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On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlev\'e 3 equation
Authors
Alexey Glutsyuk
Publication date
8 August 2023
Publisher
View
on
arXiv
Abstract
B.Josephson (Nobel Prize, 1973) predicted tunnelling effect for a system (called Josephson junction) of two superconductors separated by a narrow dielectric: 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
B
B
,
A
A
A
,
Ο
\omega
Ο
. We study its rotation number
Ο
(
B
,
A
;
Ο
)
\rho(B,A;\omega)
Ο
(
B
,
A
;
Ο
)
as a function of parameters. The three-dimensional phase-lock areas are the level sets
L
r
:
=
{
Ο
=
r
}
L_r:=\{\rho=r\}
L
r
β
:=
{
Ο
=
r
}
with non-empty interiors; they exist for
r
β
Z
r\in\mathbb Z
r
β
Z
(Buchstaber, Karpov, Tertychnyi). For every fixed
Ο
>
0
\omega>0
Ο
>
0
and
r
β
Z
r\in\mathbb Z
r
β
Z
the planar slice
L
r
β©
(
R
B
,
A
2
Γ
{
Ο
}
)
L_r\cap(\mathbb R^2_{B,A}\times\{\omega\})
L
r
β
β©
(
R
B
,
A
2
β
Γ
{
Ο
})
is a garland of domains going vertically to infinity and separated by points; those separating points for which
A
β
0
A\neq0
A
ξ
=
0
are called constrictions. In a joint paper by Yu.Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa
β
:
=
B
Ο
\ell:=\frac B\omega
β
:=
Ο
B
β
is equal to
Ο
\rho
Ο
; 2) the family of constrictions with given
β
β
Z
\ell\in\mathbb Z
β
β
Z
is an analytic submanifold
C
o
n
s
t
r
β
Constr_\ell
C
o
n
s
t
r
β
β
in
(
R
+
2
)
a
,
s
(\mathbb R^2_+)_{a,s}
(
R
+
2
β
)
a
,
s
β
,
a
=
Ο
β
1
a=\omega^{-1}
a
=
Ο
β
1
,
s
=
A
Ο
s=\frac A\omega
s
=
Ο
A
β
. Here we show that the limit points of
C
o
n
s
t
r
β
Constr_\ell
C
o
n
s
t
r
β
β
are
Ξ²
β
,
k
=
(
0
,
s
β
,
k
)
\beta_{\ell,k}=(0,s_{\ell,k})
Ξ²
β
,
k
β
=
(
0
,
s
β
,
k
β
)
, where
s
β
,
k
>
0
s_{\ell,k}>0
s
β
,
k
β
>
0
are zeros of the Bessel function
J
β
(
s
)
J_\ell(s)
J
β
β
(
s
)
, and it lands at them regularly. Known numerical pictures show that high components of
I
n
t
(
L
r
)
Int(L_r)
I
n
t
(
L
r
β
)
look similar. In his paper with Bibilo, the author introduced a candidate to the self-similarity map between neighbor components: the Poincar\'e map of the dynamical isomonodromic foliation governed by Painlev\'e 3 equation. Whenever well-defined, it preserves
Ο
\rho
Ο
. We show that the Poincar\'e map is well-defined on a neighborhood of the plane
{
a
=
0
}
β
R
β
,
a
2
Γ
(
R
+
)
s
\{ a=0\}\subset\mathbb R^2_{\ell,a}\times(\mathbb R_+)_s
{
a
=
0
}
β
R
β
,
a
2
β
Γ
(
R
+
β
)
s
β
, and it sends
Ξ²
β
,
k
\beta_{\ell,k}
Ξ²
β
,
k
β
to
Ξ²
β
,
k
+
1
\beta_{\ell,k+1}
Ξ²
β
,
k
+
1
β
for integer
β
\ell
β
.Comment: 41 page, 5 figure
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oai:arXiv.org:2308.04310
Last time updated on 12/08/2023