A Kam Theorem for Space-Multidimensional Hamiltonian PDE

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 part and as a consequence the constructed invariant tori may be unstable. $\bullet$ It applies to singular perturbation problem. In this paper we state the KAM-theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem .Comment: arXiv admin note: text overlap with arXiv:1502.0226

Fractional Lindstedt series

The parametric equations of the surfaces on which highly resonant quasi-periodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter, i.e. they are not analytic functions of the perturbation parameter. However rather generally quasi-periodic motions whose frequencies satisfy only one rational relation ("resonances of order 1") admit formal perturbation expansions in terms of a fractional power of the perturbation parameter, depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.Comment: 40 pages, 6 figure

Excitation of Longitudinal Waves in a Degenerate Isotropic Quantum Plasma

A dispersion equation, which describes the interaction of low density electron beam with a degenerate electron quantum plasma, is derived and examined for some interesting cases. In addition to the instabilities similar to those for classical plasma, due to the quantum effect a new type of instability is found. Growth rates of these new modes, which are purely quantum, are obtained. Furthermore, the excitation of Bogolyubov's type of spectrum by a strong electric field is discussed.Comment: Submitted to Journal of Plasma Physics special issu

Almost reducibility for finitely differentiable SL(2,R)-valued quasi-periodic cocycles

Quasi-periodic cocycles with a diophantine frequency and with values in SL(2,R) are shown to be almost reducible as long as they are close enough to a constant, in the topology of k times differentiable functions, with k great enough. Almost reducibility is obtained by analytic approximation after a loss of differentiability which only depends on the frequency and on the constant part. As in the analytic case, if their fibered rotation number is diophantine or rational with respect to the frequency, such cocycles are in fact reducible. This extends Eliasson's theorem on Schr\"odinger cocycles to the differentiable case

Converging Perturbative Solutions of the Schroedinger Equation for a Two-Level System with a Hamiltonian Depending Periodically on Time

We study the Schroedinger equation of a class of two-level systems under the action of a periodic time-dependent external field in the situation where the energy difference 2epsilon between the free energy levels is sufficiently small with respect to the strength of the external interaction. Under suitable conditions we show that this equation has a solution in terms of converging power series expansions in epsilon. In contrast to other expansion methods, like in the Dyson expansion, the method we present is not plagued by the presence of ``secular terms''. Due to this feature we were able to prove absolute and uniform convergence of the Fourier series involved in the computation of the wave functions and to prove absolute convergence of the epsilon-expansions leading to the ``secular frequency'' and to the coefficients of the Fourier expansion of the wave function

Instability and dynamics of two nonlinearly coupled laser beams in a plasma

We investigate the nonlinear interaction between two laser beams in a plasma in the weakly nonlinear and relativistic regime. The evolution of the laser beams is governed by two nonlinear Schroedinger equations that are coupled with the slow plasma density response. We study the growth rates of the Raman forward and backward scattering instabilities as well of the Brillouin and self-focusing/modulational instabilities. The nonlinear evolution of the instabilities is investigated by means of direct simulations of the time-dependent system of nonlinear equations.Comment: 18 pages, 8 figure

Electrostatic electron cyclotron instabilities near the upper hybrid layer due to electron ring distributions

A theoretical study is presented of the electrostatic electron cyclotron instability involving Bernstein modes in a magnetized plasma. The presence of a tenuous thermal ring distribution in a Maxwellian plasma decreases the frequency of the upper hybrid branch of the electron Bernstein mode until it merges with the nearest lower branch with a resulting instability. The instability occurs when the upper hybrid frequency is somewhat above the third, fourth, and higher electron cyclotron harmonics, and gives rise to a narrow spectrum of waves around the electron cyclotron harmonic nearest to the upper hybrid frequency. For a tenuous cold ring distribution together with a Maxwellian distribution an instability can take place also near the second electron cyclotron harmonic. Noise-free Vlasov simulations are used to assess the theoretical linear growth-rates and frequency spectra, and to study the nonlinear evolution of the instability. The relevance of the results to laboratory and ionospheric heating experiments is discussed

New Precision Electroweak Tests in Supergravity Models

We update the analysis of the precision electroweak tests in terms of 4 epsilon parameters, $\epsilon_{1,2,3,b}$, to obtain more accurate experimental values of them by taking into account the new LEP data released at the 28th ICHEP (1996, Poland). We also compute $\epsilon_1$ and $\epsilon_b$ in the context of the no-scale $SU(5)\times U(1)$ supergravity model to obtain the updated constraints by imposing the correlated constraints in terms of the experimental ellipses in the $\epsilon_1-\epsilon_b$ plane and also by imposing the new bound on the lightest chargino mass, $m_{\chi^\pm_1}\gtrsim 79$ $GeV$. Upon imposing these new experimental results, we find that the situations in the no-scale model are much more favorable than those in the standard model, and if $m_t\gtrsim 170$ $GeV$, then the allowed regions at the 95% C.~L. in the no-scale model are $\tan\beta\gtrsim 4$ and $m_{\chi^\pm_1}\lesssim 120 (82)$ $GeV$ for $\mu>0 (\mu<0)$, which are in fact much more stringent than in our previous analysis. Therefore, assuming that $m_t\gtrsim 170$ $GeV$, if the lightest chargino mass bound were to be pushed up only by a few GeV, the sign on the Higgs mixing term $\mu$ in the no-scale model could well be determined from the $\epsilon_1-\epsilon_b$ constraint to be positive at the 95% C.~L. At any rate, better accuracy in the measured $m_t$ from the Tevatron in the near future combined with the LEP data is most likely to provide a decisive test of the no-scale $SU(5)\times U(1)$ supergravity model.Comment: 15 pages, REVTEX, 1 figure (not included but available as a ps file from [email protected]