设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
PureMathematics
n
Ø
ê
Æ
,2022,12(10),1636-1648
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1210177
o
š
‚
5
Schr¨odinger
•
§
A
Ÿ
þ
Å
ð
$
K
Ž
{
www
}}}
2
À
7
K
Æ
7
K
ê
Æ
†
Ú
O
Æ
,
2
À
2
²
Â
v
F
Ï
µ
2022
c
9
13
F
¶
¹
^
F
Ï
µ
2022
c
10
12
F
¶
u
Ù
F
Ï
µ
2022
c
10
20
F
Á
‡
©
ï
Ä
o
š
‚
5
Schr¨odinger
•
§
ä
k
A
Ÿ
þ
Å
ð
˜
«
$
K
Ž
{
.
T
Ž
{
Ø
=
±
˜
Â
ñ
,
¿
…
„
ä
k
A
Ÿ
þ
Å
ð
5
.
·
‚
Ï
L
î
‚
Ø
©
Û
,
y
²
Ð
Š
á
u
H
γ
+3
(
T
d
)
ž
,
é
½
T>
0
Ú
γ>
d
2
,
•
3
~
C
=
C
(
k
u
k
L
∞
(0
,T
;
H
γ
+3
)
)
>
0
,
¦
k
u
(
t
n
)
−
u
n
k
H
γ
+3
≤
Cτ,
Ù
¥
u
n
´
o
š
‚
5
Schr¨odinger
•
§
3
t
n
=
nτ
ž
ê
Š
)
.
d
,
ê
Š
)
Ÿ
þ
M
(
u
n
)
÷
v
|
M
(
u
n
)
−
M
(
u
0
)
|≤
Cτ
3
.
'
…
c
o
š
‚
5
Schr¨odinger
•
§
§
$
K
§
˜
Â
ñ
§
Ÿ
þ
Å
ð
ALow-RegularityIntegratorforthe
Fourth-OrderNonlinearSchr¨odinger
EquationwithAlmostMassConservation
CuiNing
©
Ù
Ú
^
:
w
}
.
o
š
‚
5
Schr¨odinger
•
§
A
Ÿ
þ
Å
ð
$
K
Ž
{
[J].
n
Ø
ê
Æ
,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,forroughinitialdatain
H
γ
+3
(
T
d
)
with
γ>
d
2
,uptosomefixed
timeof
T
,thereexists
C
=
C
(
k
u
k
L
∞
(0
,T
;
H
γ
+3
)
)
>
0
,suchthat
k
u
(
t
n
)
−
u
n
k
H
γ
+3
≤
Cτ,
where
u
n
denotes thenumerical solution at
t
n
=
nτ
.Moreover,themassofthenumer-
icalsolution
M
(
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.
Ú
ó
©
•
Ä
e
ã
o
š
‚
5
Schr¨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.2022.12101771637
n
Ø
ê
Æ
w
}
Ù
¥
T
d
=(0
,
2
π
)
d
,
u
(
t,
x
)
´
(
R
+
,
T
d
)
þ
E
Š
¼
ê
,
u
0
(
x
)
∈
H
γ
(
T
d
)(
γ
≥
0)
•
‰
½
Ð
Š
,
µ
=
±
1,
¿
…
d
≥
2.
T
•
§
3
L
2
¥
)
,
÷
v
Ÿ
þ
Å
ð
½
Æ
µ
M
(
u
(
t,
x
)) =
1
2
π
Z
T
d
|
u
(
t,
x
)
|
2
d
x
=
M
(
u
0
)
.
3
n
Ø
ï
Ä
•
¡
,Zhu
[1]
y
²
à
.
o
š
‚
5
Schr¨odinger
•
§
Ÿ
þ
.
ž
Ä
)
C
©
(
±
9
)
»
5
.Guo[2]
y
²
à
.
o
š
‚
5
Schr¨odinger
•
§
Ÿ
þ
‡
.
ž
Û
·
½
5
±
9
»
•
Ð
Š
)
Ñ
5
.
Ù
¦
.
Schr¨odinger
•
§
•
2
•
ï
Ä
,
Œ
ë
„
[3–10]
.
3
ê
Š
ï
Ä
•
¡
,
é
u
ä
k
1
w
Ð
©
^
‡
•
§
,
~
„
ê
Š
•{
k
©
•
{
[11,12],
k
•
©•{
[13,14],
k
•
•{
[15,16],
Ì
•{
[17,18],
Ø
ë
Y
Galerkin
•{
[19,20],
•
ê
È
©
ì
{
[21,22]
.
•
§
)
Ø
1
w
ž
,
K
‡
m
u
$
K
5
Ž
{
,
X
StrangeSplitting
Ž
{
[23],
#.
•
ê
.
Ž
{
[24],Fourier
È
©
Ž
{
[25]
.
é
u
o
š
‚
5
Schr¨odinger
•
§
,Ning [26]
J
Ñ
˜
«
ê
Š
‚
ª
,
¦
3
›
”
n
ê
œ
¹
e
ˆ
˜
Â
ñ
,
=
Ð
©
Š
á
u
H
γ
+3
ž
•
§
)
3
H
γ
¥
˜
Â
ñ
.
(
Ü
©
z
[26],
·
‚
E
˜
«
#
$
K
Ž
{
,
Œ
±
o
š
‚
5
Schr¨odinger
•
§
(1)
˜
Â
ñ
,
d
„
÷
v
A
Ÿ
þ
Å
ð
.
3
‰
Ñ
ä
N
Ž
{
c
,
·
‚
Ä
k
½
˜
‡
^
¼
ê
.
-
ϕ
(
z
) =
e
z
−
1
z
,z
6
= 0
,
1
,z
= 0
,
¿
…
Ψ(
f
) =
e
−
iτ
(
−
∆)
2
f
−
iµτe
−
iτ
(
−
∆)
2
[
f
2
·
ϕ
(2
iτ
(
−
∆)
2
)
¯
f
]
.
½
Â
e
?
Ž
f
I
(
f
) = Ψ(
f
)
−
e
−
iτ
(
−
∆)
2
f,
Ú
J
(
f
) =
H
(
f
)
e
−
iτ
(
−
∆)
2
f,
Ù
¥
H
(
f
) =
−k
f
k
−
2
L
2
h
I
(
f
)
,e
−
iτ
(
−
∆)
2
f
i
+
1
2
k
I
(
f
)
k
2
L
2
.
y
3
,
·
‚
‰
Ñ
o
š
‚
5
Schr¨odinger
•
§
(1)
#
$
K
Ž
{
µ
u
n
= Ψ(
u
n
−
1
)+
J
(
u
n
−
1
)
,n
= 0
,
1
,
2
,...,
T
τ
.
(2)
DOI:10.12677/pm.2022.12101771638
n
Ø
ê
Æ
w
}
Ï
d
,
·
‚
Ì
‡
½
n
•
½
n
1.
u
n
´
o
š
‚
5
Schr¨odinger
•
§
(1)
÷
v
(2)
‚
ª
ê
Š
)
,
½
T>
0
.
é
?
¿
γ>
d
2
,
b
u
0
(
x
)
∈
H
γ
+3
(
T
d
)
,
•
3
~
ê
τ
0
,
C>
0
¦
é
?
¿
0
<τ
≤
τ
0
,
k
k
u
(
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
Ú
C>
0
=
•
6
u
T
Ú
k
u
k
L
∞
((0
,T
);
H
γ
+3
)
.
©
S
ü
X
e
.
3
1
2
!
¥
,
·
‚
‰
Ñ
˜
Î
ÒÚ
˜
k^
Ú
n
.
3
1
3
!
¥
,
·
‚
‰
Ñ
˜
ê
Š
‚
ª
Ì
‡
E
L
§
.
3
1
4
!
¥
,
·
‚
y
²
½
n
1.
2.
ý
•
£
3
!
¥
,
·
‚
0
˜
½
Â
!
5
Ÿ
Ú
-
‡
O
.
•
•
B
P
Ò
,
·
‚
¦
^
A
.
B
½
ö
B
&
A
5
L
«
X
e
¹
Â
µ
•
3
,
ý
é
~
ê
C>
0,
¦
A
≤
CB
,
¿
¦
^
A
∼
B
5
L
«
A
.
B
.
A
.
é
u
•
þ
ξ
:=(
ξ
1
,...,ξ
d
)
∈
Z
d
,
ξ
1
:= (
ξ
11
,...,ξ
1
d
)
∈
Z
d
,
x
:=(
x
1
,...,x
d
)
∈
T
d
,
S
È
Ú
½
Â
X
e
:
ξ
·
x
=
ξ
1
x
1
+
...
+
ξ
d
x
d
,
|
ξ
|
2
=
|
ξ
1
|
2
+
...
+
|
ξ
d
|
2
.
½
Â
h·
,
·i
•
L
2
S
È
h
f,g
i
= Re
Z
T
d
f
(
x
)
g
(
x
)
d
x
.
3
±
Ï
T
d
þ
,
·
‚
½
Â
¼
ê
f
Fourier
C
†
•
ˆ
f
ξ
=
1
(2
π
)
d
Z
T
d
e
−
i
ξ
·
x
f
(
x
)
d
x
,
Ù
Fourier
_
C
†
•
f
(
x
) =
P
ξ
∈
Z
d
e
−
i
ξ
·
x
ˆ
f
ξ
.
é
¼
ê
f
∈
L
2
(
T
d
),
·
‚
½
Â
f
(
x
)
Fourier
Ð
m
ª
•
f
(
x
) =
X
ξ
∈
Z
d
ˆ
f
ξ
e
i
ξ
·
x
.
Fourier
C
†
~
^
5
Ÿ
k
f
k
2
L
2
= 2
π
X
ξ
∈
Z
d
|
ˆ
f
ξ
|
2
,
DOI:10.12677/pm.2022.12101771639
n
Ø
ê
Æ
w
}
Ú
c
fg
(
ξ
) =
X
ξ
,
ξ
1
∈
Z
d
ˆ
f
ξ
−
ξ
1
ˆ
g
ξ
1
.
½
Â
Sobolev
˜
m
H
γ
,
γ>
0
‰
ê
•
f
2
H
γ
(
T
d
)
=
X
ξ
∈
Z
d
(1+
|
ξ
|
)
2
γ
|
ˆ
f
ξ
|
2
.
·
‚
½
Â
Ž
f
(
−
∆)
−
1
•
\
(
−
∆)
−
1
f
=
(
|
ξ
|
−
2
ˆ
f
ξ
,
ξ
6
= 0
,
0
,
ξ
= 0
.
y
3
,
·
‚
‰
Ñ
‡
^
˜
-
‡
O
.
Ú
n
2.
£
å
5
Ÿ
¤
é
f
∈
H
γ
,t
∈
R
k
k
e
−
i
(
−
∆)
2
t
f
k
H
γ
=
k
f
k
H
γ
.
Proof.
Š
â
H
γ
‰
ê
½
Â
,
Œ
•
k
e
−
i
(
−
∆)
2
t
f
k
H
γ
=
X
ξ
∈
Z
d
(1+
|
ξ
|
)
2
γ
|
\
e
−
i
(
−
∆)
2
t
f
ξ
|
2
=
X
ξ
∈
Z
d
(1+
|
ξ
|
)
2
γ
|
e
−
i
|
ξ
|
4
t
ˆ
f
ξ
|
2
=
X
ξ
∈
Z
d
(1+
|
ξ
|
)
2
γ
|
ˆ
f
ξ
|
2
=
k
f
k
H
γ
.
Ú
n
3.
(Kato-Ponce
Ø
ª
)
é
?
¿
γ>
d
2
,
f,g
∈
H
γ
,
k
e
¡
Ø
ª
¤
á
k
fg
k
H
γ
≤k
f
k
H
γ
k
g
k
H
γ
.
Proof.
T
Ú
n
y
²
Œ
ë
„
©
z
[27].
3.
ê
Š
‚
ª
E
3
!
¥
,
·
‚
0
ê
Š
‚
ª
E
L
§
.
•
{
z
Î
Ò
,
·
‚
ò
Ž
Ñ
¤
9
ž
˜
ƒ
'¼
ê
¥
˜
m
C
þ
x
,
X
u
(
t
) =
u
(
t,
x
).
,
,
·
‚
^
τ>
0
L
«ž
m
Ú
•
,
t
n
=
nτ
L
«ž
m
‚
.
DOI:10.12677/pm.2022.12101771640
n
Ø
ê
Æ
w
}
y
3
,
·
‚
£
©
z
[26]
¥
é
o
š
‚
5
Schr¨odinger
•
§
(1)
Ž
{
E
,
Ï
L
Duhamel
ú
ª
,
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
=
C
þ
v
(
t,
x
) =
e
it
(
−
∆)
2
u
(
t,
x
)
,
(3)
K
Œ
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.
Ï
L
Fourier
Ð
m
,
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
(2
iτ
(
−
∆)
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
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
x
e
it
n
α
Z
τ
0
e
is
2
|
ξ
1
|
4
(
e
isβ
−
1)
ds.
(6)
Ù
¥
·
‚
{P
¯
ˆ
v
1
=
¯
ˆ
v
ξ
1
(
t
n
),ˆ
v
2
=ˆ
v
ξ
2
(
t
n
)
Ú
ˆ
v
3
=ˆ
v
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
,
K
α
Œ
±
©
)
¤
α
= 2
|
ξ
1
|
4
+
3
X
j,k
=1
j
6
=
k
|
ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
|
ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
ξ
2
j
¯
ξ
k
¯
ξ
h
= 2
|
ξ
1
|
4
+
β,
Ù
¥
β
=
3
X
j,k
=1
j
6
=
k
|
ξ
j
|
2
ξ
j
¯
ξ
k
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
|
ξ
j
|
2
ξ
k
¯
ξ
h
+
3
X
j,k,h
=1
j
6
=
k
6
=
h
ξ
2
j
¯
ξ
k
¯
ξ
h
=
O
(
3
X
j,k
=1
j
6
=
k
|
ξ
j
|
3
|
ξ
k
|
)
.
DOI:10.12677/pm.2022.12101771641
n
Ø
ê
Æ
w
}
e
¡
,
·
‚
Ø
\
y
²
/
Û
Ñ
˜
O
,
ƒ
'
y
²
Œ
„
[26].
Ä
k
,
·
‚
‰
Ñ
Û
Ü
Ø
‘
O
.
Ú
n
4.
(
Û
Ü
Ø
)
•
3
~
ê
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
,
¦
k
v
(
t
n
+1
)
−
Φ
n
(
v
(
t
n
))
k
H
γ
≤
Cτ
2
,
Ù
g
,
·
‚
0
-
½
5
O
.
Ú
n
5.
(
-
½
5
)
f,g
∈
H
γ
,
é
u
γ>
d
2
,
K
k
k
Φ
n
(
f
)
−
Φ
n
(
g
)
k
H
γ
≤
(1+
Cτ
)
k
f
−
g
k
H
γ
+
Cτ
k
f
−
g
k
3
H
γ
,
3
þ
ã
˜
Â
ñ
Ä
:
þ
,
·
‚
F
"
E
˜
«
Œ
±
±
)
Ô
n
5
Ž
{
,
)
3
÷
v
˜
Â
ñ
c
J
e
,
„
U
±
A
Ÿ
þ
Å
ð
5
,
Ï
d
·
‚
‡
é
Ž
{
?
1?
.
½
Â
I
n
(
v
) = Φ
n
(
v
)
−
v,
(7)
ù
p
Φ
n
d
(5)
‰
Ñ
.
2
½
Â
J
n
(
v
) =
H
n
(
v
)
v,
(8)
Ù
¥
H
n
(
v
) =
−k
v
k
−
2
L
2
h
I
n
(
v
)
,v
i
+
1
2
k
I
n
(
v
)
k
2
L
2
.
k
þ
¡
?
,
·
‚
E
Ñ
÷
v
A
Ÿ
þ
Å
ð
ê
Š
‚
ª
v
n
+1
=
v
n
+
I
n
(
v
n
)+
J
n
(
v
n
)
.
(9)
ò
Û
=
C
þ
(3)
‡
L
5
“
\
(9),
=
Œ
o
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
(3)
3
Sobolev
˜
m
´
å
,
K
k
k
u
(
t
n
)
−
u
n
k
H
γ
=
k
e
−
it
n
(
−
∆)
2
v
(
t
n
)
−
e
−
it
n
(
−
∆)
2
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
Ú
A
Ÿ
þ
Å
ð
5
.
Ä
k
,
·
‚
‰
Ñ
I
n
(
v
)
Â
ñ
.
DOI:10.12677/pm.2022.12101771642
n
Ø
ê
Æ
w
}
Ú
n
6.
v
∈
H
γ
,
γ>
d
2
ž
,
•
3
~
ê
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
,
e
¡
Ø
ª
¤
á
k
I
n
(
v
)
k
L
2
≤
Cτ.
Proof.
£
I
n
(
v
)
½
Â
(7)
Œ
•
,
I
n
(
v
) =
−
iµτe
it
n
(
−
∆)
2
(
e
−
it
n
(
−
∆)
2
v
)
2
·
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
.
Ï
d
,
k
I
n
(
v
)
k
L
2
.
τ
k
(
e
−
it
n
(
−
∆)
2
v
)
2
k
L
∞
·k
ϕ
(2
iτ
(
−
∆)
2
)
e
−
it
n
(
−
∆)
2
v
k
L
2
.
τ
k
v
k
3
H
γ
.
¤
±
,
Ú
n
6
y
.
Ù
g
,
3
‰
Ñ
h
I
n
(
v
)
,v
i
Â
ñ
ƒ
c
,
·
‚
k
‡
‰
Ñ
˜
‡
‡
^
Ø
ª
.
Ú
n
7.
•
3
~
ê
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
,
¦
kR
n
2
k
H
−
γ
≤
Cτ
2
.
Proof.
Š
â
R
n
2
½
Â
(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
i
ξ
·
x
e
it
n
α
(
3
X
j,k
=1
j
6
=
k
|
ξ
j
|
3
|
ξ
k
|·
¯
ˆ
v
j
ˆ
v
k
ˆ
v
h
)
.
τ
2
X
ξ
∈
Z
d
X
ξ
1
,
ξ
2
,
ξ
3
∈
Z
d
ξ
=
ξ
1
+
ξ
2
+
ξ
3
e
i
ξ
·
x
e
it
n
α
(
|
ξ
j
|
3
¯
ˆ
v
j
·|
ξ
k
|
ˆ
v
k
·
ˆ
v
h
)
.
Ï
d
,
d
Sobolev
Ø
ª
Œ
kR
n
2
k
H
−
γ
.
τ
2
k|∇|
3
v
·|∇|
v
·
v
k
H
−
γ
.
τ
2
k|∇|
3
v
·|∇|
v
·
v
k
L
1
.
τ
2
k|∇|
3
v
k
L
2
k|∇|
v
k
L
2
k|
v
k
L
∞
.
τ
2
k
v
k
3
H
γ
+3
.
¤
±
,
T
Ú
n
y
.
Ú
n
8.
v
∈
H
γ
+3
,
γ>
d
2
ž
,
•
3
~
ê
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
,
e
¡
Ø
ª
¤
á
|h
I
n
(
v
)
,v
i|≤
Cτ
2
.
DOI:10.12677/pm.2022.12101771643
n
Ø
ê
Æ
w
}
Proof.
£
I
n
(
v
)
½
Â
(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
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
x
e
it
n
α
Z
τ
0
e
is
2
|
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
¯
ˆ
v
1
ˆ
v
2
ˆ
v
3
e
i
ξ
·
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
.
Ï
d
,
·
‚
k
h
I
n
(
v
)
,v
i
=
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,v
i
+
h−R
n
2
,v
i
.
d
S
È
½
Â
,
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,v
i
= 0
.
(
Ü
Ú
n
7,
Œ
,
|h
I
n
(
v
)
,v
i|
=
|h−R
n
2
,v
i|
.
kR
n
2
k
H
−
γ
k
v
k
H
γ
.
τ
2
k
v
k
4
H
γ
+3
.
Ï
d
,
y
²
Ú
n
(
Ø
.
e
5
,
(
Ü
Û
Ü
Ø
O
Ú
-
½
5
(
J
,
·
‚
‰
Ñ
½
n
1
y
²
.
Proof.
Ä
k
,
d
ê
Š
)
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
)
.
Š
â
Ú
n
4
Ú
Ú
n
5,
Œ
k
v
(
t
n
+1
)
−
v
n
+1
k
H
γ
≤k
Φ
n
(
v
n
)
−
Φ
n
(
v
(
t
n
))
k
H
γ
+
k
Φ
n
(
v
(
t
n
))
−
v
(
t
n
+1
)
k
H
γ
+
k
J
n
(
v
n
)
k
H
γ
≤
(1+
Cτ
)
k
v
(
t
n
)
−
v
n
k
H
γ
+
Cτ
k
v
(
t
n
)
−
v
n
k
3
H
γ
+
Cτ
2
+
k
J
n
(
v
n
)
k
H
γ
.
Š
â
J
n
½
Â
(8),
k
k
J
n
(
v
n
)
k
H
γ
≤|
H
n
(
v
n
)
|k
v
n
k
H
γ
≤|
H
n
(
v
n
)
|
(
k
v
n
−
v
(
t
n
)
k
H
γ
+
k
v
(
t
n
)
k
H
γ
)
.
DOI:10.12677/pm.2022.12101771644
n
Ø
ê
Æ
w
}
d
Ú
n
6
Ú
Ú
n
8,
Œ
|
H
n
(
v
n
)
|≤k
v
n
k
−
2
L
2
(
τ
2
k
v
n
k
4
H
γ
+3
+
1
2
τ
2
k
v
n
k
6
H
γ
)
≤
Cτ
2
(1+
k
v
n
−
v
(
t
n
)
k
4
H
γ
)
.
(10)
Ï
d
,
·
‚
k
k
J
n
(
v
n
)
k
H
γ
≤
Cτ
2
(1+
k
v
n
−
v
(
t
n
)
k
5
H
γ
)
.
(11)
(
Ü
(11),
Œ
k
v
(
t
n
+1
)
−
v
n
+1
k
H
γ
≤
Cτ
2
+(1+
Cτ
)
k
v
n
−
v
(
t
n
)
k
H
γ
+
Cτ
k
v
n
−
v
(
t
n
)
k
5
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
¥
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
.
þ
ª
¤
á
,
Ò
¿
›
X
˜
Â
ñ
…
k
O
÷
v
k
v
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
h
v
n
+1
,v
n
+1
i
=
h
v
n
+
I
n
(
v
n
)+
J
n
(
v
n
)
,v
n
+
I
n
(
v
n
)+
J
n
(
v
n
)
i
=
k
v
n
k
2
L
2
+2
h
I
n
(
v
n
)
,J
n
(
v
n
)
i
+
k
J
n
(
v
n
)
k
2
L
2
+2
h
I
n
(
v
n
)
,v
n
i
+
h
J
n
(
v
n
)
,v
n
i
+
k
I
n
(
v
n
)
k
2
L
2
.
Š
â
I
n
(
v
n
)
Ú
J
n
(
v
n
)
½
Â
Œ
2
h
I
n
(
v
n
)
,v
n
i
+
h
J
n
(
v
n
)
,v
n
i
+
k
I
n
(
v
n
)
k
2
L
2
= 0
,
K
h
v
n
+1
,v
n
+1
i
=
k
v
n
k
2
L
2
+2
h
I
n
(
v
n
)
,J
n
(
v
n
)
i
+
k
J
n
(
v
n
)
k
2
L
2
.
˜
•
¡
,
|
^
Ú
n
8
Ú
(10),
·
‚
Œ
h
I
n
(
v
n
)
,J
n
(
v
n
)
i≤|
H
n
(
v
n
)
||h
I
n
(
v
n
)
,v
n
i|
.
Cτ
4
.
,
˜
•
¡
,
(
Ü
(10)
Ú
(11),
k
k
J
n
(
v
n
)
k
2
L
2
≤|
H
n
(
v
n
)
|
2
k
v
n
k
2
L
2
≤
Cτ
4
.
DOI:10.12677/pm.2022.12101771645
n
Ø
ê
Æ
w
}
n
þ
Œ
|k
v
n
+1
k
2
L
2
−k
v
n
k
2
L
2
|≤
Cτ
4
,
=
|
M
(
v
n
+1
)
−
M
(
v
n
)
|≤
Cτ
4
.
(12)
é
(12)
?
1
S
“
Œ
|
M
(
v
n
)
−
M
(
u
0
)
|≤
Cτ
3
,
C
=
•
6
u
k
v
k
L
∞
((0
,T
);
H
γ
+3
)
.
Ï
d
,
·
‚
½
n
1
y
²
.
Ä
7
‘
8
I
[
g
,
‰
Æ
Ä
7
]
Ï
‘
8
(11901120)
"
ë
•
©
z
[1]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
[2]Guo,Q.(2016)ScatteringfortheFocusing
L
2
-Supercriticaland
˙
H
2
-SubcriticalBiharmonic
NLSEquations.
CommunicationsinPartialDifferentialEquations
,
41
,185-207.
https://doi.org/10.1080/03605302.2015.1116556
[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
[5]Wu,Y.(2013)GlobalWell-PosednessoftheDerivativeNonlinearSchr¨odingerEquationsin
EnergySpace.
Analysis&PDE
,
6
,1989-2002.https://doi.org/10.2140/apde.2013.6.1989
[6]Wu, Y. (2015)Global Well-Posednesson theDerivative NonlinearSchr¨odingerEquation.
Anal-
ysis&PDE
,
8
,1101-1113.https://doi.org/10.2140/apde.2015.8.1101
DOI:10.12677/pm.2022.12101771646
n
Ø
ê
Æ
w
}
[7]Liu,X.,Simpson,G.andSulem,C.(2013)StabilityofSolitaryWavesforaGeneralized
DerivativeNonlinearSchr¨odingerEquation.
JournalofNonlinearScience
,
23
,557-583.
https://doi.org/10.1007/s00332-012-9161-2
[8]Ning,C.(2020)InstabilityofSolitaryWaveSolutionsforDerivativeNonlinearSchr¨odinger
EquationinBorderlineCase.
NonlinearAnalysis
,
192
,ArticleID:111665.
https://doi.org/10.1016/j.na.2019.111665
[9]Feng,B.andZhu,S.(2021)StabilityandInstabilityofStandingWavesfortheFractional
NonlinearSchr¨odingerEquations.
JournalofDifferentialEquations
,
292
,287-324.
https://doi.org/10.1016/j.jde.2021.05.007
[10]Zhu,S.(2016)ExistenceandUniquenessofGlobalWeakSolutionsoftheCamassa-Holm
EquationwithaForcing.
DiscreteandContinuousDynamicalSystems
,
36
,5201-5221.
https://doi.org/10.3934/dcds.2016026
[11]Holden,H., Karlsen,K.H., Risebro,N.H. andTang,T.(2011) OperatorSplittingfor theKdV
Equation.
MathematicsofComputation
,
80
,821-846.
https://doi.org/10.1090/S0025-5718-2010-02402-0
[12]Holden,H.,Lubich,C.andRisebro,N.H.(2012)OperatorSplittingforPartialDifferential
EquationswithBurgersNonlinearity.
MathematicsofComputation
,
82
,173-185.
https://doi.org/10.1090/S0025-5718-2012-02624-X
[13]Court´es, C.,Lagouti`ere, F.andRousset,F.(2020)ErrorEstimatesofFiniteDifferenceSchemes
fortheKorteweg-deVriesEquation.
IMAJournalofNumericalAnalysis
,
40
,628-685.
https://doi.org/10.1093/imanum/dry082
[14]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
[15]Aksan, E.and
¨
Ozde¸s, A.(2006) NumericalSolutionofKorteweg-deVries EquationbyGalerkin
B-SplineFiniteElementMethod.
AppliedMathematicsandComputation
,
175
,1256-1265.
https://doi.org/10.1016/j.amc.2005.08.038
[16]Dutta,R.,Koley,U. andRisebro,N.H.(2015)ConvergenceofaHigherOrderScheme forthe
Korteweg-deVriesEquation.
SIAMJournalonNumericalAnalysis
,
53
,1963-1983.
https://doi.org/10.1137/140982532
[17]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
DOI:10.12677/pm.2022.12101771647
n
Ø
ê
Æ
w
}
[18]Shen,J.(2003)ANewDual-Petrov-GalerkinMethodforThirdandHigherOdd-OrderDif-
ferential Equations:Application totheKdV Equation.
SIAMJournalonNumericalAnalysis
,
41
,1595-1619.https://doi.org/10.1137/S0036142902410271
[19]Yan,J.andShu,C.W.(2002)ALocalDiscontinuousGalerkinMethodforKdVTypeEqua-
tions.
SIAMJournalonNumericalAnalysis
,
40
,769-791.
https://doi.org/10.1137/S0036142901390378
[20]Liu,H.andYan,J.(2006)ALocalDiscontinuousGalerkinMethodfortheKortewegdeVries
EquationwithBoundaryEffect.
JournalofComputationalPhysics
,
215
,197-218.
https://doi.org/10.1016/j.jcp.2005.10.016
[21]Hochbruck,M.andOstermann,A.(2010)ExponentialIntegrators.
ActaNumerica
,
19
,209-
286.https://doi.org/10.1017/S0962492910000048
[22]Hofmanov´a, M.andSchratz,K. (2017)An Exponential-TypeIntegratorfor theKdV Equation.
NumerischeMathematik
,
136
,1117-1137.https://doi.org/10.1007/s00211-016-0859-1
[23]Lubich,C.(2008)OnSplittingMethodsforSchr¨odinger-PoissonandCubicNonlinear
Schr¨odingerEquations.
MathematicsofComputation
,
77
,2141-2153.
https://doi.org/10.1090/S0025-5718-08-02101-7
[24]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
[25]Wu,Y.andYao,F.(2022)AFirst-OrderFourierIntegratorfortheNonlinearSchr¨odinger
EquationonTwithoutLossofRegularity.
MathematicsofComputation
,
91
,1213-1235.
https://doi.org/10.1090/mcom/3705
[26]Ning,C.(2022)Low-RegularityIntegratorfortheBiharmonicNLSEquation.Preprint.
[27]Kato,T.andPonce,G.(1988)CommutatorEstimatesandtheEulerandNavier-StokesE-
quations.
CommunicationsonPureandAppliedMathematics
,
41
,891-907.
https://doi.org/10.1002/cpa.3160410704
DOI:10.12677/pm.2022.12101771648
n
Ø
ê
Æ