Ta’rif.
( )
( )
2
u z
sh D
∈ −
funksiya
3
D
⊂
sohada maksimal deyiladi,
agarda u uchun
2 sh
−
funksiyalar sinfida ustunlik prinsipi o‘rinli bo‘lsa,
ya’ni agar
( )
( ) ( )
(
)
2
: lim
0
z
D
v
sh D
u z
v z
→∂
∈ −
−
≥
bo‘lsa, u holda
z
D
∀ ∈
uchun
( ) ( )
u z
v z
≥
.
Teorema.
( )
( )
( )
2
2
u z
sh D
C
D
∈ −
funksiya maksimal bo‘lishi
uchun
0
c
dd u
β
∧ =
bo‘lishi zarur va yetarli.
( )
{
}
2
2
2
1
1
2
2
3
3
2
u z
a z
a z
a z
sh
=
+
+
∈ −
funksiyalar sinfini qaraymiz.
Bu funksiyalar sinfiga tegishli bo‘lgan
( )
2
2
2
1
1
2
2
3
3
u z
a z
a z
a z
=
+
+
funk-
siya ta’rifidagi shartlarni qanoatlantiradi. Demak, bu funksiya uchun
0
c
dd u
β
∧ ≥
bo‘ladi.
(1), (2) va
1
1
,
2
2
j
j
j
j
j
j
u
u
u
u
u
u
i
i
z
x
y
x
y
z
∂
∂
∂
∂
∂
∂
=
−
=
+
∂
∂
∂
∂
∂
∂
ifodalarga ko‘ra
( )
2
2
2
1
1
2
2
3
3
u z
a z
a z
a z
=
+
+
funksiya uchun
(
)
1
1
1
2
2
2
3
3
3
2
c
i
dd u
a dz
d z
a dz
d z
a dz
d z
=
∧
+
∧
+
∧
va
(
)
1
1
2
2
3
3
2
i
dz
d z
dz
d z
dz
d z
β
=
∧
+
∧
+
∧
ekanligini topamiz. Endi
c
dd u
β
∧
ni hisoblaymiz.
(
)
(
)
2
1
1
1
2
2
2
3
3
3
1
1
2
2
3
3
1
1
1
1
1
2
2
2
1
1
3
3
3
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
2
2
1
1
1
3
3
4
1
(
4
c
i
dd u
a dz
d z
a dz
d z
a dz
d z
dz
d z
dz
d z
dz
d z
a dz
d z
dz
d z
a dz
d z
dz
d z
a dz
d z
dz
d z
a dz
d z
dz
d z
a dz
d z
dz
d z
a dz
d z
dz
d z
a dz
d z
dz
d z
a
β
∧ =
∧
+
∧
+
∧
∧
∧
∧
+
∧
+
∧
= −
∧
∧
∧
+
+
∧
∧
∧
+
∧
∧
∧
+
+
∧
∧
∧
+
∧
∧
∧
+
+
∧
∧
∧
+
∧
∧
∧
+
+
2
2
2
3
3
3
3
3
3
3
1
2
1
1
2
2
2
3
2
2
3
3
1
3
1
1
3
3
)
1
[(
)
(
)
4
(
)
]
dz
d z
dz
d z
a dz
d z
dz
d z
a
a dz
d z
dz
d z
a
a dz
d z
dz
d z
a
a dz
d z
dz
d z
∧
∧
∧
+
∧
∧
∧
=
= −
+
∧
∧
∧
+
+
∧
∧
∧
+
+
+
∧
∧
∧
102
Quyidagi
(
) (
) (
) (
)
2
,
1, 2,3.
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
dz
d z
d x
iy
d x
iy
dx
idy
dx
dy
idx
dy
idy
dx
idx
dy
j
∧
=
+
∧
−
=
+
∧
−
=
= −
∧
+
∧
= −
∧
=
tenglikdan foydalansak,
(
)
(
)
(
)
1
2
1
1
2
2
2
3
2
2
3
3
1
3
1
1
3
3
c
dd u
a
a dx
dy
dx
dy
a
a dx
dy
dx
dy
a
a dx
dy
dx
dy
β
∧ =
+
∧
∧
∧
+
+
∧
∧
∧
+
+
+
∧
∧
∧
(3)
bo‘lishini topamiz.
0
c
dd u
β
∧ ≥
bo‘lishi uchun (3) ifodadagi barcha koeffitsientlar mus-
bat bo‘lishi kerak, ya’ni
1
2
2
3
1
3
0
0
0
a
a
a
a
a
a
+
≥
+
≥
+
≥
(4)
Demak,
( )
{
}
2
2
2
1
1
2
2
3
3
2
u z
a z
a z
a z
sh
=
+
+
∈ −
funksiyalar sinfida
1
2
3
,
,
a a a
koeffitsientlar uchun (4) tengsizliklar o‘rinli. Bu sinfda koeefit-
sientlar yordamida tuzilgan
(
)(
)(
)
1
2
2
3
1
3
a
a
a
a
a
a
ℵ =
+
+
+
operatorni qaray-
miz.
Lemma.
( )
{
}
2
2
2
1
1
2
2
3
3
2
u z
a z
a z
a z
sh
=
+
+
∈ −
funksiyalar sinfida
0
ℵ ≥
bo‘ladi.
Isbot. Biz yuqorida
( )
{
}
2
2
2
1
1
2
2
3
3
2
u z
a z
a z
a z
sh
=
+
+
∈ −
funksiyalar
sinfida
2
sh
−
lik ta’rifiga ko‘ra, har doim (4) tengsizliklar o‘rinli ekan-
ligini topdik. Demak, bu sinfda
(
)(
)(
)
1
2
2
3
1
3
a
a
a
a
a
a
ℵ =
+
+
+
operator mus-
bat bo‘ladi.
Teorema.
( )
{
}
2
2
2
1
1
2
2
3
3
2
u z
a z
a z
a z
sh
=
+
+
∈ −
funksiyalar sinfidan
olingan
( )
u z
funksiya uchun
0
ℵ =
bo‘lsa, u holda u funksiya maksimal
funksiya bo‘ladi.
Adabiyotlar:
1. A.Sadullayev. Ko‘p argumentli golomorf funksiyalar, Urganch, 2004.
2. B.Abdullaev, A.Sadullaev, Potential theory in the class of
m
sh
−
functions. Proc. Steklov Inst. Math, 2012.
3. Le Mau Hai, Nguyen Xuan Hong, Maximal
q
-Subharmonicity in
n
,
Vietnam J. Math, 2013.
103
THE PERIODIC HARRY-DYM EQUATION WITH A SOURCE
G.Urazboev (Dr.Sc.), A.Babadjanova (PhD), I.Matmuratov (Master student,
Urgench State University)
Introduction. Harry-Dym and Martin Kruskal introduced Harry-
Dym equation as a solvable evolution equation by a spectral problem
founded on the string equation [1], moreover, they rediscovered in more
general form in the works [ 2-3 ].
In the research [4], one gap solutions of the Harry-Dym equation
has been studied, debated the geometrical features of the Harry-Dym
equation and investigated to the related string equation. Harry-Dym equ-
ation can be related to the Saffman-Taylor problem in hydrodynamics
[5]. Finite-gap densities of the acoustic operator and the suitable perio-
dic solutions of the Harry-Dym equation were studied in [6-8], while the
finite-gap densities of the acoustic operator are connected with geode-
sics on the ellipsoid [9‒12]. In the paper [13], the problem of expressing
the possible spectra of the acoustic operator with a periodic finite-gap
density has been considered and the flow preserving the periods of the
corresponding operator has been built.
The Harry-Dym equation with an integral type source has been
integrated in the work [14].
In the paper [15] the direct and inverse scattering problem for the
given below
2
2
( , )
( ) ( , ),
x
q x
x
k
φ
ξ
ξ
φ ξ
ξ
′′
= −
=
equation has been investigated and the periodic Harry-Dym equation
without source has been integrated.
In this work we have taken into account the following
N
- band
potential periodic Harry-Dym equation with a source
(
)
2
2
2
2
2
2
0
1
(
( , ))
2(1 / ( , ))
2
( , ,
)
( , )
( , ,
)
( , )
(0, )
N
t
xxx
x
n
n
x
n
q x t
q x t
f
x t
q x t
f
x t
q x t
q
t
λ
λ
=
= −
+
+
∑
(1.1)
with the initial condition
0
( ,0)
( )
q x
q x
=
, (1.2)
where
( , )
q x t
is non zero function and periodic in x with period
π
for all
time
( , )
(
, )
q x t
q x
t
π
=
+
and
0
( )
q x
is given
π
periodic function. Here,
n
λ
are nils of the function
2
( )
4
λ
∆
−
and
( , , )
f x t
λ
is Floquet-Bloch solution
for the following equation
2
2
( , , )
( , ) ( , , ),
y x t
q x t y x t
k
λ
λ
λ
λ
′′
= −
=
, (1.3)
104
which is defined by
2
( , , )
( , , )
( )
4
( , , )
( , , )
( , , )
2 ( , , )
x
s
t
c
t
f
x t
c x t
s x t
s
t
π λ
π λ
λ
λ
λ
λ
π λ
±
−
± ∆
−
=
+
.
Here,
( , , )
c x
t
λ
and
( , , )
s x
t
λ
are solutions of the equation (1.3) with the
first conditions, accordingly
(0, , ) 1,
(0, , )
0,
x
c
t
c
t
λ
λ
=
=
(0, , )
0,
(0, , ) 1
x
s
t
s
t
λ
λ
=
=
and
( )
λ
∆
is defined by
( )
( , , )
( , , )
x
c
t
s
t
λ
π λ
π λ
∆
=
+
.
Our aim is to find the solution
{ ( , ), ( , ,
)}
n
q x t f x t
λ of the considering
problem (1.1)-(1.3) via the inverse spectral technique.
Prelinimary. We consider the following eigenvalue problem
2
2
( , , )
( , ) ( , , ),
y x t
q x t y x t
k
λ
λ
λ
λ
′′
= −
=
. (2.1)
We denote the solutions of (2.1) by
(
)
, ,
c x t
λ
,
(
)
, ,
s x t
λ
which satisfy
(
)
(
)
(
)
(
)
0, ,
0, ,
1,
0, ,
0, ,
0.
x
x
c
t
s
t
t
s
t
c
λ
λ
λ
λ
=
=
=
=
(2.2)
From
(
)
( )
q x
q x
π
+
=
it is easy to show that
(
)
(
) (
)
, ,
, ,
,
Ф x
t
Ф x t
M
π λ
λ
π λ
+
=
, (2.3)
where the monodromy matrix
(
)
,
M
π λ
is defined by
(
)
(
)
,
, ,
M
Ф
t
π λ
π λ
=
,
(
)
(
)
(
)
(
)
(
)
, ,
, ,
, ,
, ,
, ,
x
x
c x t
s x t
Ф x t
c
x t
t
s
x
λ
λ
λ
λ
λ
=
.
According to the equalities (2.2), for monodromy matrix
(
)
,
M
π λ
hold
that
det
( , ) 1
M
π λ
=
.
The spectrum of the equation (2.1) is real and coincide with the set
2
1
{ :
( )
4
0}
j
j
J
λ
λ
∞
=
∆
− <
=
and the intervals
2
1
2
,
[
]
j
j
j
I
λ
λ
−
=
are called the gaps
or lacunas. The zeros
( )
,
1, 2,...
j
j
t
j
I
ξ
=
∈
of
(
)
, ,
s
t
π λ
are real.
Definition. The numbers of
( )
,
1, 2,...
j
j
t
j
I
ξ
=
∈
and the signs
( )
(
, ,
,
(
)
, ))
(
j
j
x
j
j
t
sign s
t
c
t
σ
σ
π ξ
π ξ
=
=
−
are called spectral parameter of the
equation (2.1). The spectral parameter and the spectrum bounds
j
λ
,
1, 2,...
j
=
are called spectral data of problem (2.1).
When the potential
( )
q x
is the
N
—band potential case the spectrum
of the equation (2.1) doesn’t depend on the real parameter
t
, but for the
105
time evolution of the spectral parameters hold the following systems of
equation [15]
(
)
2
0
( ) ( )
(
8
,
1,
(
)
( )
( )
( ))
N
j
j
j
j
n
j
i
i j
n
d
j
N
d
t
t
t
t
t
t
t
σ
ξ
ξ
λ
ξ
ξ
ξ
=
≠
−
−
=
=
−
∏
∏
,
which is called the system of equations Dubrovin. Using the system of
equations Dubrovin and the following trace formula:
(
)
2
0
( )
( )
( )
(
(
)
( )
)
2
( )
j
i
i j
j
N
j
j
n
n
t
t
t
t
d
q t
dt
i
t
ξ
ξ
ξ
σ
ξ
λ
≠
=
−
=
−
∏
∏
,
it will be available to solve the periodic Harry-Dym equation without
source.
|
