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PureMathematicsnØêÆ,2022,12(3),434-440
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.123048
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©Ù̇ézÄ{nØ?1\ïħ‰Ñ(1+1)ÀÚ-¤’·Ü‡6íNÅ.§
‰ÑÙš‚ 5Å•§)Û)§¿?ØfƒpŠ^1•"ÀÚ-¤’·ÜíN¥‘ÔŸ
ÅóÀ§ •)‚5Úš‚5§± 93N5NX• ›^‡¥$^zÄ{?1OŽ§
˜‡ÍÜKdV•§§?˜ÚïÄÚ?ØÙ•§ŒÈ5"
'…c
ÀÚ-¤’·ÜíN§f)§zÄ{
OntheIntegrabilityofNonlinearMixed
GasEquations
YangjieJia
DepartmentofMathematics,NationalitiesCollegeofQinghaiNormalUniversity,XiningQinghai
Received:Feb.16
th
,2022;accepted:Mar.18
th
,2022;published:Mar.25
th
,2022
Abstract
Inthispaper,thetheoryofreducedperturbationmethodiswellinvestigated.The
©ÙÚ^:\#.š‚5·ÜíN•§ŒÈ5¦)[J].nØêÆ,2022,12(3):434-440.
DOI:10.12677/pm.2022.123048
\#
solitarywavemodelof(1+1)Bose-Fermimixed superfluidgasisgiven,theanalytical
solutionofits nonlinearwaveequation isgiven,andtheinteractionbehaviorofsolitons
isdiscussed.AcoupledKdVequationisobtainedbycalculatingthetwo-dimensional
matterwavepulsesinBose-Fermimixturegas,includinglinearandnonlinear,and
byusingthereducedperturbationmethodundertheconstraintconditionsofunitary
system.Theintegrabilityoftheequationisfurtherstudiedanddiscussed.
Keywords
Bose-FermiGasMixture,SolitonSolution,ReductivePerturbationTechnique
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2022.123048435nØêÆ
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DOI:10.12677/pm.2022.123048436nØêÆ
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−
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b
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2
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c
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n
r
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b
+A
b
c
b0
n
r
b0
b
c
b1
n
r
b1
b
= 0
−
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2
(
∂
2
A
p
∂x
2
+
∂
2
A
p
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2
)+(−ξ+
1
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p
+A
p
{
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+
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2
+(
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p
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)
2
]}
+A
p
c
p0
n
r
p0
p
+A
p
c
p0
n
r
p0
p
c
p1
n
r
p1
p
= 0
(14)
DOI:10.12677/pm.2022.123048437nØêÆ
\#
?˜Úµ

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+
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)
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+c
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+1
b
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2r
b0
b
+c
b0
c
b1
A
2r
b0
+2r
b1
+1
b
G
2r
b0
+2r
b1
b
= 0
−
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2
(
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2
+
∂
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p
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2
)+(−ξ+
1
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+c
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2r
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+1
p
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2r
p0
p
+c
p0
c
p1
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+2r
p1
+1
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2
2
(Š)§¤±þªdu













−
1
2
(
∂
2
A
b
∂x
2
+
∂
2
A
b
∂y
2
)+(−ξ+
1
2
)A
b
+A
b
{
∂ϕ
b
∂t
+
1
2
[(
∂ϕ
b
∂x
)
2
+(
∂ϕ
b
∂y
)
2
]}
+c
b0
A
2r
b0
+1
b
e
−r
b0
z
2
+c
b0
c
b1
A
2r
b0
+2r
b1
+1
b
e
−(r
b0
+r
b1
)z
2
= 0
−
1
2
(
∂
2
A
p
∂x
2
+
∂
2
A
p
∂y
2
)+(−ξ+
1
2
)A
p
+A
p
{
∂ϕ
p
∂t
+
1
2
[(
∂ϕ
p
∂x
)
2
+(
∂ϕ
p
∂y
)
2
]}
+c
p0
A
2r
p0
+1
p
e
−r
p0
z
2
+c
p0
c
p1
A
2r
p0
+2r
p1
+1
p
e
−(r
p0
+r
p1
)z
2
= 0
(16)
éþªü>Ó¦±e
−z
2
,¿ézÈ©(
R
∞
−∞
e
−z
2
dz=
√
π),k













−
1
2
(
∂
2
A
b
∂x
2
+
∂
2
A
b
∂y
2
)+(−ξ+
1
2
)A
b
+A
b
{
∂ϕ
b
∂t
+
1
2
[(
∂ϕ
b
∂x
)
2
+(
∂ϕ
b
∂y
)
2
]}
+
c
b0
√
r
b0
+1
A
2r
b0
+1
b
+
c
b0
c
b1
√
r
b0
+r
b1
+1
A
2r
b0
+2r
b1
+1
b
= 0
−
1
2
(
∂
2
A
p
∂x
2
+
∂
2
A
p
∂y
2
)+(−ξ+
1
2
)A
p
+A
p
{
∂ϕ
p
∂t
+
1
2
[(
∂ϕ
p
∂x
)
2
+(
∂ϕ
p
∂y
)
2
]}
+
c
p0
√
r
p0
+1
A
2r
p0
+1
p
+
c
p0
c
p1
√
r
p0
+r
p1
+1
A
2r
p0
+2r
p1
+1
p
= 0
(17)
¤±•§|£17¤ÏLOŽ;

























∂A
b
∂t
+
∂A
b
∂x
∂ϕ
b
∂x
+
∂A
b
∂y
∂ϕ
b
∂y
+
1
2
A
b
(
∂
2
ϕ
b
∂x
2
+
∂
2
ϕ
b
∂y
2
) = 0
−
1
2
(
∂
2
A
b
∂x
2
+
∂
2
A
b
∂y
2
)+(−ξ+
1
2
)A
b
+A
b
{
∂ϕ
b
∂t
+
1
2
[(
∂ϕ
b
∂x
)
2
+(
∂ϕ
b
∂y
)
2
]}
+
c
b0
√
r
b0
+1
A
2r
b0
+1
b
+
c
b0
c
b1
√
r
b0
+r
b1
+1
A
2r
b0
+2r
b1
+1
b
= 0
∂A
p
∂t
+
∂A
p
∂x
∂ϕ
p
∂x
+
∂A
p
∂y
∂ϕ
p
∂y
+
1
2
A
p
(
∂
2
ϕ
p
∂x
2
+
∂
2
ϕ
p
∂y
2
) = 0
−
1
2
(
∂
2
A
p
∂x
2
+
∂
2
A
p
∂y
2
)+(−ξ+
1
2
)A
p
+A
p
{
∂ϕ
p
∂t
+
1
2
[(
∂ϕ
p
∂x
)
2
+(
∂ϕ
p
∂y
)
2
]}
+
c
p0
√
r
p0
+1
A
2r
p0
+1
p
+
c
p0
c
p1
√
r
p0
+r
p1
+1
A
2r
p0
+2r
p1
+1
p
= 0
(18)
2-A
j
= u
j0
+ε
2
a
j0
+ε
4
a
j1
,ϕ
j
= εϕ
j0
+ε
3
ϕ
j1
,(j= b,p)
ξ= ε(z−c
j
t)(j= b,p)§τ= ε
3
t.
rþª“\•§|£18)¥§ÏLOŽn



∂a
b
0
∂τ
−
1
8c
b
∂
3
a
b
0
∂ξ
3
+
3c
b
2u
b
0
a
b
0
∂a
b
0
∂ξ
= 0,
∂a
p
0
∂τ
−
1
8c
p
∂
3
a
p
0
∂ξ
3
+
3c
p
2u
p
0
a
p
0
∂a
p
0
∂ξ
= 0.
(19)
l·‚?˜ÚÍÜKdV•§µ



∂a
b
0
∂τ
−
1
8c
b
∂
3
a
b
0
∂ξ
3
+
3c
b
2u
b
0
a
b
0
∂a
b
0
∂ξ
= 0,
∂a
p
0
∂τ
−
1
8c
p
∂
3
a
p
0
∂ξ
3
+
3c
p
2u
p
0
a
p
0
∂a
p
0
∂ξ
= 0.
(20)
DOI:10.12677/pm.2022.123048438nØêÆ
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