Related papers: The Szeg\"o Cubic Equation

We continue the study of the following Hamiltonian equation on the Hardy space of the circle, $$i\partial _tu=\Pi(|u|^2u)\ ,$$ where $\Pi $ denotes the Szeg\"o projector. This equation can be seen as a toy model for totally non dispersive…

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…

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle $S^1$ , $$i\partial\_ t u = \Pi(|u|^ 2 u) + \alpha(u|1) , \alpha \in\mathbb{R} ,$$ where $\Pi$ is the Szeg\H{o} projector. The above equation with…

The purpose of this paper is to go further into the study of the quadratic Szeg{\"o} equation, which is the following Hamiltonian PDE : $i \partial\_t u = 2J\Pi(|u|^2)+\bar{J}u^2$, $u(0, \cdot)=u\_0$, where $\Pi$ is the Szeg{\"o} projector…

This paper is concerned with the cubic Szeg\H{o} equation $$ i\partial_t u=\Pi(|u|^2 u), $$ defined on the $L^2$ Hardy space on the one-dimensional torus $\mathbb T$, where $\Pi: L^2(\mathbb T)\rightarrow L^2_+(\mathbb T)$ is the Szeg\H{o}…

We consider the following Hamiltonian equation on a special manifold of rational functions, \[i\p\_tu=\Pi(|u|^2u)+\al (u|1),\ \al\in\R,\] where $\Pi $ denotes the Szeg\H{o} projector on the Hardy space of the circle $\SS^1$. The equation…

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…

We establish an explicit formula for the solution of the the cubic Szeg\H{o} equation on the real line. Using this formula, we prove that the evolution flow of this equation can be continuously extended to the whole Hardy class $H^2$ on the…

We consider the following degenerate half wave equation on the one dimensional torus $$\quad i\partial_t u-|D|u=|u|^2u, \; u(0,\cdot)=u_0. $$ We show that, on a large time interval, the solution may be approximated by the solution of a…

G\'erard and Grellier proposed, under the name of the cubic Szeg\H{o} equation, a remarkable classical field theory on a circle with a quartic Hamiltonian. The Lax integrability structure that emerges from their definition is so…

The Hardy spaces are defined on the quotient domain of a bounded complete Reinhardt domain by a finite subgroup of $U(n)$. The Szeg\H{o} projection on the quotient domain can be studied by lifting to the covering space. This setting builds…

We consider the 1-dimensional cubic Szeg\H{o} equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, where $s>\frac12$. We show that, for $s>1$, this measure is…

We study the influence of Szeg{\"o} projector $\Pi$ on the L 2 --critical one-dimensional non linear focusing Schr{\"o}dinger equation, leading to the quintic focusing NLS-Szeg{\"o} equation i$\partial$ t u + $\partial$ 2 x u + $\Pi$(|u| 4…

In this paper, we study a quadratic equation on the one-dimensional torus : $$i \partial_t u = 2J\Pi(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u_0,$$ where $J=\int_\mathbb{T}|u|^2u \in\mathbb{C}$ has constant modulus, and $\Pi$ is the Szeg\H{o}…

We consider the half-wave maps equation $$ \partial_t \vec{S} = \vec{S} \wedge |\nabla| \vec{S}, $$ where $\vec{S}= \vec{S}(t,x)$ takes values on the two-dimensional unit sphere $\mathbb{S}^2$ and $x \in \mathbb{R}$ (real line case) or $x…

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 Szeg{\"o} equation on the circle ${\mathbb S} ^1$,$$ i\partial \_t u=\Pi (\vert u\vert…

We study the cubic Szeg\"o equation which is an integrable nonlinear non-dispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special…

The classical Szeg\H{o}-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal…

We are interested in the influence of filtering the positive Fourier modes to the integrable non linear Schr{\"o}dinger equation. Equivalently, we want to study the effect of dispersion added to the cubic Szeg{\"o} equation, leading to the…

We give a complete classification of the traveling waves of the following quadratic Szeg{\"o} equation : $i \partial\_t u = 2J\Pi(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u\_0$, and we show that they are given by two families of rational…