2016/09/01 by Victor Matveevich Buchstaber, Buchstaber, Victor M., Alexey Glutsyuk +1 · 1 citation
Materials Science · Physics and Astronomy · #33C10 #34M05 #Dynamical Systems (math.DS) #FOS: Mathematics #Magnetism in coordination complexes #Nonlinear Photonic Systems #Physics of Superconductivity and Magnetism
paper · pdf · doi:10.48550/arxiv.1609.00244
openalex publication_date 2016/09/01 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We study a family of double confluent Heun equations of the form mathcal\nE=0, where mathcal L= mathcal L\λ,\μ,n is a family of differential\noperators of order two. They depend on complex parameters \λ, \μ,\nn. Its restriction to real parameter domain \λ+\μ2>0 is a\nlinearization of the family of nonlinear equations on two-torus modeling the\nJosephson effect in superconductivity. We show that for every b,n\∈ mathbb C\nsatisfying a certain "non-resonance condition" and every \λ,\μ\∈ mathbb\nC, \μ\≠0 there exists an entire function f\±: mathbb C\→ mathbb C\n(unique up to constant factor) such that z-b mathcal L(zb\nf\±(z\±1))=d0\±+d1\±z for some d0\±,d1\±\∈ mathbb\nC. The constants dj,\± are expressed as functions of the parameters.\nThis result has several applications. First of all, it gives the description of\nthose \λ, \μ, n, b for which the monodromy of the Heun equation\nhas eigenvalue e2\π i b. It also describes those \λ, \μ, n\nfor which the monodromy is parabolic: has multiple eigenvalue. We consider the\nrotation number \ρ of the dynamical system on two-torus as a function of\nparameters restricted to a surface \λ+\μ2=const. The phase-lock areas\nare its level sets having non-empty interiors. For general families of\ndynamical systems the problem to describe the boundaries of the phase-lock\nareas is known to be very complicated. Here we include the results in this\ndirection obtained by methods of complex variables. In our case the phase-lock\nareas exist only for integer rotation numbers (quantization effect). The result\non parabolic monodromy implies the description of the union of their boundaries\nby an explicit functional equation. For every \θ\∉ mathbb Z we get a\ndescription of the set \ρ\≡\±\θ(mod2 mathbb Z) .\n