相关论文: Birkhoff normal form for the periodic Toda lattice
This article describes a normal form algorithm for the Brieskorn lattice of an isolated hypersurface singularity. It is the basis of efficient algorithms to compute the Bernstein-Sato polynomial, the complex monodromy, and Hodge-theoretic…
In this paper we prove a KAM theorem in infinite dimension which treats the case of multiple eigenvalues (or frequencies) of finite order. More precisely, we consider a Hamiltonian normal form in infinite dimension:\begin{equation}…
The discrete Toda lattice preserves the eigenvalues of tridiagonal matrices, and convergence of dependent variables to the eigenvalues can be proved under appropriate conditions. We show that the ultradiscrete Toda lattice preserves…
We apply recent results on semi-classical trace formulae and on Birkhoff normal forms for semi-classical Fourier integral operators to a wide range of semi-classical and high energy spectral inverse problems.
We consider an undamped nonlinear hinged-hinged beam with stretching nonlinearity as an infinite dimensional hamiltonian system. We obtain analytically a quantitative Birkhoff Normal Form, via a nonlinear coordinate transformation that…
We study an initial value problem for the Toda lattice with almost periodic initial data. We consider initial data for which the associated Jacobi operator is absolutely continuous and has a spectrum satisfying a Craig-type condition, and…
Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual…
The n-particle periodic Toda chain is a well known example of an integrable but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold singularities of the Toda chain, ie points where there exist k independent linear…
In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in $H^{s}_{0}(\T, \R)$ for any $s>-1/2$ where $H^{s}_{0}(\T, \R)$ denotes the subspace of the Sobolev space…
In this paper we prove that the Benjamin-Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: it admits global Birkhoff coordinates on the space $L^2(\T)$. These are…
We obtain a parametric normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov--Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer…
Bigraded Toda hierarchy $L_1^M(n)=L_2^N(n)$ is generalized to $L_1^M(n)=L_2^{N}(n)+\sum_{j\in \mathbb Z}\sum_{i=1}^{m}q^{(i)}_n\Lambda^jr^{(i)}_{n+1}$, which is the analogue of the famous constrained KP hierarchy $L^{k}=…
In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schr{\"o}dinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even dimensional riemannian manifold M. We use this normal…
In this article, we consider solutions starting close to some linearly stable invariant tori in an analytic Hamiltonian system and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The…
In this short review we compare different ways to construct solutions of the periodic Toda lattice. We give two recipes that follow from the projection method and compare them with the algebra-geometric construction of Krichever.
The vanishing of the divergence of the total stress tensor (magnetic plus kinetic) in a neighborhood of an equilibrium plasma containing a toroidal surface of discontinuity gives boundary and jump conditions that strongly constrain…
By means of a recent Birkhoff-Kellogg type theorem, we discuss the solvability of a fairly general class of parameter-dependent fourth order retarded differential equations subject to functional boundary conditions. We seek solutions within…
We introduce a new integrable hierarchy of nonlinear differential-difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the…
Consider the Kirchhoff equation $$ \partial_{tt} u - \Delta u \Big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus $\mathbb{T}^d$. In a previous paper we proved that, after a first step of quasilinear normal…
Results on the finite nonperiodic Toda lattice are extended to some generalizations of the system: The relativistic Toda lattice, the generalized Toda lattice associated with simple Lie groups and the full Kostant-Toda lattice. The areas…