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The Cubic Szeg\h{o} Equation with a Linear Perturbation  [PDF]
Haiyan Xu
Mathematics , 2015,
Abstract: 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 $\alpha= 0$ was introduced by G{\'e}rard and Grellier as an important mathematical model [5, 7, 3]. In this paper, we continue our studies started in [22], and prove our system is completely integrable in the Liouville sense. We study the motion of the singular values of the related Hankel operators and find a necessary condition of norm explosion. As a consequence, we prove that the trajectories of the solutions will stay in a compact subset, while more initial data will lead to norm explosion in the case $\alpha>0$.
Large time blow up for a perturbation of the cubic Szeg? equation  [PDF]
Haiyan Xu
Mathematics , 2013,
Abstract: 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 with $\al=0$ was first introduced by G{\'e}rard and Grellier in \cite{GG1} as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. For $\al\textless{}0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\al\textgreater{}0$, there exist trajectories on which high Sobolev norms exponentially grow with time.
An explicit formula for the cubic Szeg? equation  [PDF]
Patrick Gérard,Sandrine Grellier
Mathematics , 2013,
Abstract: 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 operators are quasiperiodic.
The Szeg? Cubic Equation  [PDF]
Patrick Gérard,Sandrine Grellier
Mathematics , 2009,
Abstract: 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 equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.
Invariant tori for the cubic Szeg? equation  [PDF]
Patrick Gérard,Sandrine Grellier
Mathematics , 2010,
Abstract: 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 evolution equations. In a previous work, we proved that this equation admits a Lax pair, and that it is completely integrable. In this paper, we construct the action-angle variables, which reduces the explicit resolution of the equation to a diagonalisation problem. As a consequence, we solve an inverse spectral problem for Hankel operators. Moreover, we establish the stability of the corresponding invariant tori. Furthermore, from the explicit formulae, we deduce the classification of orbitally stable and unstable traveling waves.
On the Radius of Analyticity of Solutions to the Cubic Szeg? Equation  [PDF]
Patrick Gerard,Yanqiu Guo,Edriss S. Titi
Mathematics , 2013,
Abstract: 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} projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time $t\in (-\infty,\infty)$. In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the $\ell^1$ norm of Fourier transforms (the Wiener algebra).
Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szeg? equation  [PDF]
Tadahiro Oh
Mathematics , 2010,
Abstract: We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on H^s(T), s > s_0 where s_0 = -11/6 + \sqrt{61}/6 \thickapprox -0.5316 < -1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the integration formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szeg\"o equation under randomization of initial data.
Clustering phenomena for linear perturbation of the Yamabe equation  [PDF]
Angela Pistoia,Giusi Vaira
Mathematics , 2015,
Abstract: Let $(M,g)$ be a non-locally conformally flat compact Riemannian manifold with dimension $N\ge7.$ We are interested in finding positive solutions to the linear perturbation of the Yamabe problem $$-\mathcal L_g u+\epsilon u=u^{N+2\over N-2}\ \hbox{in}\ (M,g) $$ where the first eigenvalue of the conformal laplacian $-\mathcal L_g $ is positive and $\epsilon$ is a small positive parameter. We prove that for any point $\xi_0\in M$ which is non-degenerate and non-vanishing minimum point of the Weyl's tensor and for any integer $k$ there exists a family of solutions developing $k$ peaks collapsing at $\xi_0$ as $\epsilon$ goes to zero. In particular, $\xi_0$ is a non-isolated blow-up point.
Explicit formula for the solution of the Szeg? equation on the real line and applications  [PDF]
Oana Pocovnicu
Mathematics , 2010,
Abstract: 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, 01/2. 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.
Derivation of the Cubic Non-linear Schr?dinger Equation from Quantum Dynamics of Many-Body Systems  [PDF]
Laszlo Erdos,Benjamin Schlein,Horng-Tzer Yau
Mathematics , 2005, DOI: 10.1007/s00222-006-0022-1
Abstract: We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schr\"odinger equation in a suitable scaling limit. The result is extended to $k$-particle density matrices for all positive integer $k$.
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