Dispersion and localisation in structured Rayleigh beams

This paper brings a comparative analysis between dynamic models of couple-stress elastic materials and structured Rayleigh beams on a Winkler foundation. Although physical phenomena have different physical origins, the underlying equations appear to be similar, and hence mathematical models have a lot in common. In the present work, our main focus is on the analysis of dispersive waves, band-gaps and localised waveforms in structured Rayleigh beams. The Rayleigh beam theory includes the effects of rotational inertia which are neglected in the Euler–Bernoulli beam theory. This makes the approach applicable to higher frequency regimes. Special attention is given to waves in pre-stressed Rayleigh beams on elastic foundations.     

Quasi-periodic stability of normally resonant tori

We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts.     

Invariant Tori and Boundedness in Asymmetric Oscillations

In this paper,we are concerned with the boundedness of all the solutions of the equation x″ ax^ -bx- Ф(x)=p(t),where p(t) is a smooth 2π-periodic function,a and b are positive constants,and the perturbation Ф(x) is bounded.     

Tuning visible light generation based on multi-channel quasi-periodically poled LiTaO3

We demonstrate broadband tuning visible light generation based on a multi-channel quasi-periodically poled LiTaO₃ crystal, in which a quasi-phase matched optical parametric generation process and a quasi-phase matched sum-frequency mixing process were achieved simultaneously. The conversion characters on spectrum and energy were studied by using a nanosecond pulse laser at 1.064 μm as pump light. We could tune the visible light over ~ 26 nm by means of changing the crystal's channel and temperature. The ratio of the output wavelength variation to that of temperature was ~ 0.07 nm/°C. The single-pass slope efficiency was 6.3% with the maximum output energy of 25 μJ.     

Mathematical analysis of plasmonic resonance for 2-D photonic crystal

In this article, we study the plasmonic resonance of infinite photonic crystal mounted by the double negative nanoparticles in two dimensions. The corresponding physical model is described by the Helmholz equation with so called Bloch wave condition in a periodic domain. By using the quasi-periodic layer potential techniques and the spectral theorem of quasi-periodic Neumann–Poincaré operator, the quasi-static expansion of the near field in the presence of nanoparticles is derived. Furthermore, when the magnetic permeability of nanoparticles satisfies the Drude model, we give the conditions under which the plasmonic resonance occurs, and the rate of blow up of near field energy with respect to nanoparticle's bulk electron relaxation rate and filling factor are also obtained. It indicates that one can appropriately control the bulk electron relaxation rate or filling factor of nanoparticle in photonic crystal structure such that the near field energy attains its maximum, and enhancing the efficiency of energy utilization.     

Anderson localization for quasi-periodic CMV matrices and quantum walks

We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.     

Effect of quasi-periodic gravitational modulation on the convective instability in Hele-Shaw cell

The aim of the present paper is to examine the effect of a quasi-periodic gravitational modulation on the onset of convective instability in Hele-Shaw cell. The quasi-periodic modulation considered here consists in a modulation having two incommensurate frequencies. This study is an extension of a previous work by Aniss et al. [Asymptotic study of the convective parametric instability in Hele-Shaw cell, Phys. Fluids 12 (2) (2000) 262-268] in which only a periodic gravitational modulation was considered. We have shown that for Pr=O(1) or Pr?1, the gravitational modulation has no effect on the convective threshold as expected. However, for Pr=O(ε²), it turns out that a modulation with two incommensurate frequencies has a stabilizing or a destabilizing effect strongly depending on the frequencies ratio.     

Quasi-periodic solutions of mixed AKNS equations

A hierarchy of new nonlinear evolution equations, which are composed of the positive and negative AKNS flows, is proposed. On the basis of the theory of algebraic curves, the corresponding flows are straightened using the Abel-Jacobi coordinates. The meromorphic function ?, the Baker-Akhiezer vector , and the hyperelliptic curve K_n are introduced and, by using these, quasi-periodic solutions of the first three nonlinear evolution equations in the hierarchy are constructed according to the asymptotic properties and the algebro-geometric characters of ?, and K_n.     

Frequency modules and nonexistence of quasi-periodic solutions of nonlinear evolution equations

This paper gives lower estimates for the frequency modules of almost periodic solutions to equations of the form , where A generates a strongly continuous semigroup in a Banach space , F(t,x) is 2π-periodic in t and continuous in (t,x), and f is almost periodic. We show that the frequency module ℳ(u) of any almost periodic mild solution u of (*) and the frequency module ℳ(f) of f satisfy the estimate e ^{2π iℳ(f)}⊂e ^{2π iℳ(u)}. If F is independent of t, then the estimate can be improved: ℳ(f)⊂ℳ(u). Applications to the nonexistence of quasi-periodic solutions are also given.