1,721,002 research outputs found
Reducibility of Schrödinger Equation on the Sphere
No full textIn this article we prove a reducibility result for the linear Schrödinger equation on the sphere Sn with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order strictly less than 1/2 and satisfying some parity condition. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. Letus note that, surprisingly, our result does not require the use of the pseudodifferential calculus although the perturbation is unbounded
Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity
No full textWe consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations. The lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known "Benjamin-Feir resonances". We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables
Long time existence for fully nonlinear NLS with small Cauchy data on the circle
No full textIn this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schrödinger equations on the one dimensional torus. We show that, if some non-resonance conditions are fulfilled, for any N ∊ N and for any initial condition, which is even in x and size ε in an appropriate Sobolev space, the lifespan of the solution is of order ε-N. After a paralinearization of the equation we perform several para-differential changes of variables which diagonalize the system up to a very regularizing term. Once achieved the diagonalization, we construct modified energies for the solution by means of Birkhoff normal forms techniques
Time quasi-periodic traveling gravity water waves in infinite depth
Get PDFWe present the recent result [9] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations
Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
No full textWe prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions
Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori
No full textWe prove a local in time well-posedness result for quasi-linear Hamiltonian Schrödinger equations on Td for any d≥1. For any initial condition in the Sobolev space Hs, with s large, we prove the existence and uniqueness of classical solutions of the Cauchy problem associated to the equation. The lifespan of such a solution depends only on the size of the initial datum. Moreover we prove the continuity of the solution map
Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle
No full textWe consider a class of Hamiltonian Klein-Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small initial data of size epsilon showing that the corresponding solutions remain close to oscillatory motions over a time scale epsilon (-9/4 + delta )for any delta > 0 . The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach
Il sistema politico-giudiziario in Italia, dall'Unità all'integrazione europea
No full textLa ricerca affronta il tema dell'organizzazione della giustizia e dei rapporti tra governo e magistratura in Italia, dall'unificazione del paese e fondazione dello Stato fino alla prima crisi del sistema costituzionale repubblicano. Nello specifico il lavoro affronta, sula piano storico-istituzionale e politologico, le tematiche dell'autonomia e indipendenza della magistratura, la questione dello status e del reclutamento, i rapporti tra politica della giustizia e domanda sociale di diritti, i rapporti costituzionali tra poteri e i conflitti tra politica e magistratura
On the lifespan of solutions and control of high Sobolev norms for the completely resonant NLS on tori
No full textWe consider a completely resonant nonlinear Schrödinger equation on the d-dimensional torus, for any d≥1, with polynomial nonlinearity of any degree 2p+1, p≥1, which is gauge and translation invariant. We study the behaviour of high Sobolev Hs-norms of solutions, s≥s1+1>d/2+2, whose initial datum u0∈Hs satisfies an appropriate smallness condition on its low Hsjavax.xml.bind.JAXBElement@fd91d51 and L2-norms respectively. We prove a polynomial upper bound on the possible growth of the Sobolev norm Hs over finite but long time scale that is exponential in the regularity parameter s1. As a byproduct we get stability of the low Hsjavax.xml.bind.JAXBElement@6d08d417-norm over such time interval. A key ingredient in the proof is the introduction of a suitable “modified energy” that provides an a priori upper bound on the growth. This is obtained by combining para-differential techniques and suitable tame estimates
Non-Resonant Conditions for the Klein – Gordon Equation on the Circle
No full textWe consider the infinite-dimensional vector of frequencies, arising from a linear Klein – Gordon equation on the one-dimensional torus and prove that there exists a positive measure set of masses s for which satisfies a Diophantine condition similar to the one introduced by Bourgain in [14],in the context of the Schrödinger equation with convolution potential.The main difficulties we have to deal with arethe asymptotically linear nature of the (infinitely many) s and the degeneracy coming from having only one parameter at disposal for their modulation.As an application we provide estimates on the inverse of the adjoint action of the associated quadratic Hamiltonian on homogenenous polynomials of any degree in Gevrey category
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