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

Micro(RNA) Management and Mismanagement of the Islet.

Pancreatic β-cells located within the islets of Langerhans play a central role in metabolic control. The main function of these cells is to produce and secrete insulin in response to a rise in circulating levels of glucose and other nutrients. The release of insufficient insulin to cover the organism needs results in chronic hyperglycemia and diabetes development. β-cells insure a highly specialized task and to efficiently accomplish their function they need to express a specific set of genes. MicroRNAs (miRNAs) are small noncoding RNAs and key regulators of gene expression. Indeed, by partially pairing to specific sequences in the 3' untranslated regions of target mRNAs, each of them can control the translation of hundreds of transcripts. In this review, we focus on few key miRNAs controlling islet function and discuss: their differential expression in Type 2 diabetes (T2D), their regulation by genetic and environmental factors, and their therapeutic potential. Genetic and epigenetic changes or prolonged exposure to hyperglycemia and/or hyperlipidemia can affect the β-cell miRNA expression profile, resulting in impaired β-cell function and survival leading to the development of T2D. Experimental approaches permitting to correct the level of misexpressed miRNAs have been shown to prevent or treat T2D in animal models, suggesting that these small RNAs may become interesting therapeutic targets. However, translation of these experimental findings to the clinics will necessitate the development of innovative strategies allowing safe and specific delivery of compounds modulating the level of the relevant miRNAs to the β-cells

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

Nonlinear propagation of light in Dirac matter

The nonlinear interaction between intense laser light and a quantum plasma is modeled by a collective Dirac equation coupled with the Maxwell equations. The model is used to study the nonlinear propagation of relativistically intense laser light in a quantum plasma including the electron spin-1/2 effect. The relativistic effects due to the high-intensity laser light lead, in general, to a downshift of the laser frequency, similar to a classical plasma where the relativistic mass increase leads to self-induced transparency of laser light and other associated effects. The electron spin-1/2 effects lead to a frequency up- or downshift of the electromagnetic (EM) wave, depending on the spin state of the plasma and the polarization of the EM wave. For laboratory solid density plasmas, the spin-1/2 effects on the propagation of light are small, but they may be significant in super-dense plasma in the core of white dwarf stars. We also discuss extensions of the model to include kinetic effects of a distribution of the electrons on the nonlinear propagation of EM waves in a quantum plasma.Comment: 9 pages, 2 figure