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Nonexistence of Vortices for Rotating Two-Component Focusing Bose Gases
Authors
Yongshuai Gao
Yujin Guo
Yan Li
Yong Luo
Publication date
27 November 2022
Publisher
View
on
arXiv
Abstract
This paper is concerned with ground states of two-component Bose gases confined in a harmonic trap
V
(
x
)
=
x
1
2
+
Ξ
2
x
2
2
V(x)=x_1^2+\Lambda^2 x_2^2
V
(
x
)
=
x
1
2
β
+
Ξ
2
x
2
2
β
rotating at the velocity
Ξ©
>
0
\Omega >0
Ξ©
>
0
, where
Ξ
β₯
1
\Lambda\ge 1
Ξ
β₯
1
and
(
x
1
,
x
2
)
β
R
2
(x_1, x_2)\in R^2
(
x
1
β
,
x
2
β
)
β
R
2
. We focus on the case where the intraspecies interaction
(
β
a
1
,
β
a
2
)
(-a_1,-a_2)
(
β
a
1
β
,
β
a
2
β
)
and the interspecies interaction
β
Ξ²
-\beta
β
Ξ²
are both attractive, i.e,
a
1
,
a
2
a_1, a_2
a
1
β
,
a
2
β
and
Ξ²
\beta
Ξ²
are all positive. It is shown that for any
0
<
Ξ©
<
Ξ©
β
:
=
2
0<\Omega <\Omega ^*:=2
0
<
Ξ©
<
Ξ©
β
:=
2
, ground states exist if and only if
0
<
a
1
,
β
a
2
<
a
β
:
=
β₯
w
β₯
2
2
0<a_1,\, a_2<a^*:=\|w\|^2_2
0
<
a
1
β
,
a
2
β
<
a
β
:=
β₯
w
β₯
2
2
β
and
00
00
00
is the unique positive solution of
β
Ξ
w
+
w
β
w
3
=
0
-\Delta w+ w-w^3=0
β
Ξ
w
+
w
β
w
3
=
0
in
R
2
R^2
R
2
. By developing the argument of refined expansions, we further prove the nonexistence of vortices for ground states as
Ξ²
β
Ξ²
β
\beta\nearrow\beta^*
Ξ²
β
Ξ²
β
, where
0
<
Ξ©
<
Ξ©
β
0<\Omega <\Omega ^*
0
<
Ξ©
<
Ξ©
β
and
0
<
a
1
,
β
a
2
<
a
β
0<a_1,\, a_2<a^*
0
<
a
1
β
,
a
2
β
<
a
β
are fixed.Comment: 59 pages, 1 figure, and all comments are welcom
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oai:arXiv.org:2211.14808
Last time updated on 30/12/2022