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AdvancesinAppliedMathematics
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,2022,11(10),7362-7372
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.1110782
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Low-RegularityIntegratorforthe
QuadraticNLSEquation
CuiNing
SchoolofFinancialMathematicsandStatistics,GuangdongUniversityofFinance,Guangzhou
Guangdong
Received:Sep.24
th
,2022;accepted:Oct.17
th
,2022;published:Oct.26
th
,2022
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[J].
A^
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,2022,11(10):7362-7372.
DOI:10.12677/aam.2022.1110782
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}
Abstract
Inthispaper,weintroduceafirstorderlow-regularityintegratorforthequadratic
nonlinearSchr¨odingerequation.Theschemeisexplicitandefficienttoimplement.
Inparticular,ourschemedoesnotcostanyadditionalderivativeforthefirstorder
convergence.Byrigorous erroranalysis,we show that thescheme provides firstorder
accuracyin
H
γ
(
T
)
forroughinitialdatain
H
γ
(
T
)
with
γ>
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/
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|
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t
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=
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∈
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(1+
|
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)
2
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|
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=
k
f
k
H
γ
.
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n
3.
(
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Ø
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¿
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1
2
,
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,
k
e
¡
Ø
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k
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it
n
∆
∂
−
1
x
h
e
it
n
∆
∂
−
1
x
v
(
t
n
)
·
e
it
n
∆
v
(
t
n
)
i
.
DOI:10.12677/aam.2022.11107827366
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w
}
¤
±
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·
‚
Œ
±
ò
Ž
{
½
Â
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v
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=
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+
1
2
e
−
i
(
t
n
+
τ
)∆
∂
−
1
x
h
e
i
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t
n
+
τ
)∆
∂
−
1
x
v
n
·
e
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(
t
n
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τ
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v
n
i
−
1
2
e
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it
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x
h
e
it
n
∆
∂
−
1
x
v
n
·
e
it
n
∆
v
n
i
,
(7)
ù
p
n
≥
0,
v
0
=
u
0
,
¿
…
v
n
=
v
n
(
x
).
ò
Û
=
C
þ
(4)
‡
L
5
“
\
(7),
=
Œ
V
N
Ú
š
‚
5
Schr¨odinger
•
§
(1)
˜
Â
ñ
ê
Š
Ž
{
(2).
4.
½
n
1
y
²
!
·
‚
Ï
L
Û
Ü
Ø
©
Û
Ú
-
½
5
©
Û
é
˜
Â
ñ
(
J
‰
Ñ
˜
‡
î
‚
y
²
.
d
Ú
n
2
Œ
•
,
Û
=
C
þ
(4)
3
Sobolev
˜
m
´
å
,
K
k
k
u
(
t
n
)
−
u
n
k
H
γ
=
k
e
it
n
∆
v
(
t
n
)
−
e
it
n
∆
v
n
k
H
γ
=
k
v
(
t
n
)
−
v
n
k
H
γ
.
Ï
d
,
·
‚
•
I
‡
y
²
é
v
(
t
n
)
Ú
v
n
˜
Â
ñ
½
n
.
£
1
3
!
¥
°
(
)
(5)
v
(
t
n
+1
) =
v
(
t
n
)+Φ
n
(
v
(
t
n
))+
L
n
,
Ú
ê
Š
)
(7)
v
n
+1
=
v
n
+Φ
n
(
v
n
)
.
Ï
d
,
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
O
Û
Ü
Ø
L
n
Ú
-
½
5
S
n
.
Ú
n
4.
(
Û
Ü
Ø
)
γ>
1
2
.
b
u
0
∈
H
γ
,
K
•
3
~
ê
τ
0
Ú
C>
0,
¦
é
u
?
¿
0
<τ
≤
τ
0
,
k
e
Ø
ª
¤
á
kL
n
k
H
γ
≤
Cτ
2
,
(8)
Ù
¥
τ
0
Ú
C>
0
=
•
6
u
T
Ú
k
v
k
L
∞
((0
,T
);
H
γ
)
.
DOI:10.12677/aam.2022.11107827367
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Proof.
Š
â
L
n
½
Â
,
2
(
Ü
Ú
n
2
Ú
Ú
n
3,
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
k
v
(
t
n
+
s
)
−
v
(
t
n
)
k
H
γ
(
k
v
(
t
n
+
s
)
k
H
γ
+
k
v
(
t
n
)
k
H
γ
)
ds.
5
¿
∂
t
v
=
−
ie
−
it
∆
|
e
it
∆
v
(
t,x
)
|
2
,
K
Œ
k
v
(
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
k
v
k
2
L
∞
((0
,T
);
H
γ
)
dt
.
Cs
k
v
k
2
L
∞
((0
,T
);
H
γ
)
.
n
þ
Œ
•
,
kL
n
k
H
γ
≤
Cτ
2
,
Ù
¥
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
)
.
–
d
,
·
‚
¤
Ú
n
4
y
²
.
Ú
n
5.
(
-
½
5
)
γ>
1
2
.
b
u
0
∈
H
γ
,
K
•
3
~
ê
τ
0
Ú
C>
0,
¦
é
u
?
¿
0
<τ
≤
τ
0
,
k
e
Ø
ª
¤
á
kS
n
k
H
γ
≤
(1+
Cτ
)
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
2
H
γ
,
Ù
¥
τ
0
Ú
C>
0
=
•
6
u
T
Ú
k
v
k
L
∞
((0
,T
);
H
γ
)
.
Proof.
d
S
n
½
Â
Œ
•
,
kS
n
k
H
γ
≤k
v
(
t
n
)
−
v
n
k
H
γ
+
k
Φ
n
(
v
(
t
n
))
−
Φ
n
(
v
n
)
k
H
γ
.
£
Φ
n
½
Â
,
(
Ü
Ú
n
2
Ú
Ú
n
3,
Œ
k
Φ
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
ds
k
H
γ
.
Z
τ
0
k
v
(
t
n
)
−
v
n
k
H
γ
k
v
(
t
n
)+
v
n
k
H
γ
ds
.
Cτ
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
2
H
γ
.
DOI:10.12677/aam.2022.11107827368
A^
ê
Æ
?
Ð
w
}
Ï
d
,
Œ
kS
n
k
H
γ
≤
(1+
Cτ
)
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
2
H
γ
,
Ù
¥
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
)
.
Ï
d
,
·
‚
Œ
±
Ú
n
5
y
²
.
e
¡
,
(
Ü
Û
Ü
Ø
O
Ú
-
½
5
(
J
,
·
‚
‰
Ñ
½
n
1
y
²
.
Proof.
Š
â
Ú
n
4
Ú
Ú
n
5,
Œ
k
v
(
t
n
+1
)
−
v
n
+1
k
H
γ
≤
Cτ
2
+(1+
Cτ
)
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
2
H
γ
,
Ù
¥
n
= 0
,
1
,
2
,...,
T
τ
−
1
Ú
C
=
•
6
u
T
Ú
k
v
k
L
∞
((0
,T
);
H
γ
)
.
|
^
S
“
{
Ú
Gronwall’s
Ø
ª
,
·
‚
Œ
k
v
(
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
Ò
¤
˜
Â
ñ
½
n
y
²
.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
]
Ï
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8
(11901120)
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[1]Bejenaru, I.andTao, T.(2006)SharpWell-PosednessandIll-PosednessResultsfor aQuadratic
Non-LinearSchr¨odingerEquation.
JournalofFunctionalAnalysis
,
233
,228-259.
https://doi.org/10.1016/j.jfa.2005.08.004
[2]Berestycki,H.andCazenave,T.(1981)InstabilityofStationaryStatesinNonlinear
Schr¨odingerandKlein-GordonEquations.
ComptesRendusdel’Acad´emiedesSciences
,
293
,
489-492.
[3]Li,Y.,Wu,Y.andXu,G.(2011)GlobalWell-PosednessfortheMass-CriticalNonlinear
Schr¨odingerEquationon
T
.
JournalofDifferentialEquations
,
250
,2715-2736.
https://doi.org/10.1016/j.jde.2011.01.025
[4]Li,Y.,Wu,Y.andXu,G.(2011)LowRegularityGlobalSolutionsfortheFocusingMass-
CriticalNLSin
R
∗
.
SIAMJournalonMathematicalAnalysis
,
43
,322-340.
https://doi.org/10.1137/090774537
DOI:10.12677/aam.2022.11107827369
A^
ê
Æ
?
Ð
w
}
[5]Weinstein,M.I. (1986) Lyapunov Stability of Ground States of Nonlinear Dispersive Evolution
Equations.
CommunicationsonPureandAppliedMathematics
,
39
,51-68.
https://doi.org/10.1002/cpa.3160390103
[6]Guo,B.andWu,Y.(1995)OrbitalStabilityofSolitaryWavesfortheNonlinearDerivative
Schr¨odingerEquation.
JournalofDifferentialEquations
,
123
,35-55.
https://doi.org/10.1006/jdeq.1995.1156
[7]Wu,Y.(2013)GlobalWell-PosednessoftheDerivativeNonlinearSchr¨odingerEquationsin
EnergySpace.
Analysis&PDE
,
6
,1989-2002.https://doi.org/10.2140/apde.2013.6.1989
[8]Wu, Y. (2015)Global Well-Posedness ontheDerivativeNonlinear Schr¨odinger Equation.
Anal-
ysis&PDE
,
8
,1101-1113.https://doi.org/10.2140/apde.2015.8.1101
[9]Liu,X.,Simpson,G.andSulem,C.(2013)StabilityofSolitaryWavesforaGeneralized
DerivativeNonlinearSchr¨odingerEquation.
JournalofNonlinearScience
,
23
,557-583.
https://doi.org/10.1007/s00332-012-9161-2
[10]LeCoz,S.andWu,Y.(2018)StabilityofMulti-SolitonsfortheDerivativeNonlinear
Schr¨odingerEquation.
InternationalMathematicsResearchNotices
,
2018
,4120-4170.
https://doi.org/10.1093/imrn/rnx013
[11]Ning,C.(2020)InstabilityofSolitaryWaveSolutionsforDerivativeNonlinearSchr¨odinger
EquationinBorderlineCase.
NonlinearAnalysis
,
192
,ArticleID:111665.
https://doi.org/10.1016/j.na.2019.111665
[12]Feng,B.andZhu,S.(2021)StabilityandInstabilityofStandingWavesfortheFractional
NonlinearSchr¨odingerEquations.
JournalofDifferentialEquations
,
292
,287-324.
https://doi.org/10.1016/j.jde.2021.05.007
[13]Guo,Q.andZhu,S.(2018)SharpThresholdofBlow-UpandScatteringfortheFractional
HartreeEquation.
JournalofDifferentialEquations
,
264
,2802-2832.
https://doi.org/10.1016/j.jde.2017.11.001
[14]Zhu,S.(2016)ExistenceandUniquenessofGlobalWeakSolutionsoftheCamassa-Holm
EquationwithaForcing.
DiscreteandContinuousDynamicalSystems
,
36
,5201-5221.
https://doi.org/10.3934/dcds.2016026
[15]Zhu,S.,Zhang,J.andYang,H.(2010)LimitingProfileoftheBlow-UpSolutionsforthe
Fourth-OrderNonlinearSchr¨odingerEquation.
DynamicsofPartialDifferentialEquations
,
7
,
187-205.https://doi.org/10.4310/DPDE.2010.v7.n2.a4
[16]Court´es, C.,Lagouti`ere, F.andRousset,F.(2020)ErrorEstimatesofFiniteDifferenceSchemes
fortheKorteweg-deVriesEquation.
IMAJournalofNumericalAnalysis
,
40
,628-685.
https://doi.org/10.1093/imanum/dry082
DOI:10.12677/aam.2022.11107827370
A^
ê
Æ
?
Ð
w
}
[17]Holden, H.,Koley, U. and Risebro, N.(2014) Convergence of a Fully Discrete Finite Difference
SchemefortheKorteweg-deVriesEquation.
IMAJournalofNumericalAnalysis
,
35
,1047-
1077.https://doi.org/10.1093/imanum/dru040
[18]Aksan, E.and
¨
Ozde¸s, A.(2006) NumericalSolutionofKorteweg-deVriesEquation by Galerkin
B-SplineFiniteElementMethod.
AppliedMathematicsandComputation
,
175
,1256-1265.
https://doi.org/10.1016/j.amc.2005.08.038
[19]Dutta,R.,Koley,U.andRisebro,N.H.(2015)ConvergenceofaHigherOrderSchemeforthe
Korteweg-deVriesEquation.
SIAMJournalonNumericalAnalysis
,
53
,1963-1983.
https://doi.org/10.1137/140982532
[20]Holden,H., Karlsen,K.H., Risebro,N.H. andTang, T.(2011) OperatorSplitting fortheKdV
Equation.
MathematicsofComputation
,
80
,821-846.
https://doi.org/10.1090/S0025-5718-2010-02402-0
[21]Holden,H.,Lubich,C.andRisebro,N.H.(2012)OperatorSplittingforPartialDifferential
EquationswithBurgersNonlinearity.
MathematicsofComputation
,
82
,173-185.
https://doi.org/10.1090/S0025-5718-2012-02624-X
[22]Ma, H.and Sun,W. (2001) Optimal Error Estimates of theLegendre-Petrov-Galerkin Method
fortheKorteweg-deVriesEquation.
SIAMJournalonNumericalAnalysis
,
39
,1380-1394.
https://doi.org/10.1137/S0036142900378327
[23]Shen,J.(2003)ANewDual-Petrov-GalerkinMethodforThirdandHigherOdd-OrderDif-
ferential Equations:Applicationto theKdVEquation.
SIAMJournalonNumericalAnalysis
,
41
,1595-1619.https://doi.org/10.1137/S0036142902410271
[24]Yan,J.andShu,C.W.(2002)ALocalDiscontinuousGalerkinMethodforKdVTypeEqua-
tions.
SIAMJournalonNumericalAnalysis
,
40
,769-791.
https://doi.org/10.1137/S0036142901390378
[25]Liu,H.andYan,J.(2006)ALocalDiscontinuousGalerkinMethodfortheKortewegdeVries
EquationwithBoundaryEffect.
JournalofComputationalPhysics
,
215
,197-218.
https://doi.org/10.1016/j.jcp.2005.10.016
[26]Hochbruck,M.andOstermann,A.(2010)ExponentialIntegrators.
ActaNumerica
,
19
,209-
286.https://doi.org/10.1017/S0962492910000048
[27]Hofmanov´a,M.and Schratz, K.(2017)AnExponential-Type Integrator forthe KdVEquation.
NumerischeMathematik
,
136
,1117-1137.https://doi.org/10.1007/s00211-016-0859-1
[28]Lubich,C.(2008)OnSplittingMethodsforSchr¨odinger-PoissonandCubicNonlinear
Schr¨odingerEquations.
MathematicsofComputation
,
77
,2141-2153.
https://doi.org/10.1090/S0025-5718-08-02101-7
DOI:10.12677/aam.2022.11107827371
A^
ê
Æ
?
Ð
w
}
[29]Ostermann, A. and Schratz, K. (2018) Low Regularity Exponential-Type Integrators for Semi-
linearSchr¨odingerEquations.
FoundationsofComputationalMathematics
,
18
,731-755.
https://doi.org/10.1007/s10208-017-9352-1
[30]Wu,Y.andYao,F.(2022)AFirst-OrderFourierIntegratorfortheNonlinearSchr¨odinger
EquationonTwithoutLossofRegularity.
MathematicsofComputation
,
91
,1213-1235.
https://doi.org/10.1090/mcom/3705
[31]Kato,T.andPonce,G.(1988)CommutatorEstimatesandtheEulerandNavier-StokesE-
quations.
CommunicationsonPureandAppliedMathematics
,
41
,891-907.
https://doi.org/10.1002/cpa.3160410704
DOI:10.12677/aam.2022.11107827372
A^
ê
Æ
?
Ð