On the dynamical equations of a system of linearly coupled nonlinear oscillators
We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Some results concerning the existence and uniqueness of global solutions of the Cauchy problem are obtained.
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| Date: | 2006 |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509588578631680 |
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| author | Bak, S. N. Pankov, A. A. Бак, С. Н. Панков, А. А. Бак, С. Н. Панков, А. А. |
| author_facet | Bak, S. N. Pankov, A. A. Бак, С. Н. Панков, А. А. Бак, С. Н. Панков, А. А. |
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| datestamp_date | 2020-03-18T19:56:00Z |
| description | We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Some results concerning the existence and uniqueness of global solutions of the Cauchy problem are obtained. |
| first_indexed | 2026-03-24T02:43:29Z |
| format | Article |
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UDK 517.97
S. N. Bak (Vynnyc. ped. un-t),
A. A. Pankov (Vynnyc. ped. un-t; KolledΩ Vyl\qma y Mπry, SÍA)
O DYNAMYÇESKYX URAVNENYQX
SYSTEMÁ LYNEJNO SVQZANNÁX
NELYNEJNÁX OSCYLLQTOROV
A system of differential equations that describes the dynamics of an infinite chain of linearly coupled
nonlinear oscillators is considered. Results are obtained on the existence and uniqueness of global
solutions of the Cauchy problem.
Rozhlqda[t\sq systema dyferencial\nyx rivnqn\, qka opysu[ dynamiku neskinçennoho lancgha
linijno zv’qzanyx nelinijnyx oscylqtoriv. Otrymano rezul\taty wodo isnuvannq ta [dynosti
hlobal\nyx rozv’qzkiv zadaçi Koßi.
1. V nastoqwej rabote yzuçagtsq uravnenyq, opys¥vagwye dynamyku besko-
neçnoj cepoçky lynejno svqzann¥x nelynejn¥x oscyllqtorov. Pust\ qn —
obobwennaq koordynata n-ho oscyllqtora. Uravnenye eho dvyΩenyq pry ot-
sutstvyy vzaymodejstvyq s sosednymy oscyllqtoramy ymeet vyd
˙̇ ( )q U qn n n= − ′ , n ∈Z.
Predpolahaetsq, çto kaΩd¥j oscyllqtor lynejno vzaymodejstvuet s dvumq
svoymy blyΩajßymy sosedqmy. Tohda uravnenyq dvyΩenyq rassmatryvaemoj
system¥ ymegt vyd
˙̇qn = – ′ + − − −− − +U q a q q a q qn n n n n n n n( ) ( ) ( )1 1 1 , n ∈Z. (1)
Uravnenyq (1) predstavlqgt soboj beskoneçnug systemu ob¥knovenn¥x dyf-
ferencyal\n¥x uravnenyj. Rassmatryvagtsq takye reßenyq system¥ (1), çto
lim ( )
n
nq t
→±∞
= 0, (2)
t. e. oscyllqtor¥ naxodqtsq v sostoqnyy pokoq na beskoneçnosty.
Podobn¥e system¥ predstavlqgt ynteres v svqzy s mnohoçyslenn¥my fyzy-
çeskymy pryloΩenyqmy [1, 2]. V rabote [3] yzuçalys\ behuwye voln¥ v takyx
cepoçkax, a v [4, 5] — peryodyçeskye po vremeny reßenyq.
V dannoj rabote rassmatryvagtsq vopros¥ korrektnosty zadaçy Koßy dlq
system¥ (1).
2. Potencyal Un zapyßem v vyde
U r
c
r V rn
n
n( ) ( )= − +
2
2
y poloΩym
b c a an n n n= − − −1.
© S. N. BAK, A. A. PANKOV, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6 723
724 S. N. BAK, A. A. PANKOV
Tohda uravnenye (1) prymet vyd
˙̇qn = a q a q b q V qn n n n n n n n+ − −+ + − ′
1 1 1 ( ), n ∈Z. (3)
Uçyt¥vaq hranyçn¥e uslovyq (2), pry podxodqwyx predpoloΩenyqx πto uravne-
nye estestvenno rassmatryvat\ kak dyfferencyal\no-operatornoe uravnenye
˙̇ ( )q Aq B qn = − (4)
v hyl\bertovom prostranstve l
2
vewestvenn¥x dvustoronnyx posledovatel\nos-
tej q = { }qn n=−∞
∞ , hde
( )Aq n = a q a q b qn n n n n n+ − −+ +1 1 1 ,
a nelynejn¥j operator B opredelqetsq formuloj
B q V qn n n( ) ( )= ′ .
Skalqrnoe proyzvedenye y norma v l2
oboznaçagtsq ( , )⋅ ⋅ y ⋅ sootvet-
stvenno.
Sohlasno opredelenyg, reßenyem (4) sçytaetsq dvaΩd¥ neprer¥vno dyffe-
rencyruemaq funkcyq ot t so znaçenyem v l
2
.
Predpolahaetsq, çto:
1) posledovatel\nosty { }an y { }cn vewestvenn¥x çysel ohranyçen¥;
2) V rn( ) — funkcyq klassa C
1
na R, V Vn n( ) ( )0 0 0= ′ = y dlq lgboho R >
> 0 suwestvuet takoe C C R= >( ) 0 , çto dlq vsex n ∈Z
′ − ′V r V rn n( ) ( )1 2 ≤ C r r1 2− , r1 , r2 ≤ R . (5)
V πtyx uslovyqx netrudno vydet\, çto A qvlqetsq ohranyçenn¥m samosoprq-
Ωenn¥m operatorom v l
2, a operator B ohranyçen y neprer¥ven po Lypßycu
na kaΩdom ßare prostranstva l
2. Tohda, kak sledstvye standartnoho rezul\ta-
ta o lokal\noj razreßymosty, ymeet mesto sledugwaq teorema.
Teorema$1. Pust\ v¥polnen¥ uslovyq 1 y 2. Tohda dlq lgb¥x q l( )0 2∈ y
q l( )1 2∈ uravnenye (3) ymeet edynstvennoe reßenye klassa C
2 , opredelennoe
na nekotorom yntervale ( ; )− t t0 0 y udovletvorqgwee naçal\n¥m uslovyqm
q q( ) ( )0 0= , ˙( ) ( )q q0 1= . (6)
Sledugwee utverΩdenye o hlobal\noj razreßymosty v¥tekaet yz teorem¥
1.2 hl. 8 [6].
Teorema$2. Pust\ v¥polnen¥ uslovyq 1 y 2 s konstantoj C , ne zavysq-
wej ot R . Tohda dlq lgb¥x q l( )0 2∈ y q l( )1 2∈ zadaça (4), (6) ymeet edyn-
stvennoe reßenye, opredelennoe pry vsex t ∈R.
3. Uslovyq teorem¥ 2 oznaçagt, v çastnosty, çto potencyal Vn ymeet rost
na beskoneçnosty ne v¥ße vtoroj stepeny. Çtob¥ oslabyt\ πto uslovye, otme-
tym, çto uravnenye (4) moΩno zapysat\ v hamyl\tonovom vyde s hamyl\tonyanom
H p q( , ) = 1
2
2p Aq q V q
n
n n− +
=−∞
∞
∑( , ) ( ) ,
hde p q= ˙ . V predpoloΩenyqx 1 y 2 H p q( , ) — funkcyonal klassa C
1
na
l
2 × l
2, y prqmoe v¥çyslenye pokaz¥vaet, çto H — yntehral uravnenyq (4), t. e.
dlq lgboho reßenyq q ( t ) uravnenyq (4) H ( p ( t ), q ( t )) ne zavysyt ot t.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
O DYNAMYÇESKYX URAVNENYQX SYSTEMÁ LYNEJNO SVQZANNÁX … 725
Teorema$3. Dopolnytel\no k uslovyqm 1 y 2 predpoloΩym, çto operator
A nepoloΩytelen, t. e. ( , )Aq q ≤ 0 dlq lgboho q l∈ 2 . Krome toho, pust\
v¥polneno odno yz sledugwyx dvux uslovyj:
a) V rn( ) ≥ 0 dlq vsex n ∈Z y r ∈R ;
b) suwestvuet takaq neub¥vagwaq funkcyq h ( r ) , r ≥ 0 , çto
lim ( )n h r→+∞ = +∞ y V r h rn( ) ( )≥ dlq vsex n ∈Z y r ∈R .
Tohda dlq lgb¥x q l( )0 2∈ y q l( )1 2∈ zadaça (4), (6) ymeet edynstvennoe
reßenye, opredelennoe pry vsex t ∈R.
Dokazatel\stvo. Sluçaj a). Pust\ q ( t ) — lokal\noe reßenye zadaçy (4),
(6), suwestvugwee v sylu teorem¥ 1. Dlq toho çtob¥ dokazat\, çto q ( t ) opre-
deleno na vsej osy, dostatoçno pokazat\, çto q t q t( ) ˙( )+ ostaetsq ohrany-
çennoj na lgbom koneçnom yntervale ( , )−a a suwestvovanyq reßenyq (sm., na-
prymer, [7], teorema X.74).
Ymeem
H q t q t( ˙( ), ( )) = H q q( )( ) ( ),1 0 .
V sylu uslovyj teorem¥ y opredelenyq hamyl\tonyana
1
2
2
2˙( )q t
l
≤ H q q( )( ) ( ),1 0 .
Sledovatel\no, ˙( )q t ohranyçeno na ( , )−a a . Poskol\ku
q ( t ) =
0
0
t
q d q∫ +˙( ) ( )τ τ ,
otsgda sleduet ohranyçennost\ q t( ) .
Sluçaj b). Pust\ H0 0≥ takovo, çto H q q H( )( ) ( ),1 0
0≤ y r > 0 —
reßenye uravnenyq h r H( ) = 0 (ono, oçevydno, suwestvuet). Yz opredelenyq H
y uslovyj teorem¥ sleduet, çto h q Hn( )( )0
0≤ y, znaçyt, q rn
( )0 ≤ . Pust\
ψ ( r ) — nekotoraq funkcyq, opredelennaq ravenstvom
ψ ( r ) =
1 0
1 1
0 1
, ,
, ,
, .
≤ ≤
− + + ≤ ≤ +
≥ +
r r
r r r r r
r r
PoloΩym
˜ ( )V rn =
0
1
r
nV d∫ ′ + −[ ( ) ( ) ( ( ))]ψ ρ ρ ψ ρ ρ .
Netrudno proveryt\, çto modyfycyrovannoe uravnenye (3) s potencyalom Ṽn
udovletvorqet uslovyqm teorem¥ 2 y, sledovatel\no, ymeet hlobal\noe reße-
nye q ( t ) s naçal\n¥my dann¥my q( )0
, q( )1
. ∏lementarn¥e v¥çyslenyq pokaz¥-
vagt, çto
˜ ( ) ( )V r h rn ≥ , hde
˜( )h r =
h r r r
r r h r h d r r r r r r r
r h d r r r r
r
r
r
r
( ), ,
( ) ( ) ( ) , ,
( ) ( ) , .
0
1
3 2 6
1
1
3 6
1
3 2 3
2 1 2 3
≤ ≤
+ − + + − +
≤ ≤ +
+ + + +
≥ +
∫
∫
+
ρ ρ
ρ ρ
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
726 S. N. BAK, A. A. PANKOV
Dlq modyfycyrovannoho hamyl\tonyana H̃ ymeem
˜ ( ( ), ( ))H p t q t = ˜ ,( )( ) ( )H q q1 0
.
Poskol\ku q rn
( )0 ≤ , to
˜ ,( )( ) ( )H q q1 0 ≤ H 0 . Sledovatel\no,
˜( )h qn ≤ H 0 .
Dalee, tak kak
˜( )h r ≥ ˜(˜)h r = h r(˜) = H0 , to q rn ≤ ˜ , a poskol\ku na q ( t )
modyfycyrovannoe uravnenye sovpadaet s ysxodn¥m, teorema dokazana.
Pry nekotor¥x dopolnytel\n¥x predpoloΩenyqx uslovyq nepoloΩytel\-
nosty v teoreme 3 moΩno opustyt\.
Sledstvye$1. V uslovyqx teorem¥ 3 b) bez uslovyq o nepoloΩytel\nosty
operatora A predpoloΩym, çto lim ( ) /r h r r→+∞ = +∞2 . Tohda zadaça (4), (6)
ymeet edynstvennoe hlobal\noe reßenye dlq lgb¥x q l( )0 2∈ y q l( )1 2∈ .
Dokazatel\stvo. Zapyßem Un v vyde
U rn( ) = –
c
r V r rn
n
− + −2
2
2 2λ λ( ( ) )
s dostatoçno bol\ßym λ > 0. Tohda nov¥j operator A, sootvetstvugwyj ko-
πffycyentam an y cn − 2λ , budet nepoloΩytel\n¥m. V to Ωe vremq
V r rn( ) − λ 2 ≥ h r r( ) − λ 2 = h r r
h r
( ) −
( )
1
2
λ .
Otsgda sleduet
V r rn( ) − λ 2 ≥ k h r k1 2( ) −
s nekotor¥my k1 0 1∈( , ) y k2 0≥ . Teper\ dostatoçno prymenyt\ teoremu 3 s
zamenoj h ( r ) na k h r k1 2( ) − .
Sledstvye dokazano.
Sledstvye 1 prymenymo, naprymer, k potencyalam vyda V r r rn n n( ) = +α β3 4,
hde posledovatel\nosty αn y βn ohranyçen¥ y βn ≥ κ > 0.
4. Arhument¥ yz p. 3 prymenym¥ y v sluçae synhulqrn¥x potencyalov typa
Lennarda – DΩonsa [8]. Soxranym uslovye 1, a uslovye 2 zamenym sledugwym:
3) funkcyq V rn( ) predpolahaetsq prynadleΩawej C
1 na ( , )−∞ d , d > 0, y
na kaΩdom koneçnom yntervale [ , ] ( , )α β ⊂ −∞ d v¥polnqetsq neravenstvo (5) s
konstantoj C, ne zavysqwej ot n ∈Z, no, vozmoΩno, zavysqwej ot yntervala.
Teorema$4. Pust\ v¥polnen¥ uslovyq 1 y 3, operator A nepoloΩytelen y
suwestvuet takaq funkcyq h ( r ) na ( , )−∞ r , çto h ( r ) ne vozrastaet na ne-
kotorom yntervale ( , )−∞ α0 y ne ub¥vaet na ( )α0 d , pryçem lim ( )r d h r→ =
= lim ( )r h r→−∞ = + ∞ y V r h rn( ) ( )≥ dlq vsex n ∈Z y r d∈ −∞( , ). Tohda
dlq lgb¥x q l( )0 2∈ y q l( )1 2∈ takyx, çto q dn
( )0 < , zadaça (4), (6) ymeet
edynstvennoe hlobal\noe reßenye.
Dokazatel\stvo teorem¥ provodytsq analohyçno dokazatel\stvu teorem¥ 3
(sluçaj b) ). Potencyal V rn( ) skleyvaetsq s kvadratn¥m potencyalom na
nekotor¥x yntervalax ( , )α α ε0 0 + y ( , )− − −β β0 0 1 . K modyfycyrovannomu
uravnenyg prymenqetsq teorema 2 y proverqetsq, çto reßenye modyfycyrovan-
noho uravnenyq qvlqetsq na samom dele reßenyem ysxodnoho uravnenyq.
Otmetym, çto reßenyq q t dn( ) < dlq vsex t ∈R. Poskol\ku mnoΩestvo
q l q d nn∈ < ∈{ }2 : , Z ne qvlqetsq otkr¥t¥m v l2 , v rassmatryvaemom sluçae
klassyçeskye rezul\tat¥ o lokal\noj razreßymosty neprymenym¥.
5. Rassmotrym teper\ sluçaj
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
O DYNAMYÇESKYX URAVNENYQX SYSTEMÁ LYNEJNO SVQZANNÁX … 727
V r
d
rn
n( ) =
3
3,
hde dn — ohranyçennaq posledovatel\nost\. Predpolahaetsq, çto operator A
otrycatel\no opredelen, t. e. ( , )Aq q q≤ −α0
2, α0 > 0, dlq q l∈ 2 .
PoloΩym
J ( q ) = –
1
2
1
3
3( , )Aq q d qn n
n
+
∈
∑
Z
= 1
2
1
3
a q b q( ) ( )+ .
Otmetym, çto a q( ) /1 2
— norma na l2 , πkvyvalentnaq standartnoj. Tohda
H p q p J q( , ) ( )= +1
2
2 .
Poskol\ku b q c q c q
l
( ) ≤ ′ ≤ ′′3
3 3, suwestvuet takaq konstanta c > 0, çto
b q( ) /1 3 ≤ ca q( ) /1 2 , q l∈ 2 . (7)
PoloΩym
γ = inf sup ( ) : ,
λ
λ
≥
∈ ≠
0
2 0J q q l q . (8)
Lemma$1. γ ≥ 1 6 6/ ( )c .
Dokazatel\stvo. Ymeem
J q a q b q( ) ( ) ( )λ λ λ= +
2 3
2 3
.
Esly b q( ) ≥ 0, to
sup ( )
λ
λ
≥0
J q = + ∞ ,
esly Ωe b q( ) < 0, to
sup ( )
λ
λ
≥0
J q = J
a q
b q
q−
( )
( )
= 1
6
3
2
a q
b q
( )
( )
.
Yz neravenstva (7) poluçaem trebuemoe.
PoloΩym
Wγ = q l J q∈ ≤ < ∀ ∈{ }2 0 0 1: ( ) [ , ]λ γ λ . (9)
Oçevydno, çto Wγ zvezdno otnosytel\no naçala, t. e. esly q W∈ γ , to θ γq W∈
dlq lgboho θ ∈[ , ]0 1 .
Lemma$2. MnoΩestvo Wγ soderΩyt otkr¥t¥j πllypsoyd
B q l a q= ∈ <{ }2 : ( ) ρ
dlq lgboho ρ ≤ 9 4 2/ ( )c , ρ ρ γ/ /( ) /2 33 3 2+ <c .
Dokazatel\stvo. V sylu (7)
λ λ λ λ λ2 3 3
3 2
2 3 3
3 2
2 3 2 3
a q c a q J q a q c a q( ) ( ) ( ) ( ) ( )/ /− ≤ ≤ + .
Sledovatel\no, J q( )λ ≥ 0 dlq lgboho λ ∈[ , ]0 1 , esly
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
728 S. N. BAK, A. A. PANKOV
1
2 3
0
3
1 2− ≥λc
a q( ) /
∀ ∈λ [ , ]0 1 .
Znaçyt, esly a q c( ) ( )/≤ 9 4 2 , to v sylu vtoroho uslovyq na ρ J q( )λ γ< .
Lemma dokazana.
PoloΩym
W∗,γ = q l a q b q J q∈ + > <{ }2 0: ( ) ( ) , ( ) γ .
V sylu neperer¥vnosty funkcyonalov a ( q ) y b ( q ) W∗,γ — otkr¥toe mnoΩe-
stvo.
Lemma$3. Wγ = W B∗,γ ∪ .
Dokazatel\stvo. Dostatoçno pokazat\, çto W Wγ γ= ∗, { }∪ 0 .
Pust\ q W∈ γ , q ≠ 0. Esly b q( ) ≥ 0, to a q b q( ) ( )+ > 0 y J q( ) < γ . Esly
Ωe b q( ) < 0, to
sup ( ) ( )
( )
J q J
a q
b q
qλ γ= −
≥ .
Tohda − >a q b q( ) ( )/ 1 y J q( ) < γ . ∏to pokaz¥vaet, çto q W∈ ∗,γ .
Naoborot, pust\ q W∈ ∗,γ . Esly b q( ) ≥ 0, to
sup ( ) ( )
[ , ]λ
λ γ
∈
= <
0 1
J q J q
y q W∈ γ . Esly Ωe b q( ) < 0, to neravenstvo − >a q b q( ) ( )/ 1 pokaz¥vaet, çto
sup ( ) ( )
[ , ]λ
λ
∈
=
0 1
J q J q ,
otkuda y poluçaem trebuemoe.
V sylu otkr¥tosty W∗,γ y B lemma 3 pokaz¥vaet, çto mnoΩestvo Wγ ot-
kr¥to, t. e. qvlqetsq okrestnost\g nulq v l2 .
Lemma$4. Wγ — ohranyçennoe mnoΩestvo.
Dokazatel\stvo. Esly b q( ) ≥ 0, to J q a q( ) ( ) /≥ 2 y a q( ) < 2γ . Esly
Ωe b q( ) < 0, to sohlasno lemme 3 b q a q( ) ( )> − . Znaçyt, J q a q( ) ( ) /> 6 y
a q( ) < 6γ . Takym obrazom, Wγ soderΩytsq v ohranyçennom mnoΩestve
q l a q∈ <{ }2 6: ( ) γ .
Teorema$5. Pust\ V r d rn n( ) ( )/= 3 3, hde dn — ohranyçennaq posledova-
tel\nost\, operator A otrycatel\no opredelen, q W( )0 ∈ γ y q l( )1 2∈ t a -
kov¥, çto
1
2
1 2 0q J q( ) ( )( )+ < γ .
Tohda zadaça Koßy s naçal\n¥my dann¥my q( )0
y q( )1
ymeet edynstvennoe
hlobal\noe reßenye.
Dokazatel\stvo. Suwestvovanye y edynstvennost\ lokal\noho reßenyq
q ( t ) sleduet yz teorem¥ 1. Kak y pry dokazatel\stve teorem¥ 3 (sluçaj a)), dos-
tatoçno pokazat\, çto q ( t ) ostaetsq ohranyçenn¥m.
PokaΩem, çto q t W( ) ∈ γ . PredpoloΩym, çto πto ne tak, y pust\ t1 0> —
naymen\ßee znaçenye t > 0, dlq kotoroho q t W( )1 ∉ γ . Tohda q t( )1 prynadle-
Ωyt hranyce ∂ γW mnoΩestva Wγ . Poskol\ku Wγ zvezdno, to θ γq t W( )1 ∈ dlq
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
O DYNAMYÇESKYX URAVNENYQX SYSTEMÁ LYNEJNO SVQZANNÁX … 729
lgboho θ ∈[ , )0 1 . Znaçyt, J q t( )( )θ γ1 < . Perexodq k predelu pry θ → 1, po-
luçaem J q t( )( )1 ≤ γ . Esly J q t( )( )1 < γ , to v sylu opredelenyq Wγ y toho, çto
J q t( )( )θ γ1 < , ymeem q t W( )1 ∈ γ . Poslednee protyvoreçyt sdelannomu predpo-
loΩenyg. Takym obrazom, J q t( )( )1 = γ .
Poskol\ku hamyl\tonyan H soxranqetsq, to
J q t( )( )1 ≤ 1
2 1
2
1˙( ) ( )( )q t J q t+ = 1
2
1 2 0q J q( ) ( )( )+ < γ .
Poluçennoe protyvoreçye pokaz¥vaet, çto q t W( ) ∈ γ dlq vsex t > 0, dlq koto-
r¥x q opredeleno. Sledovatel\no, reßenye suwestvuet pry vsex t > 0.
Tak kak uravnenye (1) ynvaryantno otnosytel\no zamen¥ t na – t, reßenye
opredeleno pry vsex t ∈R.
Teorema 5 qvlqetsq dyskretn¥m analohom odnoho rezul\tata SπttyndΩera
[9], otnosqwehosq k nelynejnomu volnovomu uravnenyg. V çastnosty, ona daet
hlobal\nug razreßymost\ zadaçy Koßy v sluçae, kohda naçal\n¥e dann¥e dos-
tatoçno mal¥ v l2
-norme.
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Poluçeno 13.07.2004,
posle dorabotky — 09.03.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
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| id | umjimathkievua-article-3490 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:43:29Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/84/85ad5413d0e41ef42518ff3250924184.pdf |
| spelling | umjimathkievua-article-34902020-03-18T19:56:00Z On the dynamical equations of a system of linearly coupled nonlinear oscillators O динамических уравнениях системы линейно связанных нелинейных осцилляторов Bak, S. N. Pankov, A. A. Бак, С. Н. Панков, А. А. Бак, С. Н. Панков, А. А. We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Some results concerning the existence and uniqueness of global solutions of the Cauchy problem are obtained. Доведено теорему про гладкість узагальнених розв'язків диференціальних рівнянь з операторними коефіцієнтами. Institute of Mathematics, NAS of Ukraine 2006-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3490 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 6 (2006); 723–729 Український математичний журнал; Том 58 № 6 (2006); 723–729 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3490/3717 https://umj.imath.kiev.ua/index.php/umj/article/view/3490/3718 Copyright (c) 2006 Bak S. N.; Pankov A. A. |
| spellingShingle | Bak, S. N. Pankov, A. A. Бак, С. Н. Панков, А. А. Бак, С. Н. Панков, А. А. On the dynamical equations of a system of linearly coupled nonlinear oscillators |
| title | On the dynamical equations of a system of linearly coupled nonlinear oscillators |
| title_alt | O динамических уравнениях системы линейно связанных нелинейных осцилляторов |
| title_full | On the dynamical equations of a system of linearly coupled nonlinear oscillators |
| title_fullStr | On the dynamical equations of a system of linearly coupled nonlinear oscillators |
| title_full_unstemmed | On the dynamical equations of a system of linearly coupled nonlinear oscillators |
| title_short | On the dynamical equations of a system of linearly coupled nonlinear oscillators |
| title_sort | on the dynamical equations of a system of linearly coupled nonlinear oscillators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3490 |
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