Characteristic exponents of Mathieu functions

Abstract

The study of the properties and solutions of the Mathieu second-order differential equation has been very extensive in the last decades, due to the special interest presented by physical problems involving periodic media and problems separable in elliptic coordinate systems, On the other hand, numerical tables for the non-periodic Floquet-type solutions are scarce [5], [6], [7]; in addition, the increments between two successive tabulated values are relatively large for some of these tables and the accuracy is relatively low in the other tables. Consequently, the applicability of the available data is rather limited.The regions of "stable" solutions of the Mathieu equation are of particular interest in applications involving the wave equation with periodic boundary conditions, since these regions correspond to the propagating waves in the corresponding media. The first three regions of stable solutions were tabulated with this type of problem in mind, the numerical results being given in Tables I, II, and III. The canonical form of the Mathieu equation considered is given by(1) y" + (p -2q cos 2z)y = 0, with p and q real. Every non-periodic solution of (1) is a linear combination of two linearly independent Floquet-type solutions of the form (2) y = eir°P(z),

Numerical Analysis of the Transient Capture Dynamics of Ions by Radio Frequency

“…When Urf is large, the growth rate of Emax and Emean is significantly accelerated, and the value of Emax is higher than Dr when Urf = 600 V, indicating that the significant increase of Emax and Emean makes ions tend to be unstable. Combined with the first stable confinement region of ions [12], the simulation results show that when Urf = 725V, q = 0.88, basically close to the upper limit of the first stable confinement parameter, and the simulation results are consistent with the theoretical values, which verifies the effectiveness of the finite element numerical simulation.…”

Dirac materials under linear polarized light: quantum wave function time evolution and topological Berry phases as classical charged particles trajectories under electromagnetic fields

“…This solution is meant to be compared with equation (30), showing that the second term is a projection of the time evolution into the high-energy state. Notice that the Mathieu functions C a k , − q1 2 , ϕ and S a k , − q1 2 , ϕ already contain in their definition the integral in time of the quasi-energy, as they have the general form e iµ(a k ,−q1/2,ϕ) F(a k , −q 1 /2, ϕ), where F(a k , −q 1 /2, ϕ) is a polynomial and µ(a k , −q 1 /2, ϕ) is the Floquet exponent [35,48]. The small projection of the solution into the high energy state can be seen in figures 3(e) and (f) as an orbital precession due to the two frequencies involved.…”

On calculating the eigenvalues of the finite Hill's differential equation

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