相关论文: The cubic szego equation and hankel operators

The goal of this paper is to construct a nonlinear Fourier transformation on the space of symbols of compact Hankel operators on the circle. This transformation allows to solve a general inverse spectral problem involving singular values of…

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 construct inverse spectral theory for finite rank Hankel operators acting on the Hardy space of the upper half-plane. A particular feature of our theory is that we completely characterise the set of spectral data. As an application of…

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 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…

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 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 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 explores the regularity properties of an inverse spectral transform for Hilbert--Schmidt Hankel operators on the unit disc. This spectral transform plays the role of action-angles variables for an integrable infinite dimensional…

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle, $$i\partial_tu=\Pi(|u|^2u) ,$$ where $\Pi $ is the Szeg\"o projector. This equation can be seen as a toy model for totally non dispersive evolution…

The nonlinear Schr\"odinger equation NLSE(p, \beta), -iu_t=-u_{xx}+\beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space \Lp^2(\mT). For p\leq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that…

This paper is dedicated to studying matrix solutions of the cubic Szeg\H{o} equation on the line in Pocovnicu [arXiv:1001.4037, arXiv:1012.2943] and G\'erard--Pushnitski [arXiv:2307.06734], leading to the following matrix Szeg\H{o} equation…

We derive an explicit formula for the general solution of the cubic Szeg\"o equation and of the evolution equation of the corresponding hierarchy. As an application, we prove that all the solutions corresponding to finite rank Hankel…

The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere…

We provide necessary and sufficient geometric conditions for the exact observability of the Schr\"odinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for…

In the paper complete systems of exact solutions for Dirac and Weyl equations in the Lobachevsky space are constructed on the base of the method of separation of the variables in quasi-cartesian coordinates. An extended helicity operator is…

We consider the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leq…

For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to…

We consider the cubic defocusing nonlinear Schr\"odinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This weakly turbulent behavior is…

In the last three decades there has been an intense activity on the exploration of turbulent phenomena of dispersive equations, as for instance the growth of Sobolev norms since the work of Bourgain in the 90s. In general the 1D cubic…