Related papers: Finite dimensional invariant KAM tori for tame vec…
This paper investigates the application of KAM theory to the stochastic nonlinear Schr\"{o}dinger equation on infinite lattices, focusing on the stability of low-dimensional invariant tori in the sense of most probable paths. For…
We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schr\"odinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic…
For an integrable Hamiltonian with $d\ (d\geq 2)$ degrees of freedom, we show the conditions on perturbations, for which invariant tori can be destructed.
In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation $({\rm gSQG})_\alpha$ in the patch form close to Rankine vortices. We show that invariant tori survive when the…
In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for…
The object of the present is a proof of the existence of functorial resolution of tame quotient singularities for quasi-projective varieties over algebraically closed fields.
In this paper, we prove a KAM theorem in a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with $n$ degrees of freedom that depend periodically or quasi-periodically (QP) on…
Quasi-periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exists after small perturbations. The parametric equations of the invariant tori can often be computed…
The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract…
We study the existence of infinite-dimensional invariant tori in a mechanical system of infinitely many rotators weakly interacting with each other. We consider explicitly interactions depending only on the angles, with the aim of…
We present an abstract KAM theorem, adapted to space-multidimensional hamiltonian PDEs with smoothing non-linearities. The main novelties of this theorem are that: $\bullet$ the integrable part of the hamiltonian may contain a hyperbolic…
We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed…
The paper consists of two sections. In Section 1, we give a short review of KAM theory with an emphasis on Whitney smooth families of invariant tori in typical Hamiltonian and reversible systems. In Section 2, we prove a KAM-type result for…
Let X be a separated scheme of finite type over an algebraically closed field k and let m be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod m Suslin homology and the first…
The parametric equations of KAM tori for a quasi integrable system, are shown to be one point Schwinger functions of a suitable euclidean quantum field theory on the torus. KAM theorem is equivalent to a ultraviolet stability theorem. A…
We consider the classical Inverse Function Theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors. Geometrisation of tame estimates coupled with certain ideas coming from Variational Analysis when…
One can see deep-learning models as compositions of functions within the so-called tame geometry. In this expository note, we give an overview of some topics at the interface of tame geometry (also known as o-minimality), optimization…
In the patching setting, given a factorization inverse system of fields over which patching for finite-dimensional vector spaces holds, together with a crossed module over the inverse limit field, the corresponding six-term Mayer--Vietoris…
In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schr\"odinger equation \begin{eqnarray}\label{maineq0} \mathbf{i}u_{t}-u_{xx}+V*u+\epsilon f(x)|u|^{4}u=0,\…
In this paper, we investigate the sharp regularity properties of a special weighted Sobolev space defined on the $ n $-dimensional torus, which is of independent interest. As a key application, we show that for almost all $ n $-dimensional…