1,721,030 research outputs found
Reducible quasi-periodic solutions for the non linear Schrödinger equation
Full text linkThe present paper is devoted to the construction of small reducible quasi-periodic solutions for the completely resonant NLS equations on a d-dimensional torus Td. The main point is to prove that the normal form is reducible, block diagonal and satisifies the second Melnikov conditon block wise. From this we deduce the result by a KAM algorithm
About Linearization of Infinite-Dimensional Hamiltonian Systems
No full textThis article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of its analytic higher order perturbations. We first define the subset of elements which are formally symplectically conjugated to a (formal) Birkhoff normal form. We prove that if the quadratic Hamiltonian satisfies a Diophantine-like condition and if such a perturbation is formally symplectically conjugated to the quadratic Hamiltonian, then it is also analytically symplectically conjugated to it. Of course what is an analytic symplectic change of variables depends strongly on the choice of the phase space. Here we work on periodic functions with Gevrey regularity
Quasi-periodic oscillations for wave equations under periodic forcing
Full text linkExistence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number
Quasi-periodic solutions of completely resonant forced wave equations
No full textWe prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases when the forcing frequency is: (Case A) a rational number and (Case B) an irrational number
Linear Schrödinger equation with an almost periodic potential
Full text linkWe study the reducibility of a linear Schrodinger equation subject to a small unbounded almost periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analyticity, and on the frequency of the almost periodic perturbation, we prove that such an equation is reducible to constant coefficients via an analytic almost periodic change of variables. This implies control of both Sobolev and analytic norms for the solution of the corresponding Schrödinger equation for all times
A note on the construction of Sobolev almost periodic invariant tori for the 1d NLS
Full text linkWe announce a method for the construction of almost periodic solutions of the one dimensional analytic NLS with only Sobolev regularity both in time and space. This is the first result of this kind for PDEs
Exponential and sub-exponential stability times for the NLS on the circle
No full textIn this note we study stability times for a family of parameter dependent nonlinear Schrodinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we state a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively. Complete proofs are given elsewhere (see [BMP18])
An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS
Full text linkWe study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively
Exponential and sub-exponential stability times for the NLS on the circle
Full text linkIn this note we study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we state a rather flexible Birkho¤ Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively. Complete proofs are given elsewhere (see [BMP18])
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