Explicit formula for the solution of the Szeg\"o equation on the real line and applications

We consider the cubic Szeg\"o equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szeg\"o projector on positive frequencies. It is a model for totally non-dispersive evolution equations and is completely integrable in the sense that it admits a Lax pair. We find an explicit formula for solutions of the Szeg\"o equation. As an application, we prove soliton resolution in H^s for all s>0, for generic data. As for non-generic data, we construct an example for which soliton resolution holds only in H^s, 0<s<1/2, while the high Sobolev norms grow to infinity over time, i.e. \lim_{t\to\pm\infty}|u(t)|_{H^s}=\infty if s>1/2. As a second application, we construct explicit generalized action-angle coordinates by solving the inverse problem for the Hankel operator H_u appearing in the Lax pair. In particular, we show that the trajectories of the Szeg\"o equation with generic data are spirals around Lagrangian toroidal cylinders T^N \times R^N.Comment: Small modifications in the proof of Proposition 1.3, changed the order in the proof of Theorem 1.9, replaced the proof of \chi proper mapping in Theorem 1.

Traveling waves for the cubic Szego equation on the real line

We consider the cubic Szego equation i u_t=Pi(|u|^2u) on the real line, with solutions in the Hardy space on the upper half-plane, where Pi is the Szego projector onto the non-negative frequencies. This equation was recently introduced by P. Gerard and S. Grellier as a toy model for totally non-dispersive evolution equations. We show that the only traveling waves are rational functions with one simple pole. Moreover, they are shown to be orbitally stable, in contrast to the situation of the circle S^1 studied by the above authors, where some traveling waves were shown to be unstable.Comment: 24 pages, added references, small revision of the second part of the proof of Theorem 2.1 (p.8-9

Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-K\"ahler theorem. We consider a linear partial differential operator $P$ given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that $P$ is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor $R$ of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds

The Critical Theory of Artistic Capitalism

This article takes up Lipovetsky‟s discussion on artistic capitalism in L’esthétisation du monde. Vivre à l’âge du capitalisme artiste, to trace its definitions and methodological construction, but also in order to create a critical theory of artistic capitalism, based on the following working-hypothesis: the production of art and the production of self, understood in the sense of a Foucauldian project of the aesthetics of existence, represent correspondent purposes in artistic capitalism. My research will be focused on examining previous attempts of developing such a critical inquiry, claimed by Luc Boltanski, Eve Chiapello, and Luc Ferry. It is my thesis that the failure of a homogeneous critical theory of artistic capitalism is owed to different inconsistent interpretations of contaminating ethics with aesthetics in order to create an ideal of morality and authenticity for the existence of the individual inspired by contemporary techniques of art production, aspects that were conceived by Lipovetsky as parts of the process of the “aestheticization of the world”