Собственные функции монодромии уравнений Гойна и границы зон фазового захвата в модели сильношунтированного эффекта Джозефсона

Труды Математического института им. В.А. Стеклова РАН. 2017. Т. 297. С. 62–104.

Бухштабер В. М., Glutsyuk A.

Abstract—We study a family of double confluent Heun equations of the form LE = 0, where
L = L(λ,μ,n) is a family of second-order differential operators acting on germs of holomorphic
functions of one complex variable. They depend on complex parameters λ, μ, and n. The
restriction of the family to real parameters satisfying the inequality λ + μ^2>0 is a linearization
of the family of nonlinear equations on the two-torus that model the Josephson effect in
superconductivity. The main result of the paper gives the description of those values λ, μ, n, and b for which the monodromy
operator of the corresponding Heun equation has eigenvalue exp(2πib). It also gives the description
of those values λ, μ, and n for which the monodromy is parabolic, i.e., has a multiple eigenvalue.
We consider the rotation number ρ of the dynamical system on the two-torus as a function of
parameters restricted to a surface λ + μ^2 = const. The phase-lock areas are its level sets with
nonempty interior. For general families of dynamical systems, the problem of describing the
boundaries of the phase-lock areas is known to be very complicated. In the present paper
we include the results in this direction that were obtained by methods of complex variables.
In our case the phase-lock areas exist only for integer rotation numbers (quantization effect),
and their complement is an open set. On their complement the rotation number function is
an analytic submersion that induces its fibration by analytic curves. The above-mentioned
result on parabolic monodromy implies the explicit description of the union of boundaries of
the phase-lock areas as solutions of an explicit transcendental functional equation. For every non-integer
θ  we get a description of the set {ρ ≡ ±θ (mod 2Z)}.

Keywords: эффект Джозефсона, monodromy, монодромия, Josephson effect, перемычки, число вращения, rotation number, динамические системы на торе, double confluent Heun equation, dynamical systems on torus, phase-lock areas, constrictions, зоны фазового захвата, дважды конфлюэнтное уравнение Гойна