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PureMathematicsnØêÆ,2022,12(10),1636-1648
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1210177
oš‚5Schr¨odinger•§AŸþÅð
$KŽ{
www}}}
2À7KÆ7KêƆÚOÆ,2À2²
ÂvFϵ2022c913F¶¹^Fϵ2022c1012F¶uÙFϵ2022c1020F
Á‡
©ïÄoš‚5Schr¨odinger•§äkAŸþÅð˜«$KŽ{.TŽ{Ø=
±˜Âñ,¿…„äkAŸþÅð5.·‚ÏLî‚Ø©Û,y²ЊáuH
γ+3
(T
d
)
ž,é½T>0Úγ>
d
2
,•3~C= C(kuk
L
∞
(0,T;H
γ+3
)
) >0,¦
ku(t
n
)−u
n
k
H
γ+3
≤Cτ,
Ù¥u
n
´oš‚5Schr¨odinger•§3t
n
= nτžêŠ).d,êŠ)ŸþM(u
n
)÷v
|M(u
n
)−M(u
0
)|≤Cτ
3
.
'…c
oš‚5Schr¨odinger•§§$K§˜Âñ§ŸþÅð
ALow-RegularityIntegratorforthe
Fourth-OrderNonlinearSchr¨odinger
EquationwithAlmostMassConservation
CuiNing
©ÙÚ^:w}.oš‚5Schr¨odinger•§AŸþÅð$KŽ{[J].nØêÆ,2022,12(10):1636-1648.
DOI:10.12677/pm.2022.1210177
w}
SchoolofFinancialMathematicsandStatistics,GuangdongUniversityofFinance,Guangzhou
Guangdong
Received:Sep.13
th
,2022;accepted:Oct.12
th
,2022;published:Oct.20
th
,2022
Abstract
Inthispaper,weintroducealow-regularityintegratorforthefourth-ordernonlinear
Schr¨odingerequationwithalmostmassconservation.Thealgorithmcannotonly
achievefirst-orderconvergence,butalsoobeyalmostmassconservationlaw.Byrig-
orouserroranalysis,forroughinitialdatainH
γ+3
(T
d
)withγ>
d
2
,uptosomefixed
timeofT,thereexistsC= C(kuk
L
∞
(0,T;H
γ+3
)
) >0,suchthat
ku(t
n
)−u
n
k
H
γ+3
≤Cτ,
where u
n
denotes thenumerical solution att
n
= nτ.Moreover,themassofthenumer-
icalsolutionM(u
n
)verifies
|M(u
n
)−M(u
0
)|≥Cτ
3
.
Keywords
Fourth-OrderNonlinearSchr¨odingerEquation,Low-RegularityIntegrator,
First-OrderConvergent,MassConversation
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
©•Äeãoš‚5Schr¨odinger•§



iu
t
= (−∆)
2
u+µ|u|
2
u,(t,x) ∈(R
+
,T
d
),
u(0,x) = u
0
(x),
(1)
DOI:10.12677/pm.2022.12101771637nØêÆ
w}
Ù¥T
d
=(0,2π)
d
,u(t,x)´(R
+
,T
d
)þEŠ¼ê,u
0
(x)∈H
γ
(T
d
)(γ≥0)•‰½ÐŠ,µ=±1,
¿…d≥2.T•§3L
2
¥),÷vŸþÅð½Æµ
M(u(t,x)) =
1
2π
Z
T
d
|u(t,x)|
2
dx= M(u
0
).
3nØïÄ•¡,Zhu[1]y²à.oš‚ 5Schr¨odinger•§Ÿþ.žÄ)C©
(±9)»5.Guo[2]y²à.oš‚5Schr¨odinger•§Ÿþ‡.žÛ·½
5±9»•ÐŠ)Ñ5.Ù¦.Schr¨odinger•§•2•ïÄ,Œë„[3–10].
3êŠïÄ•¡,éuäk1wЩ^‡•§,~„ꊕ{k©•{[11,12],k•
©•{[13,14],k••{[15,16],Ì•{[17,18],ØëYGalerkin•{[19,20],•êÈ©ì
{[21,22].•§)Ø1wž,K‡mu$K5Ž{,XStrangeSplittingŽ{[23],#.
•ê.Ž{[24],FourierÈ© Ž{[25].éuoš‚5Schr¨odinger•§,Ning [26]JÑ˜«ê
Š‚ª,¦3›”nêœ¹eˆ˜Âñ,=ЩŠáuH
γ+3
ž•§)3H
γ
¥˜
Âñ.
(Ü©z[26],·‚E˜«#$KŽ{,Œ±oš‚5Schr¨odinger•§(1)˜
Âñ,d„÷vAŸþÅð.3‰ÑäNŽ{c,·‚Äk½Â˜‡^¼ê.-
ϕ(z) =



e
z
−1
z
,z6= 0,
1,z= 0,
¿…
Ψ(f) = e
−iτ(−∆)
2
f−iµτe
−iτ(−∆)
2
[f
2
·ϕ(2iτ(−∆)
2
)
¯
f].
½Âe?Žf
I(f) = Ψ(f)−e
−iτ(−∆)
2
f,
Ú
J(f) = H(f)e
−iτ(−∆)
2
f,
Ù¥
H(f) = −kfk
−2
L
2

hI(f),e
−iτ(−∆)
2
fi+
1
2
kI(f)k
2
L
2

.
y3,·‚‰Ñoš‚5Schr¨odinger•§(1)#$KŽ{µ
u
n
= Ψ(u
n−1
)+J(u
n−1
),n= 0,1,2,...,
T
τ
.(2)
DOI:10.12677/pm.2022.12101771638nØêÆ
w}
Ïd,·‚̇½n•
½n1.u
n
´oš‚5Schr¨odinger•§(1)÷v(2)‚ªêŠ),½T>0.é?¿γ>
d
2
,
bu
0
(x) ∈H
γ+3
(T
d
),•3~êτ
0
,C>0¦é?¿0 <τ≤τ
0
,k
ku(t
n
)−u
n
k
H
γ+3
≤Cτ,n= 0,1,2,...,
T
τ
.
d,
|M(u
n
)−M(u
0
)|≤Cτ
3
,
Ù¥τ
0
ÚC>0=•6uTÚkuk
L
∞
((0,T);H
γ+3
)
.
©SüXe.312!¥,·‚‰Ñ˜ÎÒÚ˜k^Ún.313 !¥,·‚‰Ñ˜
ꊂª̇EL§.314 !¥,·‚y²½n1.
2.ý•£
3!¥,·‚0˜½Â!5ŸÚ-‡O.••BPÒ,·‚¦^A.B½öB&A
5L«Xe¹Âµ•3,ýé~êC>0,¦A≤CB,¿¦^A∼B5L«A.B.A.
éu•þξ:=(ξ
1
,...,ξ
d
) ∈Z
d
,ξ
1
:= (ξ
11
,...,ξ
1d
) ∈Z
d
,x:=(x
1
,...,x
d
) ∈T
d
,SÈÚ½ÂX
e:
ξ·x= ξ
1
x
1
+...+ξ
d
x
d
,|ξ|
2
= |ξ
1
|
2
+...+|ξ
d
|
2
.
½Âh·,·i•L
2
SÈ
hf,gi= Re
Z
T
d
f(x)g(x)dx.
3±ÏT
d
þ,·‚½Â¼êfFourierC†•
ˆ
f
ξ
=
1
(2π)
d
Z
T
d
e
−iξ·x
f(x)dx,
ÙFourier_C†•f(x) =
P
ξ∈Z
d
e
−iξ·x
ˆ
f
ξ
.
é¼êf∈L
2
(T
d
),·‚½Âf(x)FourierÐmª•
f(x) =
X
ξ∈Z
d
ˆ
f
ξ
e
iξ·x
.
FourierC†~^5Ÿ
kfk
2
L
2
= 2π
X
ξ∈Z
d
|
ˆ
f
ξ
|
2
,
DOI:10.12677/pm.2022.12101771639nØêÆ
w}
Ú
c
fg(ξ) =
X
ξ,ξ
1
∈Z
d
ˆ
f
ξ−ξ
1
ˆg
ξ
1
.
½ÂSobolev˜mH
γ
,γ>0‰ê•


f


2
H
γ
(T
d
)
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|
ˆ
f
ξ
|
2
.
·‚½ÂŽf(−∆)
−1
•
\
(−∆)
−1
f=
(
|ξ|
−2
ˆ
f
ξ
, ξ6= 0,
0, ξ= 0.
y3,·‚‰Ñ‡^˜-‡O.
Ún2.£å5Ÿ¤éf∈H
γ
,t∈Rk
ke
−i(−∆)
2
t
fk
H
γ
= kfk
H
γ
.
Proof.ŠâH
γ
‰ê½Â,Œ•
ke
−i(−∆)
2
t
fk
H
γ
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|
\
e
−i(−∆)
2
t
f
ξ
|
2
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|e
−i|ξ|
4
t
ˆ
f
ξ
|
2
=
X
ξ∈Z
d
(1+|ξ|)
2γ
|
ˆ
f
ξ
|
2
= kfk
H
γ
.
Ún3.(Kato-PonceØª)é?¿γ>
d
2
,f,g∈H
γ
,ke¡Øª¤á
kfgk
H
γ
≤kfk
H
γ
kgk
H
γ
.
Proof.TÚny²Œë„©z[27].
3.ꊂªE
3!¥,·‚0ꊂªEL§.•{zÎÒ, ·‚òŽÑ¤9ž˜ƒ'¼ê¥
˜mCþx,Xu(t) = u(t,x).,,·‚^τ>0L«žmÚ•,t
n
= nτL«žm‚.
DOI:10.12677/pm.2022.12101771640nØêÆ
w}
y3,·‚£©z[26]¥éoš‚5Schr¨odinger•§(1)Ž{E,ÏLDuhamelúª,k
u(t,x) = e
−it(−∆)
2
u
0
−iµ
Z
t
0
e
−i(t−s)(−∆)
2
(|u(s,x)|
2
u(s,x))ds.
-Û=Cþ
v(t,x) = e
it(−∆)
2
u(t,x),(3)
KŒ
v(t
n
+τ) = v(t
n
)−iµ
Z
τ
0
e
i(t
n
+s)(−∆)
2
h
|e
−i(t
n
+s)(−∆)
2
v(t
n
+s)|
2
e
−i(t
n
+s)(−∆)
2
v(t
n
+s)
i
ds.
ÏLFourierÐm,Œ
v(t
n
+τ) = Φ
n
(v(t
n
))+R
n
1
+R
n
2
,(4)
Ù¥
Φ
n
(v(t
n
)) = v(t
n
)−iµτe
it
n
(−∆)
2

(e
−it
n
(−∆)
2
v(t
n
))
2
··ϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
v(t
n
)

,(5)
Ú
R
n
1
= −iµ
Z
τ
0
e
i(t
n
+s)(−∆)
2

|e
−i(t
n
+s)(−∆)
2
v(t
n
+s)|
2
e
−i(t
n
+s)(−∆)
2
v(t
n
+s)
−|e
−i(t
n
+s)(−∆)
2
v(t
n
)|
2
e
−i(t
n
+s)(−∆)
2
v(t
n
)

ds,
¿…
R
n
2
= −iµ
X
ξ∈Z
d
X
ξ
1
,ξ
2
,ξ
3
∈Z
d
ξ=ξ
1
+ξ
2
+ξ
3
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
e
it
n
α
Z
τ
0
e
is2|ξ
1
|
4
(e
isβ
−1)ds.(6)
Ù¥·‚{P
¯
ˆv
1
=
¯
ˆv
ξ
1
(t
n
),ˆv
2
=ˆv
ξ
2
(t
n
)Úˆv
3
=ˆv
ξ
3
(t
n
),…α=|ξ|
4
+ |ξ
1
|
4
−|ξ
2
|
4
−|ξ
3
|
4
.|
^ξ= ξ
1
+ξ
2
+ξ
2
+ξ
3
,KαŒ±©)¤
α= 2|ξ
1
|
4
+
3
X
j,k=1
j6=k
|ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h=1
j6=k6=h
|ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h=1
j6=k6=h
ξ
2
j
¯
ξ
k
¯
ξ
h
= 2|ξ
1
|
4
+β,
Ù¥
β=
3
X
j,k=1
j6=k
|ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h=1
j6=k6=h
|ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h=1
j6=k6=h
ξ
2
j
¯
ξ
k
¯
ξ
h
= O(
3
X
j,k=1
j6=k
|ξ
j
|
3
|ξ
k
|).
DOI:10.12677/pm.2022.12101771641nØêÆ
w}
e¡,·‚Ø\y²/ÛјO,ƒ'y²Œ„[26].Äk,·‚‰ÑÛÜØ‘O.
Ún4.(ÛÜØ)•3~êC=•6ukvk
L
∞
((0,T);H
γ+3
)
,¦
kv(t
n+1
)−Φ
n
(v(t
n
))k
H
γ
≤Cτ
2
,
Ùg,·‚0-½5O.
Ún5.(-½5)f,g∈H
γ
,éuγ>
d
2
,Kk
kΦ
n
(f)−Φ
n
(g)k
H
γ
≤(1+Cτ)kf−gk
H
γ
+Cτkf−gk
3
H
γ
,
3þã˜ÂñÄ:þ,·‚ F"E˜«Œ±±)Ôn5Ž{,)3÷v˜Â
ñcJe,„U±AŸþÅð5,Ïd·‚‡éŽ{?1?.½Â
I
n
(v) = Φ
n
(v)−v,(7)
ùpΦ
n
d(5)‰Ñ.2½Â
J
n
(v) = H
n
(v)v,(8)
Ù¥
H
n
(v) = −kvk
−2
L
2

hI
n
(v),vi+
1
2
kI
n
(v)k
2
L
2

.
kþ¡?,·‚EÑ÷vAŸþÅðꊂª
v
n+1
= v
n
+I
n
(v
n
)+J
n
(v
n
).(9)
òÛ=Cþ(3)‡L5“\(9),=Œoš‚5Schr¨odinger•§(1)˜Âñꊎ
{(2).
4.½n1y²
!·‚ÏLÛÜØ©ÛÚ-½5©Ûé˜Âñ(J‰Ñ˜‡î‚y².dÚn2Œ•,
Û=Cþ(3)3Sobolev˜m´å,Kk
ku(t
n
)−u
n
k
H
γ
= ke
−it
n
(−∆)
2
v(t
n
)−e
−it
n
(−∆)
2
v
n
k
H
γ
= kv(t
n
)−v
n
k
H
γ
.
Ïd,·‚•I‡y²év(t
n
)Úv
n
˜Âñ½nÚAŸþÅð5.
Äk,·‚‰ÑI
n
(v)Âñ.
DOI:10.12677/pm.2022.12101771642nØêÆ
w}
Ún6.v∈H
γ
,γ>
d
2
ž,•3~êC=•6ukvk
L
∞
((0,T);H
γ+3
)
,e¡Øª¤á
kI
n
(v)k
L
2
≤Cτ.
Proof.£I
n
(v)½Â(7)Œ•,
I
n
(v) = −iµτe
it
n
(−∆)
2

(e
−it
n
(−∆)
2
v)
2
·ϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
v

.
Ïd,
kI
n
(v)k
L
2
.τk(e
−it
n
(−∆)
2
v)
2
k
L
∞
·kϕ(2iτ(−∆)
2
)e
−it
n
(−∆)
2
vk
L
2
.τkvk
3
H
γ
.
¤±,Ún6y.
Ùg,3‰ÑhI
n
(v),viÂñƒc,·‚k‡‰Ñ˜‡‡^Øª.
Ún7.•3~êC=•6ukvk
L
∞
((0,T);H
γ+3
)
,¦
kR
n
2
k
H
−γ
≤Cτ
2
.
Proof.ŠâR
n
2
½Â(6),Œ
|R
n
2
|.τ
2
X
ξ∈Z
d
X
ξ
1
,ξ
2
,ξ
3
∈Z
d
ξ=ξ
1
+ξ
2
+ξ
3
e
iξ·x
e
it
n
α
(
3
X
j,k=1
j6=k
|ξ
j
|
3
|ξ
k
|·
¯
ˆv
j
ˆv
k
ˆv
h
)
.τ
2
X
ξ∈Z
d
X
ξ
1
,ξ
2
,ξ
3
∈Z
d
ξ=ξ
1
+ξ
2
+ξ
3
e
iξ·x
e
it
n
α
(|ξ
j
|
3
¯
ˆv
j
·|ξ
k
|ˆv
k
·ˆv
h
).
Ïd,dSobolevØªŒ
kR
n
2
k
H
−γ
.τ
2
k|∇|
3
v·|∇|v·vk
H
−γ
.τ
2
k|∇|
3
v·|∇|v·vk
L
1
.τ
2
k|∇|
3
vk
L
2
k|∇|vk
L
2
k|vk
L
∞
.τ
2
kvk
3
H
γ+3
.
¤±,TÚny.
Ún8.v∈H
γ+3
,γ>
d
2
ž,•3~êC=•6ukvk
L
∞
((0,T);H
γ+3
)
,e¡Øª¤á
|hI
n
(v),vi|≤Cτ
2
.
DOI:10.12677/pm.2022.12101771643nØêÆ
w}
Proof.£I
n
(v)½Â(7)
I
n
(v) =−iµ
X
ξ∈Z
d
X
ξ
1
,ξ
2
,ξ
3
∈Z
d
ξ=ξ
1
+ξ
2
+ξ
3
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
e
it
n
α
Z
τ
0
e
is2|ξ
1
|
4
ds
=−iµ
X
ξ∈Z
d
X
ξ
1
,ξ
2
,ξ
3
∈Z
d
ξ=ξ
1
+ξ
2
+ξ
3
¯
ˆv
1
ˆv
2
ˆv
3
e
iξ·x
Z
τ
0
e
i(t
n
+s)α
ds−R
n
2
=−iµ
Z
τ
0
e
i(t
n
+s)(−∆)
2
h
|e
−i(t
n
+s)(−∆)
2
v|
2
e
−i(t
n
+s)(−∆)
2
v
i
ds−R
n
2
.
Ïd,·‚k
hI
n
(v),vi= h−iµ
Z
τ
0
e
i(t
n
+s)(−∆)
2
h
|e
−i(t
n
+s)(−∆)
2
v|
2
e
−i(t
n
+s)(−∆)
2
v
i
ds,vi+h−R
n
2
,vi.
dSÈ½Â,Œ
h−iµ
Z
τ
0
e
i(t
n
+s)(−∆)
2
h
|e
−i(t
n
+s)(−∆)
2
v|
2
e
−i(t
n
+s)(−∆)
2
v
i
ds,vi= 0.
(ÜÚn7,Œ,
|hI
n
(v),vi|= |h−R
n
2
,vi|.kR
n
2
k
H
−γ
kvk
H
γ
.τ
2
kvk
4
H
γ+3
.
Ïd,y²Ún(Ø.
e5,(ÜÛÜØOÚ-½5(J,·‚‰Ñ½n1y².
Proof.Äk,dêŠ)E(9),·‚k
v
n+1
−v(t
n+1
) =v
n
+I
n
(v
n
)+J
n
(v
n
)−v(t
n+1
)
=Φ
n
(v
n
)−Φ
n
(v(t
n
))+Φ
n
(v(t
n
))−v(t
n+1
)+J
n
(v
n
).
ŠâÚn4ÚÚn5,Œ
kv(t
n+1
)−v
n+1
k
H
γ
≤kΦ
n
(v
n
)−Φ
n
(v(t
n
))k
H
γ
+kΦ
n
(v(t
n
))−v(t
n+1
)k
H
γ
+kJ
n
(v
n
)k
H
γ
≤(1+Cτ)kv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
3
H
γ
+Cτ
2
+kJ
n
(v
n
)k
H
γ
.
ŠâJ
n
½Â(8),k
kJ
n
(v
n
)k
H
γ
≤|H
n
(v
n
)|kv
n
k
H
γ
≤|H
n
(v
n
)|(kv
n
−v(t
n
)k
H
γ
+kv(t
n
)k
H
γ
).
DOI:10.12677/pm.2022.12101771644nØêÆ
w}
dÚn6ÚÚn8,Œ
|H
n
(v
n
)|≤kv
n
k
−2
L
2
(τ
2
kv
n
k
4
H
γ+3
+
1
2
τ
2
kv
n
k
6
H
γ
) ≤Cτ
2
(1+kv
n
−v(t
n
)k
4
H
γ
).(10)
Ïd,·‚k
kJ
n
(v
n
)k
H
γ
≤Cτ
2
(1+kv
n
−v(t
n
)k
5
H
γ
).(11)
(Ü(11),Œ
kv(t
n+1
)−v
n+1
k
H
γ
≤Cτ
2
+(1+Cτ)kv
n
−v(t
n
)k
H
γ
+Cτkv
n
−v(t
n
)k
5
H
γ
.
|^S“{ÚGronwall’sØª,Œ
kv(t
n+1
)−v
n+1
k
H
γ
≤Cτ
2
n
X
j=0
(1+Cτ)
j
≤Cτ,n= 0,1,2,...,
T
τ
−1,
Ù¥C=•6ukvk
L
∞
((0,T);H
γ+3
)
.þª¤á,Ò¿›X˜Âñ…kO÷v
kv
n
k
H
γ
≤C,n= 0,1,2,...,
T
τ
−1.
Ï•v
n
= v
n
+I
n
(v
n
)+J
n
(v
n
),k
hv
n+1
,v
n+1
i=hv
n
+I
n
(v
n
)+J
n
(v
n
),v
n
+I
n
(v
n
)+J
n
(v
n
)i
=kv
n
k
2
L
2
+2hI
n
(v
n
),J
n
(v
n
)i+kJ
n
(v
n
)k
2
L
2
+2hI
n
(v
n
),v
n
i+hJ
n
(v
n
),v
n
i+kI
n
(v
n
)k
2
L
2
.
ŠâI
n
(v
n
)ÚJ
n
(v
n
)½ÂŒ
2hI
n
(v
n
),v
n
i+hJ
n
(v
n
),v
n
i+kI
n
(v
n
)k
2
L
2
= 0,
K
hv
n+1
,v
n+1
i= kv
n
k
2
L
2
+2hI
n
(v
n
),J
n
(v
n
)i+kJ
n
(v
n
)k
2
L
2
.
˜•¡,|^Ún8Ú(10),·‚Œ
hI
n
(v
n
),J
n
(v
n
)i≤|H
n
(v
n
)||hI
n
(v
n
),v
n
i|.Cτ
4
.
,˜•¡,(Ü(10)Ú(11),k
kJ
n
(v
n
)k
2
L
2
≤|H
n
(v
n
)|
2
kv
n
k
2
L
2
≤Cτ
4
.
DOI:10.12677/pm.2022.12101771645nØêÆ
w}
nþŒ
|kv
n+1
k
2
L
2
−kv
n
k
2
L
2
|≤Cτ
4
,
=
|M(v
n+1
)−M(v
n
)|≤Cτ
4
.(12)
é(12)?1S“Œ
|M(v
n
)−M(u
0
)|≤Cτ
3
,
C=•6ukvk
L
∞
((0,T);H
γ+3
)
.
Ïd,·‚½n1y².
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11901120)"
ë•©z
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