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 $\mu\neq0$ this happens exactly when $l\in\mathbb N$ and the parameters $(\lambda,\mu)$ lie on an algebraic curve $\Gamma_l\subset\mathbb C^2_{(\lambda,\mu)}$ called the $l$-th spectral curve and defined as zero locus of determinant of a remarkable three-diagonal $l\times l$-matrix. They studied the real spectral curves and obtained important results with applications to phase-lock areas in model of Josephson junction, which is a family of dynamical systems on 2-torus. In the present paper we prove irreducibility of complex spectral curves. We also calculate their genera for $l\leqslant20$ and present a conjecture on general genus formula. We apply the irreducibility result to the phase-lock areas, which are those level sets of the rotation number function $\rho$ on the parameter space of the above-mentioned family of dynamical systems that have non-empty interiors. The family of their boundaries is a countable union of analytic surfaces. We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four irreducible components, and we describe them. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits and a partial positive result towards its confirmation.