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PureMathematicsnØêÆ,2020,10(9),914-920
PublishedOnlineSeptember2020inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2020.109106
2ÂCXêK(m,n)•§°()
´´´æææxxx§§§‡‡‡ùùù
∗
HuŒÆênÆ§Hï
Email:
∗
418495670@qq.com
ÂvFϵ2020c830F¶¹^Fϵ2020c920F¶uÙFϵ2020c927F
Á‡
©8´|^zE|z{pš‚5•§gŽ5ïÄ2ÂCX êK(m,n)•§°()"
ÏLÎÒOŽ¼T•§#°()"
'…c
°()§2ÂK(m,n)•§§ÎÒOŽ
ExactSolutionsfortheGeneralizedK(m,n)
EquationwithVariableCoefficients
YatingYi,ChaohongPan
∗
SchoolofMathematicsandPhysics,UniversityofSouthChina,HengyangHunan
Email:
∗
418495670@qq.com
Received:Aug.30
th
,2020;accepted:Sep.20
th
,2020;published:Sep.27
th
,2020
Abstract
The objective of thispaperis toinvestigate exact solutionsfor thegeneralizedK(m,n)
∗ÏÕŠö"
©ÙÚ^:´æx,‡ù.2ÂCXêK(m,n)•§°()[J].nØêÆ,2020,10(9):914-920.
DOI:10.12677/pm.2020.109106
´æx§‡ù
withvariablecoefficients.Anextendedapproachisproposedforreducingtheorder
oftheequationswithhigherordernonlinearity.Newexactsolutionsarefoundby
symboliccomputation.
Keywords
ExactSolutions,GeneralizedK(m,n)Equation,SymbolicComputation
Copyright
c
2020byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
¯¤±•§Korteweg-deVries(KdV)•§
u
t
+auu
x
+bu
xxx
= 0,(1.1)
´˜‡^5£ãfYÅ3šÊ56NL¡$Ä[1],Ù¥aÚb´?¿š"~ê"
Cc5§duK(n,n)•§2• A^§˜ïÄÆöò 81=•CXêKdV.•§[2–6]"
ÏLe¡C†
u(x,t) = v
1
n−1
(x,t),(1.2)
†9Ï•§E|§Lv<[7]¼XeK(n,n)•§NõwªäkØÓ(1Å)
u
t
+a(t)u
x
+b(t)(u
n
)
x
+k(t)(u
n
)
xxx
= 0,n6= 0,1.(1.3)
ù)•)n¼ê±ÏÅ)ÚV-¼ê)"
ùŸ©Ù§É©Ù[7]éu§·‚ÏLXeC†ïÄ2ÂKdV•§µ
u
t
+a(t)u
x
+b(t)(u
m
)
x
+k(t)(u
n
)
xxx
= 0,m,n6= 0,1,(1.4)
Ù¥a(t)b(t)k(t) 6= 0.
ùp§·‚•é•§(1.4)compacton)a,"compacton)(Ò´äk;|8áÅ))"
Compacton)®²3©z[8–12]2• ïÄ"´éum6=nž•§(1.4)compacton)„vk
ïÄ"
DOI:10.12677/pm.2020.109106915nØêÆ
´æx§‡ù
2.•§(1.4)z
•¼compacton)§·‚òe¡•§
u(x,t) = c
1
+c
2
z(ξ),ξ= p(t)x+q(t),(2.1)
“\(1.4)¥,Ù¥z
0
=
√
a
1
+a
2
z+a
3
z
2
,=±1,a
1
,a
2
,a
3
,c
1
†c
2
´¢~ê§p(t)†q(t)
´™•¼ê"4∆=a
2
2
−4a
1
a
3
,Ò¬ke¡¯¢[13]"a
3
<0†∆>0,·‚kz(ξ)=

√
∆
2a
3
sin(
√
−a
3
ξ)−
a
2
2a
3
†z(ξ) =

√
∆
2a
3
cos(
√
−a
3
ξ)−
a
2
2a
3
.
5Ÿ1.m= n,|^C†
u= v
p
,(3.1)
Ù¥p=
α
n−1
,α∈Z†α6= 0,•§(1.4)UC¤e¡ü«œ/"
(i) XJα≥2,•§(1.4)UC†¤
v
t
+a(t)v
x
+b(t)nv
α
v
x
+k(t)n(np−1)(np−2)v
α−2
v
3
x
+3k(t)n(np−1)v
α−1
v
x
v
xx
+k(t)nv
α
v
xxx
= 0.
(3.2)
(ii) XJα<2,•§(1.4)UC†¤
v
2−α
v
t
+a(t)v
2−α
v
x
+b(t)nv
2
v
x
+k(t)n(np−1)(np−2)v
3
x
+3k(t)n(np−1)vv
x
v
xx
+k(t)nv
2
v
xxx
= 0.
(3.3)
5Ÿ2.m6= nÚ
n−1
m−1
=
β+2
α
6= 1,|^C†
u= v
p
,(3.4)
Ù¥α,β∈Z,α6=0,β6= −2andp=
α
m−1
=
β+2
n−1
,•§(1.4)Uz{¤
v
t
+a(t)v
x
+b(t)mv
α
v
x
+k(t)n(np−1)(np−2)v
β
v
3
x
+3k(t)n(np−1)v
β+1
v
x
v
xx
+k(t)nv
β+2
v
xxx
= 0.
(3.5)
5Ÿ1Ú2y²|^C†(3.1),·‚k











u
t
= pv
p−1
v
t
,
u
x
= pv
p−1
v
x
,
(u
m
)
x
= mpv
mp−1
v
x
,
(u
n
)
xxx
= np(np−1)(np−2)v
np−3
v
3
x
+3np(np−1)v
np−2
v
x
v
xx
+npv
np−1
v
xxx
.
(3.6)
ò(3.6)“\•§(1.4),·‚
pv
p−1
v
t
+a(t)pv
p−1
v
x
+b(t)mpv
mp−1
v
x
+k(t)np(np−1)(np−2)v
np−3
v
3
x
+3k(t)np(np−1)v
np−2
v
x
v
xx
+k(t)npv
np−1
v
xxx
= 0.
(3.7)
DOI:10.12677/pm.2020.109106916nØêÆ
´æx§‡ù
þªü঱v
s
,·‚k
v
s
[pv
p−1
v
t
+a(t)pv
p−1
v
x
+b(t)mpv
mp−1
v
x
+k(t)np(np−1)(np−2)v
np−3
v
3
x
+3k(t)np(np−1)v
np−2
v
x
v
xx
+k(t)npv
np−1
v
xxx
] = 0,
(3.8)
½ö
pv
s+p−1
v
t
+a(t)pv
s+p−1
v
x
+b(t)mpv
s+mp−1
v
x
+k(t)np(np−1)(np−2)v
s+np−3
v
3
x
+3k(t)np(np−1)v
s+np−2
v
x
v
xx
+k(t)npv
s+np−1
v
xxx
= 0,
(3.9)
Ù¥s´˜‡?¿~ê"4







s+p−1 = A,
s+mp−1 = B,
s+np−3 = C,
(3.10)
Ù¥A,BÚC´?¿~ê"l(3.10)·‚Œ±
p=
B−A
m−1
=
C−A+2
n−1
.(3.11)
e¡§·‚ò©m= nÚm6= n5©Oy²5Ÿ1Ú2"
Case(1)m= n,i.e.B= C+2,ÏL|^(3.10)Ú(3.11),•§(3.9)Œ±z•
v
A
v
t
+a(t)v
A
v
x
+b(t)mv
B
v
x
+k(t)n(np−1)(np−2)v
B−2
v
3
x
+3k(t)n(np−1)v
B−1
v
x
v
xx
+k(t)nv
B
v
xxx
= 0.
(3.12)
B−2 ≥A,þªü঱v
−A
,·‚
v
t
+a(t)v
x
+b(t)mv
B−A
v
x
+k(t)n(np−1)(np−2)v
B−A−2
v
3
x
+3k(t)n(np−1)v
B−A−1
v
x
v
xx
+k(t)nv
B−A
v
xxx
= 0.
(3.13)
•{ü儧·‚4B−A= α.•§(3.13)Us¤•§(3.2)"B−2 <A,þªü঱v
−B+2
,
·‚
v
A−B+2
v
t
+a(t)v
A−B+2
v
x
+b(t)mv
2
v
x
+k(t)n(np−1)(np−2)v
3
x
+3k(t)n(np−1)vv
x
v
xx
+k(t)nv
2
v
xxx
= 0.
(3.14)
?˜Ú•§(3.14)ÒUz¤•§(3.3).u´·‚Ò¤5Ÿ1y²"
Case(2)m6= n,|^(3.10)†(3.11),•§(3.9)U¤
v
t
+a(t)v
x
+b(t)mv
B−A
v
x
+k(t)n(np−1)(np−2)v
C−A
v
3
x
+3k(t)n(np−1)v
C−A+1
v
x
v
xx
+k(t)nv
C−A+2
v
xxx
= 0.
(3.15)
4B−A= αÚC−A= β,·‚Ò¤5Ÿ2y²"
DOI:10.12677/pm.2020.109106917nØêÆ
´æx§‡ù
3.•§(1.4)Compacton)
3.1 m= n
4α= 2,•§(3.2)C¤
v
t
+a(t)v
x
+nb(t)v
2
v
x
+
2n(n+1)
(n−1)
2
k(t)v
3
x
+
3n(n+1)
n−1
k(t)vv
x
v
xx
+nk(t)v
2
v
xxx
= 0.(4.1)
ò(2.1)“\(4.1)4Xêx
s
z
i
(ξ)
√
a
1
+a
2
z+a
3
z
2
(s=0,1,i=0,1,2)u0,·‚ÒŒ±'
uc
2
,p(t)†q(t)“ꕧ"ÏLêÆ^‡Mathematica¦)ù•§§·‚













c
2
=
2a
3
c
1
a
2
,
p(t) = ±
n−1
2n
q
−b(t)
a
3
k(t)
,
p
0
(t) = 0,
q
0
(t) = ∓
n−1
4n
2
a
2
2
q
−b(t)
a
3
k(t)
(2na(t)a
2
2
+(n+1)b(t)(a
2
2
−4a
1
a
3
)c
2
1
),
(4.2)
Ù¥a
2
a
3
c
1
6= 0,a
1
,a
2
,a
3
†c
1
´?¿~ê"3(4.2),·‚‡¦
b(t)
k(t)
7L´˜‡~ê"
(i) XJa
3
<0,∆ >0…
b(t)
k(t)
>0,·‚
u
1
(x,t) =
"
c
1
√
∆
a
2
sin
n−1
2n
s
b(t)
k(t)
x+
√
−a
3
q(t)
!#
2
n−1
,
Ú
u
2
(x,t) =
"
c
1
√
∆
a
2
cos
n−1
2n
s
b(t)
k(t)
x+
√
−a
3
q(t)
!#
2
n−1
.
3.2 m6= n
m6= n,•§(3.5)Œ±¤
v
3
v
t
+a(t)v
3
v
x
+mb(t)v
4
v
x
+k(t)(2−m)

(3−2m)(4−3m)
(m−1)
2
v
3
x
+
9−6m
m−1
vv
x
v
xx
+v
2
v
xxx

= 0.
(4.7)
ò(2.1)“\(4.7)¿4Xêx
s
z
i
(ξ)
√
a
1
+a
2
z+a
3
z
2
(s=0,1,i=0,1,2,3)u0§·‚Œ±
¼'ua
3
,c
2
,p(t)Úq(t)“ꕧ"ÏL^‡Mathematica§·‚Œ±

















q
0
(t) =
−16M
3
2
M
5
a(t)a
3
1
−mM
3
3
b(t)a
3
2
c
4
1
4M
3
2
M
6
a
3
1
,
c
2
=
M
3
a
2
c
1
4M
2
a
1
,
p(t) = ±
4M
5
M
6
,
p
0
(t) = 0,
a
3
=
3M
4
a
2
2
8(3m−4)M
2
1
a
1
,
(4.8)
DOI:10.12677/pm.2020.109106918nØêÆ
´æx§‡ù
Ù¥M
1
= 14m
2
−41m+29, M
2
= −116+251m−179m
2
+42m
3
, M
3
= −164+374m−281m
2
+69m
3
,
M
4
= −2460+8726m−12305m
2
+8618m
3
−2997m
4
+414m
5
,M
5
=
p
(m−1)
2
m(3m−4)M
2
1
b(t)a
1
c
2
1
,
M
6
=
p
(m−2)
3
M
4
k(t)a
2
2
,a
1
a
2
a
3
c
1
6=0.3(4.8)¥,·‚‡¦Ïf
b(t)
k(t)
7L´˜‡~ê"?˜Ú
§·‚kµ
(i) a
3
<0†∆ >0,·‚¼•§(1.4))
u
6
(x,t) =
"
c
1
+c
2

√
∆
2a
3
sin(
√
−a
3
(p(t)x+q(t)))−
a
2
2a
3
!#
1
m−1
,
†
u
7
(x,t) =
"
c
1
+c
2

√
∆
2a
3
cos(
√
−a
3
(p(t)x+q(t)))−
a
2
2a
3
!#
1
m−1
.
4.(Ø
ùŸ©Ù¥·‚ÏL˜‡{üE|¼2ÂK(m,n)•§°()"¯¢þ§·‚ÏL
e¡C†
u(x,t) =
n
X
i=0
c
i
z
i
(ξ),ξ= p(t)x+q(t),
Ù¥c
i
(i= 0,1,2,3,···)´¢~ê"Óž·‚Ú\9Ï•§
z
0
= 
p
a
1
+a
2
z+a
3
z
2
+a
4
z
3
+a
5
z
4
,
Ù¥a
i
(i= 1,2,3,4,5)´¢~ê"·‚Œ±¼2ÂK(m,n)•§Ù{°()"
Ä7‘8
HŽ˜e]Ï‘8(17C1363)"
ë•©z
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´æx§‡ù
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