论文标题
关于过度阻尼约瑟夫森连接和painlevé3方程的模型中收缩的家族
On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation
论文作者
论文摘要
B.Josephson预测的隧道效应（诺贝尔奖，1973年）涉及约瑟夫森交界处：两个由狭窄介电隔开的超导体。它说明了通过它的超电流的存在，并控制了它的方程式。过度阻尼的约瑟夫森交界处是由2个螺纹的一个微分方程家族建模的，具体取决于3个参数：$ b $（abscissa），$ a $（坐标），$ω$（频率）。我们研究其旋转号$ρ（b，a;ω）$作为$（b，a）$带固定$ω$的函数。相锁区域是$ l_r：= \ {ρ= r \} $，带有非空内部；它们以$ r \ in \ mathbb z $（buchstaber，karpov，tertychnyi）的形式存在。每个$ L_R $都是无限的域链，垂直到无穷大，并被称为收缩的点分开（期望$ a = 0 $的人）。我们表明：1）$ l_r $中的所有收缩都在其轴上$ \ {b =ωr\} $（确认了tertychnyi的猜想，kleptsyn，kleptsyn，schurov filimonov）; 2）每个收缩都是阳性的：垂直线中的一些刺穿邻域位于$ \ permatatorName {int}（l_r）$（确认另一个猜想）中。我们首先证明，每个收缩的可变形性，其任意小$ω$，相同的$ρ$，$ \ ell：= \ fracbΩ$和类型（正面或无），使用$ \ bar {\ bar {\ nathb c} $（buchstaber，karpov and teryos and terys and ternyi and ternyi and）的模型对模型进行等效描述由Painlevé3方程式描述。然后，通过缓慢快速的方法证明，幽灵收缩的不存在（即用给定的$ \ ell $的$ρ\ neq \ ell $，或非阳性类型的收缩）证明了$ \ ell $。在第6节中，我们介绍了结果，详细方法和开放问题的应用。
The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate), $ω$ (frequency). We study its rotation number $ρ(B,A;ω)$ as a function of $(B,A)$ with fixed $ω$. The phase-lock areas are the level sets $L_r:=\{ρ=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$ (Buchstaber, Karpov, Tertychnyi). Each $L_r$ is an infinite chain of domains going vertically to infinity and separated by points called constrictions (expect for those with $A=0$). We show that: 1) all the constrictions in $L_r$ lie in its axis $\{ B=ωr\}$ (confirming a conjecture of Tertychnyi, Kleptsyn, Filimonov, Schurov); 2) each constriction is positive: some its punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$ (confirming another conjecture). We first prove deformability of each constriction to another one, with arbitrarily small $ω$, of the same $ρ$, $\ell:=\frac Bω$ and type (positive or not), using equivalent description of model by linear systems of differential equations on $\bar{\mathbb C}$ (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with $ρ\neq\ell$, or of non-positive type) with a given $\ell$ for small $ω$ is proved by slow-fast methods. In Section 6 we present applications of results and elaborated methods and open problems.
