Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients
We propose an algorithm of the construction of asymptotic two-phase soliton-type solutions of the Korteweg - de Vries equation with a small parameter at the higher derivative.
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| Дата: | 2008 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3162 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509205022113792 |
|---|---|
| author | Samoilenko, V. G. Samoilenko, Yu. I. Самойленко, В. Г. Самойленко, Юл. І. |
| author_facet | Samoilenko, V. G. Samoilenko, Yu. I. Самойленко, В. Г. Самойленко, Юл. І. |
| author_sort | Samoilenko, V. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:10Z |
| description | We propose an algorithm of the construction of asymptotic two-phase soliton-type solutions of the Korteweg - de Vries equation with a small parameter at the higher derivative. |
| first_indexed | 2026-03-24T02:37:24Z |
| format | Article |
| fulltext |
UDK 517.9
V. H. Samojlenko, G. I. Samojlenko ( Ky]v. nac. un-t im. T. Íevçenka)
ASYMPTOTYÇNI DVOFAZOVI SOLITONOPODIBNI
ROZV’QZKY SYNHULQRNO ZBURENOHO RIVNQNNQ
KORTEVEHA – DE FRIZA ZI ZMINNYMY KOEFICI{NTAMY
We propose an algorithm of the construction of asymptotic two-phase soliton-type solutions of the
Korteweg – de Vries equation with a small parameter at the higher derivative.
PredloΩen alhorytm postroenyq asymptotyçeskyx dvuxfazn¥x solytonopodobn¥x reßenyj
uravnenyq Korteveha – de Fryza s mal¥m parametrom pry starßej proyzvodnoj.
1. Vstup. Rivnqnnq Korteveha – de Friza [1] [ fundamental\nym rivnqnnqm su-
çasno] fizyky. Znaçna uvaha do joho vyvçennq prydilq[t\sq zavdqky isnuvanng
tak zvanyx solitonnyx [2, 3] rozv’qzkiv c\oho rivnqnnq. Qk vidomo [4], dlq riv-
nqnnq Korteveha – de Friza spoçatku bulo pobudovano v qvnomu vyhlqdi odnoso-
litonni rozv’qzky, a potim, za dopomohog metodu oberneno] zadaçi teori] rozsig-
vannq, — dvo- ta bahatosolitonni rozv’qzky.
Qk pravylo, solitonni rozv’qzky ne moΩna pobuduvaty u vypadku naqvnosti
zminnyx koefici[ntiv u rivnqnni Korteveha – de Friza, a tomu dlq pobudovy po-
dibnyx rozv’qzkiv u bahat\ox pracqx vykorystovuvalysq asymptotyçni metody.
Zokrema, v [5 – 12] rozhlqnuto pytannq pro pobudovu asymptotyçnyx rozv’qzkiv
rivnqnnq Korteveha – de Friza zi zminnymy koefici[ntamy dlq riznyx typiv syn-
hulqrnostej. Taki rozv’qzky pry pevnyx znaçennqx koefici[ntiv [ solitonnymy,
a tomu vony otrymaly [5, 6] nazvu solitonopodibnyx rozv’qzkiv.
U statti [5] pokazano, wo odnosolitonnyj rozv’qzok rivnqnnq Korteveha – de
Friza z malog dyspersi[g vyhlqdu
ut + 6uux + ε2ux x x = 0 (1)
pry prqmuvanni maloho parametra ε do nulq svo[g hranyceg ma[ rozryvnu
funkcig zminnyx ( , )x t ∈R2
. Tomu pry pobudovi asymptotyçnyx odnofazovyx
solitonopodibnyx rozv’qzkiv synhulqrno zburenoho rivnqnnq Korteveha – de
Friza zi zminnymy koefici[ntamy [5 – 12] krim rozrobky alhorytmu pobudovy
asymptotyçnoho rozv’qzku ta dovedennq ocinky joho toçnosti dovodyt\sq dodat-
kovo znaxodyty deqku funkcig, wo vyznaça[ tak zvanu linig rozryvu. Analo-
hiçna sytuaciq ma[ misce i u vypadku zadaçi pro znaxodΩennq asymptotyçnyx
dvofazovyx solitonopodibnyx rozv’qzkiv rivnqnnq Korteveha – de Friza.
Ce lehko pomityty z analizu formuly [4] dlq (toçnoho) dvosolitonnoho
rozv’qzku rivnqnnq (1):
u x t( , , )ε = 2
18 24 2
4
6 4
16
2
16 4
3 2
16 4 2
+
+
+
+
ch ch
ch ch
x t x t
x t x t x t x t
– –
– – –
–
–
ε ε
ε ε ε ε
, (2)
qkyj pry t T> nablyΩeno moΩna podaty u vyhlqdi sumy dvox solitoniv
u x t( , , )ε =
8
2 16
2
4
8
2 16
2
4
ch ch
ch ch
–2 –2
–2 –2
( – ) ( – )
, – ,
( – )
–
( – )
– , ,
x t
C
x t
C t T
x t
C
x t
C t T
ε ε
ε ε
+
+ +
<
+
>
de C = ln 3 , T > 0 — deqka stala.
Oçevydno, wo pry ε → 0 rozv’qzok u x t( , , )ε , wo vyznaça[t\sq formu-
© V. H. SAMOJLENKO, G. I. SAMOJLENKO, 2008
388 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
ASYMPTOTYÇNI DVOFAZOVI SOLITONOPODIBNI ROZV’QZKY … 389
logK(2), potoçkovo prqmu[ do rozryvno] funkci] vyhlqdu
u x t( , ) =
6 16 3
2
4 0
6 0
η η
η
( – ) ( – ), ,
( ), ,
x t x t t
x t
+ ≠
=
de
η ξ( ) =
0 0
1 0
, ,
, .
ξ
ξ
≠
=
U danij statti rozhlqda[t\sq pytannq pro pobudovu dvofazovyx solitonopo-
dibnyx asymptotyçnyx rozv’qzkiv rivnqnnq Korteveha – de Friza zi zminnymy
koefici[ntamy vyhlqdu
ε2ux x x = a x ut( , )ε + b x uux( , )ε , (3)
de
a x( , )ε = a xk
k
k( )
=
∞
∑
0
ε , b x( , )ε = b xk
k
k( )
=
∞
∑
0
ε ,
funkci] a xk ( ), b xk ( ) ∈ C∞( )R1
, k ≥ 0.
Na moΩlyvist\ pobudovy takyx rozv’qzkiv u vypadku, koly funkci] a x( , )ε
ta b x( , )ε ne zaleΩat\ vid maloho parametra, bulo vkazano u [5]. NyΩçe zapro-
ponovano alhorytm pobudovy asymptotyçnoho dvofazovoho solitonopodibnoho
rozv’qzku rivnqnnq (3) ta navedeno tverdΩennq pro porqdok toçnosti pobudova-
noho asymptotyçnoho rozv’qzku.
2. Osnovni prypuwennq i poznaçennq. Budemo prypuskaty, wo nezburena
(ε = 0) dlq (3) zadaça
a x ut0( ) + b x uux0( ) = 0
ma[ rozv’qzok u x t0( , ) , qkyj [ rozryvnym lyße na deqkyx hladkyx kryvyx x =
= ϕs t( ) , t T∈[ ]0; , s = 1, 2, dlq qkyx vykonu[t\sq umova uzhodΩenosti ϕ1 0( ) =
= ϕ2 0( ) .
Analohiçno do [5] poznaçymo çerez G1 = G T1 0R R× [ ] ×( ); linijnyj prostir
neskinçenno dyferencijovnyx funkcij f = f x t( , , )τ , ( , , )x t τ ∈ R × 0; T[ ] × R ,
takyx, wo dlq dovil\nyx nevid’[mnyx cilyx çysel n, m , q, p rivnomirno wodo
( , )x t na koΩnij kompaktnij mnoΩyni K � R × 0; T[ ] vykonugt\sq dvi taki umo-
vy:
1) spravdΩu[t\sq spivvidnoßennq
lim ( , , )
τ
τ ∂
∂τ
∂
∂
∂
∂
τ
→ +∞
n
m
m
q
q
p
pt x
f x t = 0, ( , )x t K∈ ,
2) isnu[ taka neskinçenno dyferencijovna funkciq f x t−( , ) , wo
lim ( , , ) – ( , )
–τ
α
ατ ∂
∂τ
∂
∂
∂
∂
τ
→ ∞
−( )n
m
m
q
qt x
f x t f x t = 0, ( , )x t K∈ .
Nexaj G1
0 = G T1
0 0R R× [ ] ×( ); — linijnyj pidprostir prostoru G 1 =
= G1 R( × 0; T[ ] × R) neskinçenno dyferencijovnyx funkcij f = f x t( , , )τ , ( x, t,
τ ) ∈ R × 0; T[ ] × R, takyx, wo rivnomirno wodo zminnyx ( , )x t na koΩnomu kom-
pakti K � R × 0; T[ ] vykonu[t\sq umova
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
390 V. H. SAMOJLENKO, G. I. SAMOJLENKO
lim ( , , )
–τ
τ
→ ∞
f x t = 0.
Poznaçymo çerez G2
0 = G2
0 R( × 0; T[ ] × R × R) linijnyj prostir neskinçen-
no dyferencijovnyx funkcij f = f x t( , , , )τ τ1 2 , ( , , , )x t τ τ1 2 ∈ R × 0; T[ ] × R × R ,
dlq qkyx isnugt\ funkci] f1
± = f x t1 2
±( , , )τ , f2
± = f x t2 1
±( , , )τ ∈ G1
0
taki, wo
dlq dovil\nyx nevid’[mnyx cilyx çysel α, q, p1, p2, β 1, β2 magt\ misce spiv-
vidnoßennq
lim ( , , , ) – ( , , )
τ
β
β
β
β
α
ατ ∂
∂τ
∂
∂τ
∂
∂
∂
∂
τ τ τ
1
1
1
1
2
21
1 2
1 2 1 2→ ±∞
±( )p q
qx t
f x t f x t = 0,
lim ( , , , )
τ
β
β
β
β
α
ατ ∂
∂τ
∂
∂τ
∂
∂
∂
∂
τ τ
2
1
1
1
2
22
1 2
1 2→ ±∞
(p q
qx t
f x t – f x t2 1
± )( , , )τ = 0, ( , )x t K∈ .
Ci rivnosti i podibni (analohiçni) dali slid rozumity qk okremi spivvidnoßennq,
uKqkyx zminna, za qkog obçyslg[t\sq hranycq, prqmu[ lyße abo do + ∞ abo Ω
doKK– ∞.
Nexaj G2 = G2 R( × 0; T[ ] × R × R) — linijnyj prostir neskinçenno dyfe-
rencijovnyx funkcij f = f x t( , , , )τ τ1 2 , ( , , , )x t τ τ1 2 ∈ R × 0; T[ ] × R × R , takyx,
wo isnugt\ funkci] f1
± = f x t1 2
±( , , )τ , f2
± = f x t2 1
±( , , )τ ∈ G1 ta neskinçenno
dyferencijovni funkci] u x t1
±( , ) ta u x t2
±( , ) taki, wo dovil\nyx nevid’[mnyx ci-
lyx çysel α, q, p1, p2, β1, β2 vykonugt\sq spivvidnoßennq
lim ( , , , )
τ
β
β
β
β
α
ατ ∂
∂τ
∂
∂τ
∂
∂
∂
∂
τ τ
k
k
k
p q
qx t
f x t
→ ±∞
(
1
1
2
2
1 2
1 2 – f x t u x tk k
± ± )( , , ) – ( , )τ2 = 0,
( , )x t K∈ , k = 1, 2.
Oznaçennq [5]. Funkciq u x t( , , )ε nazyva[t\sq asymptotyçno dvofazovog
solitonopodibnog, qkwo dlq dovil\noho ciloho çysla N ≥ 0 ]] moΩna zobra-
zyty u vyhlqdi
u x t( , , )ε = Y x t
S x t S x t
N , ,
( , )
,
( , )
,1 2
ε ε
ε
+ O Nε +( )1
,
de
Y x tN ( , , , , )τ τ ε1 2 =
j
N
j
j ju x t V x t
=
∑ +( )
0
1 2ε τ τ( , ) ( , , , ) ,
τ1 =
S x t1( , )
ε
, τ2 =
S x t2( , )
ε
,
Sk = S x tk ( , ), k = 1, 2, — neskinçenno dyferencijovni funkci] zminnyx ( , )x t ∈
∈ R × 0; T[ ], taki, wo
∂
∂
S
x
k
kΓ
≠ 0, de Γk = ( , )x t{ ∈ R × 0; T[ ], S x tk ( , ) = 0} , k
= 1, 2; u x tj ( , ), ( , )x t ∈ R × 0; T[ ], j = 1, N , — neskinçenno dyferencijovni funk-
ci]; V x t0 1 2( , , , )τ τ ∈ G2
0
, V x tj ( , , , )τ τ1 2 ∈ G2.
Asymptotyçnyj dvofazovyj solitonopodibnyj rozv’qzok zadaçi (3) ßuka[mo
u vyhlqdi asymptotyçnoho rqdu
u x t( , , )ε = Y x tN ( , , )ε + O Nε +( )1
, (4)
de
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
ASYMPTOTYÇNI DVOFAZOVI SOLITONOPODIBNI ROZV’QZKY … 391
Y x tN ( , , )ε =
j
N
j
j ju x t V t
=
∑ +( )
0
1 2ε τ τ( , ) ( , , ) ,
τ1 =
x t– ( )ϕ
ε
1 , τ2 =
x t– ( )ϕ
ε
2 ,
N — dovil\ne (fiksovane) natural\ne çyslo.
Funkciq U x tN ( , , )ε =
j
N j
ju x t=∑ 0
ε ( , ) nazyva[t\sq rehulqrnog çastynog
asymptotyky (4), a funkciq
V tN ( , , , )τ τ ε1 2 =
j
N
j
jV t
=
∑
0
1 2ε τ τ( , , ),
— synhulqrnog çastynog asymptotyky (4). Pry c\omu Y x tN ( , , )ε = U xN ( , t,
ε) + V tN ( , , , )τ τ ε1 2 .
Kryvi x = ϕs t( ) , s = 1, 2, nazyvagt\sq kryvymy rozryvu i vyznaçagt\sq u
procesi pobudovy asymptotyçnoho rozv’qzku.
Vidpovidno do zahal\no] metodolohi] asymptotyçnyx metodiv dlq vyznaçennq
koefici[ntiv asymptotyçnyx rozkladiv (4), vraxovugçy vyhlqd poxidnyx u xt( , t,
ε), u x tx( , , )ε , u x tx x x( , , )ε , pislq ]x pidstanovky v rivnqnnq (3) znaxodymo
ε ∂
∂ ε
∂
∂τ
∂
∂τ ∂τ
∂
∂τ ∂τ
∂
∂τ
2
3
3 3
3
1
3
3
1
2
2
3
1 2
2
3
2
3
1U
x
V V V VN N N N N+ + + +
=
= a x
U
t
V
t
V
t
V
tN N N N( , ) – ( ) – ( )ε ∂
∂
∂
∂ ε
∂
∂τ
ϕ
ε
∂
∂τ
ϕ+ ′ ′
1 1
1
1
2
2 +
+ b x
U
x
V V
U VN N N
N N( , )ε ∂
∂ ε
∂
∂τ ε
∂
∂τ
+ +
+( )1 1
1 2
K+K g x tN ( , , , , )τ τ ε1 2 ,
de g x tN ( , , , )τ ε = O Nε +( )1
— deqka neskinçenno dyferencijovna funkciq svo]x
arhumentiv, wo vyznaça[t\sq rekurentnym (wodo j) çynom za funkciqmy Y xj ( , t,
τ τ ε1 2, , ), j = 0 1, –N .
Sprqmovugçy τ1 do + ∞ ta τ2 do + ∞, dlq rehulqrno] çastyny asympto-
tyky ma[mo
a x
u
t0
0( )
∂
∂
+ b x u
u
x0 0
0( )
∂
∂
= 0,
a x
u
t
j
0( )
∂
∂
+ b x u
u
xj0
0( )
∂
∂
+ b x u
u
x
j
0 0( )
∂
∂
= F x tj ( , ) , j = 1, N ,
de funkci] F x tj ( , ) vyznaçagt\sq rekurentnym çynom pislq znaxodΩennq
funkcij u x t0( , ), u x t1( , ), … , u x tj – ( , )1 , j = 1, N .
3. Vyznaçennq synhulqrno] çastyny asymptotyky. Analohiçno [7, 8]
synhulqrna çastyna asymptotyky (4) vyznaça[t\sq na koΩnij iz kryvyx x =
= ϕs t( ) , s = 1, 2. Dlq znaxodΩennq funkcij V t0 1 2( , , )τ τ ma[mo kvazilinijne od-
noridne rivnqnnq z çastynnymy poxidnymy vyhlqdu
∂
∂τ
∂
∂τ ∂τ
∂
∂τ ∂τ
∂
∂τ
3
0
1
3
3
0
1
2
2
3
0
1 2
2
3
0
2
33 3
V V V V+ + + =
= a x t b x u x t
V
0 1 0 0
0
1
( ) – ( ) ( ) ( , )′( ) +[ ]ϕ ∂
∂τ
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
392 V. H. SAMOJLENKO, G. I. SAMOJLENKO
+ a x t b x u x t
V
0 2 0 0
0
2
( ) – ( ) ( ) ( , )′( ) +[ ]ϕ ∂
∂τ
+ b x V
V V
0 0
0
1
0
2
( )
∂
∂τ
∂
∂τ
+
, (5)
de znaçennq funkci] V t0 1 2( , , )τ τ ta ]] poxidnyx obçyslg[t\sq takym çynom:
dlq vypadku kryvo] x = ϕ1( )t v toçci t
t t
, ,
( ) – ( )
0 1 2ϕ ϕ
ε
, a dlq vypadku kryvo]
x = ϕ2( )t v toçci t
t t
,
( ) – ( )
,
ϕ ϕ
ε
2 1 0
, t T∈[ ]0; . Pry c\omu znaçennq funkcij
a x0( ), b x0( ) , u x t0( , ) vyznaçagt\sq na vidpovidnyx kryvyx x = ϕs t( ) , s = 1, 2.
Funkci] V tj ( , , )τ τ1 2 , j = 1, N , vyznaçagt\sq iz systemy linijnyx neodnorid-
nyx dyferencial\nyx rivnqn\ z çastynnymy poxidnymy vyhlqdu
∂
∂τ
∂
∂τ ∂τ
∂
∂τ ∂τ
∂
∂τ
3
1
3
3
1
2
2
3
1 2
2
3
2
33 3
V V V Vj j j j+ + + =
= a x t b x u x t
Vj
0 1 0 0
1
( ) – ( ) ( ) ( , )′( ) +[ ]ϕ
∂
∂τ
+ a x t b x u x t
Vj
0 2 0 0
2
( ) – ( ) ( ) ( , )′( ) +[ ]ϕ
∂
∂τ
+
+ b x V
V
V
V
V
V
V
V
j j
j j
0
0
1
0
2
0
1
0
2
( )
∂
∂τ
∂
∂τ
∂
∂τ
∂
∂τ
+ + +
+ F j t( , , )τ τ1 2 , j = 1, N , (6)
de znaçennq funkcij a x0( ), b x0( ) , u x t0( , ) vyznaçagt\sq vidpovidno na kryvyx
x = ϕs t( ) , s = 1, 2, a znaçennq funkci] F j t( , , )τ τ1 2 znaxodqt\sq rekurentnym
çynom pislq vyznaçennq znaçen\ funkcij V t0 1 2( , , )τ τ , V t1 1 2( , , )τ τ , … , V tj – ( ,1
τ1, τ2) na vidpovidnyx kryvyx.
Qkwo poznaçyty çerez F j
s t( , , )τ τ1 2 znaçennq funkci] F j t( , , )τ τ1 2 na kryvij
x = ϕs t( ) , s = 1, 2, to, zokrema, budemo maty
F1 1 2
s t( , , )τ τ = b t V
u t t
xs
s s
0 0
0ϕ ∂ ϕ
∂
( )
( ),( ) ( )
+ a t
V
ts
s
0
0ϕ ∂
∂
( )( ) +
+ τ ϕs s
sb t V′ ( )
0 0( ) + b t Vs
s
1 0ϕ ( )( ) + τ ϕ ϕs s sb t u t t′ ( ) ( )0 0( ) ( ), + b t u t ts s1 0ϕ ϕ( ) ( ),( ) ( ) +
+ τ ϕ ∂ ϕ
∂s s
sb t
u t t
x0
0( )
( ),( ) ( )
+ b t u t ts s0 1ϕ ϕ( ) ( ),( ) ( ) – ′ ( ) ′a t ts s0 1ϕ τ ϕ( ) ( ) –
– a t t
V
s
s
1 1
0
1
ϕ ϕ ∂
∂τ
( ) ( )( ) ′
+ τ ϕs s
sb t V′ ( )
0 0( ) + b t Vs
s
1 0ϕ ( )( ) + τ ϕ ϕs s sb t u t t′ ( ) ( )0 0( ) ( ), +
+ b t u t ts s1 0ϕ ϕ( ) ( ),( ) ( ) + τ ϕ ∂ ϕ
∂s s
sb t
u t t
x0
0( )
( ),( ) ( )
+ b t u t ts s0 1ϕ ϕ( ) ( ),( ) ( ) –
– ′ ( ) ′a t ts s0 2ϕ τ ϕ( ) ( ) – a t t
V
s
s
1 2
0
2
ϕ ϕ ∂
∂τ
( ) ( )( ) ′
,
V s
0 = V t ts0 ϕ ( ),( ), s = 1, 2.
Pobudova asymptotyçnoho dvofazovoho solitonopodibnoho rozv’qzku [ moΩ-
lyvog, qkwo vykonugt\sq umovy uzhodΩennq vyhlqdu
a t0 1ϕ ( )( ) = a t0 2ϕ ( )( ), b t0 1ϕ ( )( ) = b t0 2ϕ ( )( ),
(7)
u t t0 1ϕ ( ),( ) = u t t0 2ϕ ( ),( ) ,
Fk t1
1 2( , , )τ τ = Fk t2
1 2( , , )τ τ , k = 1, N , (8)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
ASYMPTOTYÇNI DVOFAZOVI SOLITONOPODIBNI ROZV’QZKY … 393
de t T∈[ ]0; , τ1, τ2 ∈ R.
Zaznaçymo, wo rivnist\ (8) pry k = 1 [ moΩlyvog, napryklad, u vypadku
vykonannq takyx umov:
′ ( )b t0 1ϕ ( ) = ′ ( )b t0 2ϕ ( ) = 0, b t1 1ϕ ( )( ) = b t1 2ϕ ( )( ),
′ ( )a t0 1ϕ ( ) = ′ ( )a t0 2ϕ ( ) = 0, a t1 1ϕ ( )( ) = a t1 2ϕ ( )( ) , (9)
∂ ϕ
∂
u t t
x
0 1( ),( )
=
∂ ϕ
∂
u t t
x
0 2( ),( )
= 0, t T∈[ ]0; .
Pry c\omu, qkwo vykonugt\sq umovy (7), (8), to V tj
1
1 2( , , )τ τ = V tj
2
1 2( , , )τ τ dlq
vsix t T∈[ ]0; , τ1, τ2 ∈ R.
Slid takoΩ zauvaΩyty, wo dlq dvosolitonnoho rozv’qzku (2) rivnqnnq Kor-
teveha – de Friza umovy (7) – (9) vykonugt\sq.
Nadali prypuska[mo, wo umovy (7), (8) magt\ misce.
Rozhlqnemo pytannq pro isnuvannq rozv’qzku rivnqnnq (5). Vykona[mo v sys-
temi dyferencial\nyx rivnqn\ (5), (6) zaminu zminnyx
ξ =
γ τ γ τ
γ γ
2 1 1 2
2 1
( ) – ( )
( ) – ( )
t t
t t
, η =
τ τ
γ γ
1 2
2 1
–
( ) – ( )t t
, (10)
de
γ j t( ) = – a t tj0 1ϕ ϕ( ) ( )( ) ′ + b t u t t0 1 0 1ϕ ϕ( ) ( ),( ) ( ), j = 1, 2.
V rezul\tati rivnqnnq (5) nabere vyhlqdu dyferencial\noho rivnqnnq
∂
∂ξ
3
0
3
V
– b
V
Vs0
0
0( )ϕ ∂
∂ξ
+
∂
∂η
V0 = 0,
qke za dopomohog zaminy zminnyx
ξ1 = 1
6 0
1
2b ( )ϕ ξ
, η1 = 1
6 0
3
2b ( )ϕ η
zvodyt\sq do rivnqnnq Korteveha – de Friza z postijnymy koefici[ntamy vy-
hlqdu
∂
∂ξ
3
0
1
3
V
– 6 0
0
1
V
V∂
∂ξ
+
∂
∂η
V0
1
= 0. (11)
Qk vidomo [6], dvosolitonnyj rozv’qzok rivnqnnq (11) zapysu[t\sq za dopomo-
hog formuly
V0 1 1( , )ξ η = – ln det ( )2
2
1
2
∂
∂ξ
E G+ , (12)
de E — odynyçna ( )2 2× -matrycq, a matrycq G ma[ vyhlqd
G =
c c c
c c c
1
2
1
1 1
1
1 1 2 1
1 2 1
1 2
1 1 2 1
1 2 1
1 2
2
2
1
2 1
2
2
2
2
2
( )
exp(– )
( ) ( )
exp –( )
( ) ( )
exp –( )
( )
exp(– )
η κ ξ
κ
η η κ κ ξ
κ κ
η η κ κ ξ
κ κ
η κ ξ
κ
+( )
+
+( )
+
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
394 V. H. SAMOJLENKO, G. I. SAMOJLENKO
κ j t( ) =
γ
ϕ
j t
b t
( )
( )1
6 0
1
2( )
, cj ( )η1 = c ej
tj( ) ( )0
3
1κ η
,
cj ( )0 , j = 1, 2, — dovil\ni stali.
Z (12) znaxodymo
V0 1 1( , )ξ η = −
2 2 1 1
2 2 1 1κ κ ξc e– + 2 2 2
2 2 2 1κ κ ξc e– –
– 2 1
2
2
2 1 2
2
1 2
2 1 2 1c c e
( – ) – ( )κ κ
κ κ
κ κ ξ+ –
c c
e1
4
2
2
1 2
2
2
1
2
1 2
2
4 2
2
1 2 1
( – )
( )
(– – )κ κ κ
κ κ κ
κ κ ξ
+
–
–
c c
e1
2
2
4
1 2
2
1
2
2
1 2
2
2 4
2
1 2 1
( – )
( )
(– – )κ κ κ
κ κ κ
κ κ ξ
+
×
× 1
2 2 4
1
2
1
2 2
2
2
2 1
2
2
2
1 2
2
1 2 1 2
2
2
2
1 1 2 1 1 2 1+ + +
+
+c
e
c
e
c c
e
κ κ
κ κ
κ κ κ κ
κ ξ κ ξ κ κ ξ– – – ( )
–
( – )
( )
.
Neskladno perekonatysq v tomu, wo dlq funkci] V0 1 1( , )ξ η vykonugt\sq
spivvidnoßennq
lim ( , , )
τ
τ τ
1
0 1 2→ +∞
V t = f t01 2
+ ( , )τ = –
( )
( )
–
–
4 0
1
0
2
2 2
2 2
2
2
2
2
2
2 2
2 2
κ
κ
τ κ
τ κ
c e
c
e
A
A+
,
lim ( , , )
–τ
τ τ
1
0 1 2→ ∞
V t = f t01 2
− ( , )τ =
4 0
1
0
2
2 2
2
1 2
2 2
1 2
2 2
2
1 2
2
2 1 2
2
2
2
2 2
2 2
κ κ κ
κ κ κ κ
κ κ κ
τ κ
τ κ
c e
c
e
A
A
( )( – )
( )
( )( – )
( )
–
–+ +
+
,
lim ( , , )
τ
τ τ
2
0 1 2→ +∞
V t = f t02 2
+ ( , )τ = –
( )
( )
–
–
4 0
1
0
2
1 1
2 2
1
2
1
2
2
1 1
1 1
κ
κ
τ κ
τ κ
c e
c
e
A
A+
,
lim ( , , )
–τ
τ τ
2
0 1 2→ ∞
V t = f t02 2
– ( , )τ =
4 0
1
0
2
1 1
2
1 2
2 2
1 2
2 1
2
1 2
2
1 1 2
2
2
2
1 1
1 1
κ κ κ
κ κ κ κ
κ κ κ
τ κ
τ κ
c e
c
e
A
A
( )( – )
( )
( )( – )
( )
–
–+ +
+
,
A = 1
6 0
1
2b tϕ( )( )
.
Tut funkci] f t01 1
± ( , )τ ta f t02 2
± ( , )τ , oçevydno, naleΩat\ prostoru G1
0
, a otΩe,
funkciq V t0 1 2( , , )τ τ naleΩyt\ prostoru G2
0
.
Rozhlqnemo teper pytannq pro isnuvannq rozv’qzku systemy linijnyx dyfe-
rencial\nyx rivnqn\ (6). Ma[ misce taka lema.
Lema. Neobxidnog umovog rozv’qznosti systemy linijnyx dyferencial\nyx
rivnqn\ (6) u prostori G2 [ umovy ortohonal\nosti vyhlqdu
–
lim ( , , ) ( , , )
∞
+∞
→ ±∞∫ ( )
τ
τ τ τ τ τ
1
1 2 0 1 2 2F j t V t d = 0, (13)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
ASYMPTOTYÇNI DVOFAZOVI SOLITONOPODIBNI ROZV’QZKY … 395
–
lim ( , , ) ( , , )
∞
+∞
→ ±∞∫ ( )
τ
τ τ τ τ τ
2
1 2 0 1 2 1F j t V t d = 0. (14)
Dovedennq. PomnoΩymo obydvi çastyny koΩnoho z dyferencial\nyx riv-
nqn\ systemy (6) na V t0 1 2( , , )τ τ ta perejdemo do hranyci v otrymanij rivnosti
pry τ1 → + ∞. Vraxovugçy, wo rozv’qzok V tj ( , , )τ τ1 2 naleΩyt\ prostoru G2,
perekonu[mos\, wo vykonu[t\sq umova
lim ( , , ) – ( , ) – ( , )–τ
∂
∂τ ∂τ
τ τ τ
1
3
1 2
3 1 2 1 2 1→ +∞
+( )k k j jV t f t u x t = 0,
de f tj1 2
+( , )τ ∈ G1, u x t1( , ) ∈ C T∞ × [ ]( )R 0; .
Oskil\ky funkciq V t0 1 2( , , )τ τ naleΩyt\ prostoru G2
0
i dlq ne] spravdΩu-
[t\sq rivnist\
lim ( , , )
τ
τ τ
1
0 1 2→ +∞
V t = f t G01 2 1
0+ ∈( , )τ ,
dlq koΩnoho rivnqnnq systemy (6) ma[mo spivvidnoßennq
f t tj01 2 1 2
1
+
→ +∞
( , ) lim ( , , )τ τ τ
τ
F =
d f t
d
f tj
3
1 2
2
3 01 2
+
+( , )
( , )
τ
τ
τ –
– – ( ) ( ) ( ) ( ), ( , )
( , )
a t t b t u t t f t
df t
d
j
0 1 2 0 1 0 1 01 2
1 2
2
ϕ ϕ ϕ ϕ τ
τ
τ
( ) ′ + ( ) ( )[ ] +
+
–
– b t f t
df t
d
f t
df t
d
f tj
j0 1 01 2
1 2
2
1 2
01 2
2
01 2ϕ τ
τ
τ
τ τ
τ
τ( ) ( , )
( , )
( , )
( , )
( , )( ) +
+
+
+
+
+ , j = 1, N . (15)
Zintehruvavßy spivvidnoßennq (15) po τ2 v meΩax vid – ∞ do + ∞ , otry-
ma[mo
f t t dj01 2 1 2 2
1
+
→ +∞
∞
+∞
∫ ( , ) lim ( , , )
–
τ τ τ τ
τ
F =
= – ( , )
( , )
( ) ( ) – ( ) ( ),
( , )
–∞
+∞
+
+ +
∫
+ ( ) ′ ( ) ( )[ ]f t
d f t
d
a t t b t u t t
df t
dj1 2
3
01 2
2
3 0 1 2 0 1 0 1
01 2
2
τ τ
τ
ϕ ϕ ϕ ϕ τ
τ
–
– b t f t
df t
d
d0 1 01 2
01 2
2
2ϕ τ τ
τ
τ( ) ( , )
( , )( )
+
+
= 0.
Takym çynom, ma[mo rivnist\
–
lim ( , , ) ( , , )
∞
+∞
→ +∞∫ ( )
τ
τ τ τ τ τ
1
1 2 0 1 2 2F j t V t d = 0,
qka, oçevydno, [ neobxidnog umovog isnuvannq rozv’qzku systemy linijnyx dy-
ferencial\nyx rivnqn\ z çastynnymy poxidnymy (6).
Analohiçno dovodqt\sq umova (13) pry τ1 → – ∞ ta umova (14) pry τ2 → ± ∞.
Lemu dovedeno.
Umovy ortohonal\nosti (13), (14) moΩna vykorystaty dlq znaxodΩennq spiv-
vidnoßen\ dlq vyznaçennq kryvyx rozryvu x = ϕs t( ) , s = 1, 2. Zokrema, z cyx
rivnostej pry j = 1 znaxodymo ßukani spivvidnoßennq
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
396 V. H. SAMOJLENKO, G. I. SAMOJLENKO
′ ( )b t0 1ϕ ( ) = ′ ( )b t0 2ϕ ( ) = 0,
b t
u t t
x
a t0 1
0 1
0 1 2ϕ ∂ ϕ
∂
ϕ ϕ( )
( ),
– ( )( ) ( ) ′ ( ) ′ = 0,
2 0 1
0 1 2
3
b t
u t t
x
t
A t
ϕ ∂ ϕ
∂
κ
( )
( ), ( )
( )
( ) ( )
+ a t d
dt
t
A t0 1
2
3
ϕ κ
( )
( )
( )
( )
= 0,
b t
u t t
x
t
A t0 1
0 1 1
3
ϕ ∂ ϕ
∂
κ
( )
( ), ( )
( )
( ) ( )
+ ′ ( ) ′a t t
t
A t0 1 1
1
3
ϕ ϕ κ
( ) ( )
( )
( )
+ a t d
dt
t
A t0 1
1
3
ϕ κ
( )
( )
( )
( )
= 0,
(16)
b t
u t t
x0 2
0 2ϕ ∂ ϕ
∂
( )
( ),( ) ( )
+ ′ ( ) ′a t0 2 1ϕ ϕ( ) = 0,
2 0 2
0 2 1
3
b t
u t t
x
t
A t
ϕ ∂ ϕ
∂
κ
( )
( ), ( )
( )
( ) ( )
+ a t d
dt
t
A t0 2
1
3
ϕ κ
( )
( )
( )
( )
= 0,
b t
u t t
x
t
A t0 2
0 2 2
3
ϕ ∂ ϕ
∂
κ
( )
( ), ( )
( )
( ) ( )
+ ′ ( ) ′a t t
t
A t0 2 2
2
3
ϕ ϕ κ
( ) ( )
( )
( )
+ a t d
dt
t
A t0 2
2
3
ϕ κ
( )
( )
( )
( )
= 0,
de t T∈[ ]0; .
Zokrema, qkwo vykonugt\sq umovy (9), to systema spivvidnoßen\ (16) dlq
vyznaçennq kryvyx rozryvu ϕs t( ) , s = 1, 2, ekvivalentna systemi vyhlqdu
d
dt
t
A t
κ1
3( )
( )
= 0, d
dt
t
A t
κ2
3( )
( )
= 0,
qku moΩna zapysaty u vyhlqdi systemy nelinijnyx zvyçajnyx dyferencial\nyx
rivnqn\
– ( ) ( ) ( ) ( ),a t t b t u t t0 1 1 0 1 0 1
3
2ϕ ϕ ϕ ϕ( ) ′ + ( ) ( )( ) = C b t1 0 1
2ϕ ( )( )( ) ,
(17)
– ( ) ( ) ( ) ( ),a t t b t u t t0 2 2 0 2 0 2
3
2ϕ ϕ ϕ ϕ( ) ′ + ( ) ( )( ) = C b t2 0 2
2ϕ ( )( )( ) ,
de C1 ta C2 — dovil\ni stali.
Pry znaxodΩenni rozv’qzku systemy (17) apriori prypuska[t\sq, wo ]] roz-
v’qzok zadovol\nq[ umovy uzhodΩenosti (7). Zaznaçymo, wo taki rozv’qzky sys-
temy rivnqn\ (17) isnugt\, wo vyplyva[ z naqvnosti dvosolitonnoho rozv’qzku (2)
dlq rivnqnnq Korteveha – de Friza vyhlqdu (1), qke [ çastynnym vypadkom riv-
nqnnq (3), asymptotyçni rozv’qzky qkoho budugt\sq.
Na zaverßennq rozhlqnemo pytannq pro dostatni umovy isnuvannq rozv’qzku
systemy linijnyx dyferencial\nyx rivnqn\ z çastynnymy poxidnymy (6). V re-
zul\tati zaminy zminnyx (10) systemaK(6) nabyra[ vyhlqdu
∂
∂ξ
3
0
3
V
– b
V
V
V
Vj
j
0
0
0( )ϕ ∂
∂ξ
∂
∂ξ
+
+
∂
∂η
Vj = F j ( , )ξ η , j = 1, N . (18)
Qkwo prypustyty, wo funkciq F j ( , )ξ η naleΩyt\ prostoru ßvydko spad-
nyx funkcij za zminnog ξ pry koΩnomu znaçenni η ≥ 0, to vidomo, wo systema
linijnyx dyferencial\nyx rivnqn\ (18) ma[ [13] rozv’qzok Vj t( , , )ξ η , vyznaçenyj
dlq vsix ξ ∈R ta znaçen\ η, wo naleΩyt\ deqkomu skinçennomu intervalu.
Qk pidsumok vykladenyx vywe mirkuvan\, moΩna sformulgvaty taki tverd-
Ωennq.
Teorema 1. Prypustymo, wo funkci] a xk ( ), b xk ( ) [ neskinçenno dyferen-
cijovnymy na R1( ) pry k ≥ 0, vykonugt\sq umovy (7) ta (9), systemaK(17)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
ASYMPTOTYÇNI DVOFAZOVI SOLITONOPODIBNI ROZV’QZKY … 397
ma[ rozv’qzok ϕ1( )t , ϕ2( )t dlq t T∈[ ]0; takyj, wo ϕ1 0( ) = ϕ2 0( ) .
Todi funkciq
Y x t0( , , )ε = u x t0( , ) + V t0 1 2( , , )τ τ , τs =
x ts– ( )ϕ
ε
, s = 1, 2,
[ nul\ovym çlenom asymptotyçnoho rozvynennq dlq dvofazovoho solitonopo-
dibnoho rozv’qzku zadaçi (3) pry t T∈[ ]0; .
Teorema 2. Prypustymo, wo funkci] a xk ( ), b xk ( ) [ neskinçenno dyferen-
cijovnymy na R1( ) pry k ≥ 0, vykonugt\sq umova (7), umova (8) pry k = 1, N
ta umovy (9), systemaK(17) ma[ rozv’qzok ϕ1( )t , ϕ2( )t dlq t T∈[ ]0; takyj,
wo ϕ1 0( ) = ϕ2 0( ) , funkciq V tj ,
–
–
,
–
–
γ τ γ τ
γ γ
τ τ
γ γ
2 1 1 2
2 1
1 2
2 1
, j = 1, N , wo [ rozv’qz-
kom rivnqnnq (18), naleΩyt\ prostoru G2.
Todi funkciq
Y x tN ( , , )ε = ε τ τj
j j
j
N
u x t V t( , ) ( , , )+( )
=
∑ 1 2
0
,
τs =
x ts– ( )ϕ
ε
, s = 1, 2, ( , )x t ∈R × 0; T[ ],
[ asymptotyçnym dvofazovym solitonopodibnym rozv’qzkom rivnqnnq (3).
Vysnovky. V danij statti zaproponovano alhorytm pobudovy asymptotyçno-
ho rozv’qzku rivnqnnq Korteveha – de Friza zi zminnymy koefici[ntamy ta malym
parametrom.
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ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
|
| id | umjimathkievua-article-3162 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:24Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a5/4ef13e1667980c5c37356fafd15a32a5.pdf |
| spelling | umjimathkievua-article-31622020-03-18T19:47:10Z Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients Асимптотичні двофазові солітоноподібні розв'язки сингулярно збуреного рівняння Кортевега - де Фріза зі змінними коефіцієнтами Samoilenko, V. G. Samoilenko, Yu. I. Самойленко, В. Г. Самойленко, Юл. І. We propose an algorithm of the construction of asymptotic two-phase soliton-type solutions of the Korteweg - de Vries equation with a small parameter at the higher derivative. Предложен алгоритм построения асимптотических двухфазных солитоноподобных решений уравнения Кортевега - де Фриза с малым параметром при старшей производной. Institute of Mathematics, NAS of Ukraine 2008-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3162 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 3 (2008); 388–397 Український математичний журнал; Том 60 № 3 (2008); 388–397 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3162/3072 https://umj.imath.kiev.ua/index.php/umj/article/view/3162/3073 Copyright (c) 2008 Samoilenko V. G.; Samoilenko Yu. I. |
| spellingShingle | Samoilenko, V. G. Samoilenko, Yu. I. Самойленко, В. Г. Самойленко, Юл. І. Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients |
| title | Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients |
| title_alt | Асимптотичні двофазові солітоноподібні розв'язки сингулярно збуреного рівняння Кортевега - де Фріза зі змінними коефіцієнтами |
| title_full | Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients |
| title_fullStr | Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients |
| title_full_unstemmed | Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients |
| title_short | Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients |
| title_sort | asymptotic two-phase solitonlike solutions of the singularly perturbed korteweg-de vries equation with variable coefficients |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3162 |
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