The cubic szego equation and hankel operators

2015/08/27 by Grellier, Sandrine, Gerard, Patrick · 2 citations
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.06814

Abstract

This monograph is an expanded version of the preprint arXiv:1402.1716 or hal-00943396v1.It is devoted to the dynamics on Sobolev spaces of the cubic Szegö equation on the circle \mathbb S ¹,i∂ _t u=Π(\vert u\vert ²u) .Here Π denotes the orthogonal projector from L²(\mathbb S ¹) onto the subspace L²_+(\mathbb S ¹) of functions with nonnegative Fourier modes.We construct a nonlinear Fourier transformation on H^1/2(\mathbb S ¹)∩ L²_+(\mathbb S ¹) allowing to describe explicitly the solutions of this equationwith data in H^1/2(\mathbb S ¹)∩ L²_+(\mathbb S ¹). This explicit description implies almost-periodicity of every solution in H^{\frac 12}_+. Furthermore, it allows to display the following turbulence phenomenon. For a dense G_δ subset of initial data in C^∞ (\mathbb S ¹)∩ L²_+(\mathbb S ¹), the solutions tend to infinity in H^s for every s\textgreater\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.

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