)\) onto
the subspace \(L^{2}_+(\mathbb{S}^1)\) of functions with
nonnegative Fourier modes. The authors construct a nonlinear Fourier
transformation on
\(H^{1/2}(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\), allowing
them to describe explicitly the solutions of
this equation with data in
\(H^{1/2}(\mathbb{S}^1)\cap L^2_+(\mathbb{S}^1)\).

This explicit description implies almost-periodicity of every
solution in this space. Furthermore, it allows the authors to display
the following turbulence phenomenon. For a dense \(G_\delta\)
subset of initial data in \(C^\infty(\mathbb{S}^1)\cap
L^2_+(\mathbb{S}^1)\), the solutions tend to infinity in
\(H^s\) for every \(s>\frac 12\) with super-polynomial
growth on some sequence of times, while they go back to their initial
data on another sequence of times tending to infinity.

This transformation is defined by solving a general inverse
spectral problem involving singular values of a Hilbert–Schmidt
Hankel operator and of its shifted Hankel operator.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians.