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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(10),7362-7372
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.1110782
š‚5Schr¨odinger•§$KŽ{
www}}}
2À7KÆ§7KêƆÚOÆ§2À2²
ÂvFϵ2022c924F¶¹^Fϵ2022c1017F¶uÙFϵ2022c1026F
Á‡
©ïÄgš‚5Schr¨odinger•§äk˜Âñ˜«$KÈ©ì§ù«È©ì´wª
¿…¯„k"A O/§·‚Ž{ØI‡›”êÒŒ±¢y˜Âñ"ÏLî‚Ø
©Û§·‚y²ЊáuH
γ
(T)ž§gš‚5Schr¨odinger•§3H
γ
(T)þäk˜Â
ñ§Ù¥γ>
1
2
"
'…c
gš‚5Schr¨odinger•§§$K§˜Âñ
Low-RegularityIntegratorforthe
QuadraticNLSEquation
CuiNing
SchoolofFinancialMathematicsandStatistics,GuangdongUniversityofFinance,Guangzhou
Guangdong
Received:Sep.24
th
,2022;accepted:Oct.17
th
,2022;published:Oct.26
th
,2022
©ÙÚ^:w}.š‚5Schr¨odinger•§$KŽ{[J].A^êÆ?Ð,2022,11(10):7362-7372.
DOI:10.12677/aam.2022.1110782
w}
Abstract
Inthispaper,weintroduceafirstorderlow-regularityintegratorforthequadratic
nonlinearSchr¨odingerequation.Theschemeisexplicitandefficienttoimplement.
Inparticular,ourschemedoesnotcostanyadditionalderivativeforthefirstorder
convergence.Byrigorous erroranalysis,we show that thescheme provides firstorder
accuracyinH
γ
(T)forroughinitialdatainH
γ
(T)withγ>
1
2
.
Keywords
QuadraticNonlinearSchr¨odingerEquation,Low-RegularityIntegrator,FirstOrder
Convergent
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
3©·‚•Äeãgš‚5Schr¨odinger•§
iu
t
+∂
xx
u= |u|
2
,(t,x) ∈(R
+
,T).(1)
Ù¥T=(0,2π),u(t,x)´(R
+
,T)þEŠ¼ê,u
0
(x)∈H
γ
(T)(γ≥0)•‰½ÐŠ.T•§·½
5!-½5!»5•¡2•ïÄ.Tao[1]y²gš‚5Schr¨odinger•§3H
s
(R)˜m,
s≥−1ž´ÛÜ·½;3s<−1ž´Ø·½.Ù¦a.Schr¨odinger•§•2•ïÄ,X
²;š‚5Schr¨odinger•§Œë„[2–5].êš‚5Schr¨odinger•§Œë„[6–11].©ê
š‚5Schr¨odinger•§Œë„[12–15].
·‚Ø=Œ±lêÆnØþïÄš‚5Schr¨odinger•§,Ù¢,·‚„Œ± ^ꊕ{&
?š‚5Schr¨odinger•§,~Xk•©•{[16][17],k••{[18][19],©•{[20][21],Ì•
{[22][23],ØëYGalerkin•{[24][25],•êÈ©ì{[26][27].
DOI:10.12677/aam.2022.11107827363A^êÆ?Ð
w}
êŠÂñ5ïÄ8I´¦°()u(t
n
)ÚêŠ)u
n
÷veªµ
ku(t
n
)−u
n
k
H
γ
≤Cτ
α
,
Ù¥,τL«žmÚ•,αL«Âñê,Њu
0
(x)=u(0,x)áu,‡H
γ+s
˜m.·‚F"s¦þ,
=ЩŠ¦ŒUk$K5,¢yαÂñ.þãꊕ{ч¦ Њäkép1w5,
8cŒ[•'5´$K5Ž{.Ï•,3¢S)¹¥,duD(6Ä!ÿþØ5!‘Å5
Ï,ЩêâÏ~Ã{÷vp1w5‡¦.$KŽ{•¡,éu²;š‚5Schr¨odinger•
§,Lubich[28]|^StrangeSplittingŽ{,ЊáuH
γ+2
ž,T•§3H
γ
¥˜Âñ.
5,OstermannÚSchratz[29]|^#.•ê.Ž{,ЊáuH
γ+1
ž,3H
γ
¥˜Â
ñ.•C,WuÚYao[30]‰Ñ˜«FourierÈ©Ž{,ЊáuH
γ
ž,y²3H
γ
¥˜Âñ,
TŽ{ØI‡›”?Ûê.
©ïÄ´gš‚5Schr¨odinger•§(1)$KŽ{±ˆ˜Âñ,©ïŽ{
•µ
u
n
= e
iτ∆
u
n−1
+
1
2
∂
−1
x
h
e
iτ∆
∂
−1
x
u
n−1
·e
iτ∆
u
n−1
i
−
1
2
∂
−1
x
h
∂
−1
x
u
n−1
·u
n−1
i
.(2)
·‚̇½n•
½n1.u
n
•gš‚5Schr¨odinger•§(1)÷v(2)‚ªêŠ),½T>0.é?¿γ>
1
2
,
bu
0
(x) ∈H
γ+3
(T),•3~êτ
0
,C>0¦é?¿0 <τ≤τ
0
,k
ku(t
n
)−u
n
k
H
γ
≤Cτ,n= 0,1,2,...,
T
τ
,(3)
Ù¥τ
0
ÚC>0=•6uTÚkuk
L
∞
((0,T);H
γ
)
.
©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.
3±ÏTþ,·‚½Â¼êfFourierC†•
ˆ
f
ξ
=
1
(2π)
d
Z
T
d
e
−iξ·x
f(x)dx,
ÙFourier_C†•f(x) =
P
ξ∈Z
e
−iξ·x
ˆ
f
ξ
.
é¼êf∈L
2
(T),·‚½Âf(x)FourierÐmª•
f(x) =
X
ξ∈Z
ˆ
f
ξ
e
iξ·x
.
DOI:10.12677/aam.2022.11107827364A^êÆ?Ð
w}
FourierC†~^5Ÿ
kfk
2
L
2
= (2π)
X
ξ∈Z
|
ˆ
f
ξ
|
2
,
Ú
c
fg(ξ) =
X
ξ,ξ
1
∈Z
ˆ
f
ξ−ξ
1
ˆg
ξ
1
½ÂSobolev˜mH
γ
,γ>0‰ê•


f


2
H
γ
(T)
=
X
ξ∈Z
(1+|ξ|)
2γ
|
ˆ
f
ξ
|
2
.
y3,·‚‰Ñ‡^˜-‡O.
Ún2.(å5Ÿ)éf∈H
γ
,t∈Rk
ke
−i∆t
fk
H
γ
= kfk
H
γ
.
Proof.ŠâH
γ
‰ê½Â,Œ•
ke
−i∆t
fk
H
γ
=
X
ξ∈Z
(1+|ξ|)
2γ
|
\
e
−i∆t
f
ξ
|
2
=
X
ξ∈Z
(1+|ξ|)
2γ
|e
−i|ξ|
2
t
ˆ
f
ξ
|
2
=
X
ξ∈Z
(1+|ξ|)
2γ
|
ˆ
f
ξ
|
2
= kfk
H
γ
.
Ún3.(Kato-PonceØª)é?¿γ>
1
2
,f,g∈H
γ
,ke¡Øª¤á
kfgk
H
γ
≤kfk
H
γ
kgk
H
γ
.
Proof.TÚny²Œë„©z[31].
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‚.g š‚
5Schr¨odinger•§(1),ÏLDuhamelúª,Œ
u(t,x) = e
it∆
u
0
−i
Z
t
0
e
i(t−s)∆
|u(s,x)|
2
ds.
DOI:10.12677/aam.2022.11107827365A^êÆ?Ð
w}
Ú\Û=Cþ
v(t,x) = e
−it∆
u(t,x),(4)
k
v(t,x) = v
0
−i
Z
t
0
e
−is∆
|e
is∆
v(s,x)|
2
ds.(5)
d,Û=Cþ÷v
∂
t
v= −ie
−it∆
|e
it∆
v(t,x)|
2
.
Ï•·‚•I‡˜Âñ,¤±Œ±^v(t
n
+s) ≈v(t
n
){zþオ,
v(t
n
+τ) = v(t
n
)+Φ(v(t
n
))+L
n
,
Ù¥
Φ
n
(v(t
n
)) = −i
Z
τ
0
e
−i(t
n
+s)∆
|e
i(t
n
+s)∆
v(t
n
,x)|
2
ds,
Ú
L
n
= −i
Z
τ
0
e
−i(t
n
+s)∆

|e
i(t
n
+s)∆
v(t
n
+s,x)|
2
−|e
i(t
n
+s)∆
v(t
n
,x)|
2

ds.(6)
Ù¥,L
n
Œ±w¤˜‡p‘,Œ±3Ž{E¥ï,ùpج›”K5.
e¡,·‚UYéΦ(v(t
n
))?1©Û,|^FourierÐmŒ
Φ
n
(v(t
n
)) =−i
X
ξ=ξ
1
+ξ
2
ˆ
¯v
1
ˆv
2
e
iξ·x
e
it
n
(|ξ|
2
+|ξ
1
|
2
−|ξ
2
|
2
)
Z
τ
0
e
is(|ξ|
2
+|ξ
1
|
2
−|ξ
2
|
2
)
ds,
Ù¥·‚{P
¯
ˆv
1
=
¯
ˆv
ξ
1
(t
n
)Úˆv
2
=ˆv
ξ
2
(t
n
).
Šâξ= ξ
1
+ξ
2
,Kk|ξ|
2
+|ξ
1
|
2
−|ξ
2
|
2
= 2ξξ
1
.Ïd,Φ(v(t
n
))¥È©‘Œ±¤
Z
τ
0
e
is(|ξ|
2
+|ξ
1
|
2
−|ξ
2
|
2
)
ds=
Z
τ
0
e
is2ξξ
1
ds=
e
iτ2ξξ
1
−1
2iξξ
1
.
Ïd,Φ(v(t
n
))Œ±w«£Ôn˜m
Φ
n
(v(t
n
)) =−i
X
ξ=ξ
1
+ξ
2
ˆ
¯v
1
ˆv
2
e
iξ·x
e
it
n
(|ξ|
2
+|ξ
1
|
2
−|ξ
2
|
2
)
e
iτ2ξξ
1
−1
2iξξ
1
=
1
2
e
−i(t
n
+τ)∆
∂
−1
x
h
e
i(t
n
+τ)∆
∂
−1
x
v(t
n
)·e
i(t
n
+τ)∆
v(t
n
)
i
−
1
2
e
−it
n
∆
∂
−1
x
h
e
it
n
∆
∂
−1
x
v(t
n
)·e
it
n
∆
v(t
n
)
i
.
DOI:10.12677/aam.2022.11107827366A^êÆ?Ð
w}
¤±,·‚Œ±òŽ{½Â¤
v
n+1
= v
n
+
1
2
e
−i(t
n
+τ)∆
∂
−1
x
h
e
i(t
n
+τ)∆
∂
−1
x
v
n
·e
i(t
n
+τ)∆
v
n
i
−
1
2
e
−it
n
∆
∂
−1
x
h
e
it
n
∆
∂
−1
x
v
n
·e
it
n
∆
v
n
i
,(7)
ùpn≥0,v
0
= u
0
,¿…v
n
= v
n
(x).
òÛ=Cþ(4)‡L5“\(7),=ŒVNÚš‚5Schr¨odinger•§(1)˜Âñꊎ
{(2).
4.½n1y²
!·‚Ï LÛÜØ©ÛÚ-½5©Ûé˜Âñ(J‰Ñ˜ ‡î‚y².dÚn2Œ•,
Û=Cþ(4)3Sobolev˜m´å,Kk
ku(t
n
)−u
n
k
H
γ
= ke
it
n
∆
v(t
n
)−e
it
n
∆
v
n
k
H
γ
= kv(t
n
)−v
n
k
H
γ
.
Ïd,·‚•I‡y²év(t
n
)Úv
n
˜Âñ½n.
£13!¥°()(5)
v(t
n+1
) = v(t
n
)+Φ
n
(v(t
n
))+L
n
,
ÚêŠ)(7)
v
n+1
= v
n
+Φ
n
(v
n
).
Ïd,Œü‡)Ø
v(t
n+1
)−v
n+1
= L
n
+S
n
,
Ù¥
S
n
= v(t
n
)−v
n
+Φ
n
(v(t
n
))−Φ
n
(v
n
).
e¡,·‚òOÛÜØL
n
Ú-½5S
n
.
Ún4.(ÛÜØ)γ>
1
2
.bu
0
∈H
γ
,K•3~êτ
0
ÚC>0,¦éu?¿0<τ≤τ
0
,k
eØª¤á
kL
n
k
H
γ
≤Cτ
2
,(8)
Ù¥τ
0
ÚC>0=•6uTÚkvk
L
∞
((0,T);H
γ
)
.
DOI:10.12677/aam.2022.11107827367A^êÆ?Ð
w}
Proof.ŠâL
n
½Â,2(ÜÚn2ÚÚn3,Œ
kL
n
k
H
γ
.
Z
τ
0
k

|e
i(t
n
+s)∆
v(t
n
+s)|
2
−|e
i(t
n
+s)∆
v(t
n
)|
2

k
H
γ
ds
.
Z
τ
0
kv(t
n
+s)−v(t
n
)k
H
γ
(kv(t
n
+s)k
H
γ
+kv(t
n
)k
H
γ
)ds.
5¿∂
t
v= −ie
−it∆
|e
it∆
v(t,x)|
2
,KŒ
kv(t
n
+s)−v(t
n
)k
H
γ
.
Z
s
0
k∂
t
v(t
n
+t)k
H
γ
dt
.
Z
s
0
k−ie
−it∆
|e
it∆
v(t)|
2
k
H
γ
dt
.
Z
s
0
kvk
2
L
∞
((0,T);H
γ
)
dt
.Cskvk
2
L
∞
((0,T);H
γ
)
.
nþŒ•,
kL
n
k
H
γ
≤Cτ
2
,
Ù¥C=•6ukvk
L
∞
((0,T);H
γ
)
.–d,·‚¤Ún4y².
Ún5.(-½5)γ>
1
2
.bu
0
∈H
γ
,K•3~êτ
0
ÚC>0,¦éu?¿0<τ≤τ
0
,ke
Øª¤á
kS
n
k
H
γ
≤(1+Cτ)kv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
2
H
γ
,
Ù¥τ
0
ÚC>0=•6uTÚkvk
L
∞
((0,T);H
γ
)
.
Proof.dS
n
½ÂŒ•,
kS
n
k
H
γ
≤kv(t
n
)−v
n
k
H
γ
+kΦ
n
(v(t
n
))−Φ
n
(v
n
)k
H
γ
.
£Φ
n
½Â,(ÜÚn2ÚÚn3,Œ
kΦ
n
(v(t
n
))−Φ
n
(v
n
)k
H
γ
= k−i
Z
τ
0
e
−i(t
n
+s)∆

|e
i(t
n
+s)∆
v(t
n
,x)|
2
−|e
i(t
n
+s)∆
v
n
|
2

dsk
H
γ
.
Z
τ
0
kv(t
n
)−v
n
k
H
γ
kv(t
n
)+v
n
k
H
γ
ds
.Cτkv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
2
H
γ
.
DOI:10.12677/aam.2022.11107827368A^êÆ?Ð
w}
Ïd,Œ
kS
n
k
H
γ
≤(1+Cτ)kv(t
n
)−v
n
k
H
γ
+Cτkv(t
n
)−v
n
k
2
H
γ
,
Ù¥C=•6ukvk
L
∞
((0,T);H
γ
)
.Ïd,·‚Œ±Ún5y².
e¡,(ÜÛÜØOÚ-½5(J,·‚‰Ñ½n1y².
Proof.ŠâÚn4ÚÚn5,Œ
kv(t
n+1
)−v
n+1
k
H
γ
≤Cτ
2
+(1+Cτ)kv(t
n
)−v
n
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