On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction

Abstract

The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are \(l,\lambda ,\mu \in \mathbb {R}\). Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a polynomial solution. They have shown that for μ ≠ 0 this happens exactly when \(l\in \mathbb {N}\) and the parameters (λ, μ) lie on an algebraic curve \({\Gamma }_{l}\subset \mathbb {C}^{2}_{(\lambda ,\mu )}\) called the l-spectral curve and defined as zero locus of determinant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dynamical systems on 2-torus depending on real parameters (B, A; ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in \(\mathbb {R}^{2}_{(B,A)}\) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every \(l\in \mathbb {N}\). We also calculate its genus for \(l\leqslant 20\) and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in \(\mathbb {R}^{3}_{(B,A,\omega ^{-1})}\). We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.