Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice
Using the inverse scattering transform, we investigate an initial boundary-value problem with zero boundary condition for the Toda lattice. We prove the existence and uniqueness of a rapidly decreasing solution and determine a class of initial data that guarantees the existence of a rapidly decreasi...
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| Дата: | 2005 |
|---|---|
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| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509799482916864 |
|---|---|
| author | Khanmamedov, A. Kh. Ханмамедов, А. Х. Ханмамедов, А. Х. |
| author_facet | Khanmamedov, A. Kh. Ханмамедов, А. Х. Ханмамедов, А. Х. |
| author_sort | Khanmamedov, A. Kh. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:01:36Z |
| description | Using the inverse scattering transform, we investigate an initial boundary-value problem with zero boundary condition for the Toda lattice. We prove the existence and uniqueness of a rapidly decreasing solution and determine a class of initial data that guarantees the existence of a rapidly decreasing solution. |
| first_indexed | 2026-03-24T02:46:51Z |
| format | Article |
| fulltext |
UDK 530.1
Ah. X. Xanmamedov (Bakyn. un-t, AzerbajdΩan)
BÁSTROUBÁVAGWEE REÍENYE
NAÇAL|NO-KRAEVOJ ZADAÇY DLQ CEPOÇKY TODÁ
By using the method of inverse scattering problem, we investigate an initial boundary-value problem
with zero boundary condition for the Toda lattice. We prove the existence and uniqueness of a rapidly
decreasing solution. We determine a class of initial data which guarantees the existence of rapidly
decreasing solution.
Metodom oberneno] zadaçi rozsiqnnq doslidΩu[t\sq poçatkovo-krajova zadaça z nul\ovog kra-
jovog umovog dlq lancgΩka Tody. Dovedeno isnuvannq ta [dynist\ ßvydkospadnoho rozv’qzku.
Vkazano klas poçatkovyx danyx, qkyj zabezpeçu[ isnuvannq ßvydkospadnoho rozv’qzku.
Vvedenye. Dlq posledovatel\nostej an = an ( t ) > 0, bn = bn ( t ) , n ≥ 1, vewe-
stvennoznaçn¥x funkcyj an , bn ∈ C( )[ , )1 0 ∞ rassmotrym naçal\no-kraevug za-
daçu dlq cepoçky Tod¥
˙ ( ), ,
˙ , , , ,
a a b b
d
dt
b a a n
n n n n
n n n
= − ⋅ =
= − = …
+
−
1
2
1 2
1
1
2 2
(1)
an ( 0 ) = ân, bn ( 0 ) = b̂n , (2)
a0 = 0. (3)
V rabote [1] predloΩen sposob yntehryrovanyq zadaçy (1) – (3), osnov¥vag-
wyjsq na obratnoj zadaçe po spektral\noj funkcyy. Tam Ωe ustanovleno su-
westvovanye reßenyq v klasse ohranyçenn¥x lokal\no ravnomerno po t posle-
dovatel\nostej an , bn . S druhoj storon¥, v rabotax [2, 3] metodom obratnoj za-
daçy rasseqnyq (MOZR) poluçena sxema postroenyq b¥stroub¥vagweho reße-
nyq zadaçy Koßy dlq cepoçky Tod¥, a v rabote [4] ustanovlen¥ formul¥ dlq
naxoΩdenyq b¥stroub¥vagweho reßenyq nekotoroj naçal\no-kraevoj zadaçy
dlq lenhmgrovskoj cepoçky. Odnako v πtyx rabotax predpolahaetsq, çto b¥-
stroub¥vagwye reßenyq suwestvugt y naçal\n¥e dann¥e ub¥vagt dostatoçno
b¥stro. Vmeste s tem vopros suwestvovanyq b¥stroub¥vagweho reßenyq os-
talsq otkr¥t¥m.
Reßenye an ( t ), b n ( t ) zadaçy (1) – (3) nazovem b¥stroub¥vagwym, esly
an ( t ) – 1 y bn — b¥stroub¥vagwye funkcyy, t. e. udovletvorqgt uslovyg
sup ( )
0
1
≤ ≤t T
Q t < ∞ , hde Qr ( t ) =
n
r
n nn a t b t
=
∞
∑ − +( )
1
1( ) ( ) , r = 1 ( yly r = 3 ) .
V nastoqwej rabote s pomow\g MOZR dokazano suwestvovanye b¥stroub¥-
vagweho reßenyq zadaçy (1) – (3). Krome toho, ukazan bolee ßyrokyj klass na-
çal\n¥x dann¥x, obespeçyvagwyx suwestvovanye takoho reßenyq. Rassmotre-
nye naçal\no-kraevoj zadaçy svqzano s tem, çto s pomow\g MOZR v obwem slu-
çae naçal\no-kraevug zadaçu dlq nelynejn¥x uravnenyj ne udaetsq reßyt\
stol\ Ωe πffektyvno, kak zadaçu Koßy (sm. [1, 5]).
Osnovn¥m rezul\tatom dannoj rabot¥ qvlqetsq sledugwaq teorema.
Teorema(1. B¥stroub¥vagwee reßenye zadaçy (1) – (3) suwestvuet y edyn-
stvenno, esly naçal\n¥e dann¥e udovletvorqgt uslovyg Q3 ( 0 ) < ∞ .
1. Predvarytel\n¥e svedenyq. V πtom punkte m¥ sformulyruem nekoto-
r¥e vspomohatel\n¥e fakt¥, mnohye yz kotor¥x soderΩatsq v [6]. Rassmotrym
hranyçnug zadaçu
© Ah. X. XANMAMEDOV, 2005
1144 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
BÁSTROUBÁVAGWEE REÍENYE NAÇAL|NO-KRAEVOJ ZADAÇY … 1145
ˆ ˆ ˆa y b y a yn n n n n n− − ++ +1 1 1 = λ yn , â0 = 1, n = 1, 2, … , (4)
y0 = 0, (5)
hde predpolahaetsq, çto Q1 ( 0 ) < ∞ . V uravnenyy (4) poloΩym λ = 2 cos z, z =
= ξ + i τ . Pry Im z ≥ 0 opredelym reßenyq fn ( z ) typa Josta uravnenyq (4), po-
loΩyv lim ( )
n n
inzf z e
→∞
− = 1. Takoe reßenye suwestvuet y opredelqetsq odno-
znaçno. Spravedlyvo [6] predstavlenye çerez operator preobrazovanyq
fn ( z ) = αn
inz
m
nm
imze A e1
1
+
=
∞
∑ , (6)
pryçem
αn = 1 + o ( 1 ) , n → ∞ , An m = O a b
k n m
k k
= +
∞
∑ − +( )
[ / ]
ˆ ˆ
2
1 , n + m → ∞ ,
hde [ x ] — celaq çast\ x.
Velyçyn¥ αn
, An m y koπffycyent¥ ân, b̂n uravnenyq (4) svqzan¥ sootno-
ßenyqmy
b̂n = A An n1 1 1− − , , ân =
α
α
n
n
+1
, ân
2 = 1 2 1 2 1+ − −−A A b An n n n,
ˆ
, (7)
ˆ ˆ
, , , ,a A A b A A An n m nm n n m n m n m
2
1 1 1 2 2+ + − + +− + + − = 0. (8)
Çerez ϕn = ϕn ( z ) oboznaçym reßenye uravnenyq (4), udovletvorqgwee us-
lovyqm ϕ0 = 0, ϕ1 = 1. Tohda [6] verno sootnoßenye
– 2i nsin ( )ξϕ ξ = f f f fn n0 0( ) ( ) ( ) ( )ξ ξ ξ ξ− − − , ξ π≠ k , k = 0, ± 1, ± 2, . (9)
Funkcyq f0 ( z ) rehulqrna v poluploskosty Im z ≥ 0 y tam v polupolose
Π = { / / }: ,z i= + − ≤ ≤ >ξ τ π ξ π τ2 3 2 0
moΩet ymet\ tol\ko koneçnoe çyslo prost¥x nulej v toçkax z ik k= τ , k =
= 1, … , N1 , z ik k= +π τ , k = N1 + 1, … , N (sm. [6]). Hranyçnaq zadaça (4), (5)
poroΩdaet v prostranstve l2 1( , )∞ ohranyçenn¥j samosoprqΩenn¥j operator
L̂ . Pry πtom sobstvenn¥e znaçenyq operatora L̂ qvlqgtsq prost¥my y sovpa-
dagt s toçkamy λ j jz= 2cos , j = 1, … , N.
Vvedem oboznaçenyq
S ( ξ ) =
f
f
0
0
( )
( )
− ξ
ξ
, M j
−2 =
n
n jf z
=
∞
∑
1
2( ), j = 1, … , N.
Kak pokazano v [6], vektor¥ { }( )un ξ 1
∞ , { }( )u zn j 1
∞ , opredelenn¥e po formulam
un( )ξ = f S fn n( ) ( ) ( )− −ξ ξ ξ , 0 ≤ ξ ≤ π ,
u zn j( ) = M f zj n j( ), j = 1, … , N,
obrazugt poln¥j nabor normyrovann¥x sobstvenn¥x vektorov operatora L̂ ,
t. e. ymeet mesto formula
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1146 Ah. X. XANMAMEDOV
j
N
n j m j n mu z u z u u d
=
∑ ∫+
1 0
1
2
( ) ( ) ( ) ( )
π
ξ ξ ξ
π
= δnm , (10)
hde δnm — symvol Kronekera.
Nabor velyçyn { ( ) }; ; , , ,S z M j Nj jξ > = …0 1 nazovem dann¥my rasseqnyq
dlq zadaçy (4), (5). V [6] ustanovlen¥ xarakterystyçeskye svojstva dann¥x ras-
seqnyq, pozvolqgwye po nym vosstanovyt\ koπffycyent¥ ân, b̂n uravnenyq
(4). PoloΩym
Fn =
j
N
j
inz inM e S e dj
= −
∑ ∫−
1
2 1
2π
ξ ξ
π
π
ξ( ) . (11)
Spravedlyvo sledugwee utverΩdenye (sm. [6]).
UtverΩdenye(1. Dlq toho çtob¥ nabor velyçyn b¥l dann¥my rasseqnyq dlq
nekotoroj zadaçy vyda (4), (5) s koneçn¥m perv¥m momentom Q1 ( 0 ) < ∞ , ne-
obxodymo y dostatoçno, çtob¥ v¥polnqlys\ sledugwye uslovyq:
a) funkcyq S ( ξ ) neprer¥vna na vewestvennoj osy y
S ( ξ + 2 π ) = S ( ξ ) , S( )ξ = S ( – ξ ) = S−1( )ξ ;
b)
m
m mm F F
=
∞
+∑ −
1
2 < ∞ ;
v) yzmenenye arhumenta funkcyy S ( ξ ) svqzano s çyslom N formuloj
N =
ln ( ) ln ( ) ( ) ( )S S
i
S S+ − − − − −0 0
2
2 0
4
π
π
π
.
Pry v¥polnenyy uslovyj πtoho utverΩdenyq obratnaq zadaça rasseqnyq re-
ßaetsq sledugwym obrazom. Snaçala po dann¥m rasseqnyq postroym velyçynu
Fn po formule (11). Tohda koπffycyent¥ ân, b̂n vosstanavlyvagtsq po lg-
b¥m yz formul (7), hde velyçyn¥ Anm y αn naxodqtsq yz sootnoßenyj
F A A Fn m nm
k
nk n m k2
1
2+
=
∞
+ ++ + ∑ = 0, n ≥ 0, m ≥ 1, (12)
αn
−2 = 1 2
1
2+ +
=
∞
+∑F A Fn
k
nk n k , (13)
pervoe yz kotor¥x ymeet edynstvennoe reßenye v lp( , )1 ∞ , p = 1, 2, otnosy-
tel\no Anm .
Uravnenye (12) naz¥vaetsq osnovn¥m uravnenyem typa Marçenko. Ono yhra-
et central\nug rol\ pry yssledovanyy obratnoj zadaçy y daet vozmoΩnost\ po-
luçyt\ nekotor¥e ocenky otnosytel\no Anm . Dejstvytel\no, yz uslovyq b) ut-
verΩdenyqT1 sleduet, çto uravnenye (12) poroΩdaetsq vpolne neprer¥vn¥m ope-
ratorom v l1 1( , )∞ , t. e. operator F( )n , dejstvugwyj po formule ( )( )F n my =
=
k n m k kF y=
∞
+ +∑ 1 2 , vpolne neprer¥ven v l1 1( , )∞ . Poskol\ku uravnenye (12) pry
kaΩdom n ymeet edynstvennoe reßenye v l1 1( , )∞ , operator I n+ F( ) dlq kaΩ-
doho n ymeet ohranyçenn¥j obratn¥j. Lehko vydet\, çto πtot obratn¥j opera-
tor ohranyçen v l1 1( , )∞ po norme ravnomerno otnosytel\no n. Tohda, perepy-
sav osnovnoe uravnenye (12) y uravnenye
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
BÁSTROUBÁVAGWEE REÍENYE NAÇAL|NO-KRAEVOJ ZADAÇY … 1147
( ) ( ), , ( )A A A A Fnm n m
k
nk n k n m k− + −−
=
∞
− − + +∑1
1
1 2 1 =
= ( ) ( )( ) ( )F F A F Fn m n m
k
nk n m k n m k2 1 2
1
2 1 2− + +
=
∞
− + + + +− + −∑
v vyde operatorn¥x uravnenyj y vospol\zovavßys\ ravnomernoj ohranyçenno-
st\g semejstva operatorov ( )( )I n+ −F 1, a takΩe uslovyem b), poluçym ocenky
Anm ≤ C n mσ ( )2 + ,
(14)
( ) ( ), ( )A A F Fnm n m n m n m− + −− + − +1 2 2 1 ≤ C n m nσ σ( ) ( )2 1 2 1− + − ,
hde
σ ( n ) =
k n
k kF F
=
∞
+∑ −2 .
2. Preobrazovanyq dann¥x rasseqnyq. PredpoloΩym, çto v uravnenyy (4)
stoqt koπffycyent¥ an ( t ) , bn ( t ) , n ≥ 1, qvlqgwyesq b¥stroub¥vagwym re-
ßenyem zadaçy (1) – (3). Tohda spravedlyvo sootnoßenye (9), hde funkcyy
ϕ ξn( ), fn( )ξ uΩe zavysqt ot t tn n: ( , )ϕ ϕ ξ= , f f tn n= ( , )ξ .
Lemma(1. Esly v uravnenyy (4) s koπffycyentamy an = an ( t ) , bn = bn ( t ) ,
n ≥ 1, poslednye qvlqgtsq b¥stroub¥vagwym reßenyem zadaçy (1) – (3), to
πvolgcyq dann¥x rasseqnyq opys¥vaetsq formulamy
S ( ξ, t ) = S e it( , ) sinξ ξ0 2− , zk ( t ) = zk ( 0 ) , M tk
2( ) = M ek
it zk2 20( ) sin− ,
(15)
k = 1, … , N.
Dokazatel\stvo. Vvedem, kak ob¥çno, [1], operator¥ L y A, poloΩyv
( Ly )1 = b t y a t y1 1 1 2( ) ( )+ , ( Ly )n = a t y b t y a t yn n n n n n− − ++ +1 1 1( ) ( ) ( ) ,
( Ay )1 = –
1
2 1 2a t y( ) , ( Ay )n =
1
2
1
21 1 1a t y a t yn n n n− − +−( ) ( ) , n ≥ 2.
Operator¥ L y A obrazugt [1] paru Laksa, y systema uravnenyj (1) s uçe-
tom hranyçnoho uslovyq (3) πkvyvalentna operatornomu uravnenyg
L̇ = AL – LA, (16)
hde toçka oznaçaet dyfferencyrovanye po t. Zametym, çto operator¥ L y A
qvlqgtsq sootvetstvenno symmetryçeskym y kososymmetryçeskym v l2 1( , )∞ :
L* = L, A* = – A.
Yz (16) y ravenstva
˙ ˙L Lψ ψ+ = λψ̇ sleduet, çto operator M =
d
dt
– A pe-
revodyt reßenyq uravnenyq Lψ = λψ v reßenyq πtoho Ωe uravnenyq. Rassmot-
rym ravenstvo (9). Oçevydno, çto ϕ = { }( , )ϕ ξn nt =
∞
1 qvlqetsq reßenyem uravne-
nyq Lϕ = λϕ, hde λ = 2 cos ξ . S druhoj storon¥, ( )Mϕ 1 = a1 2 2ϕ / =
= ( )/λ − b1 2, otkuda sleduet M ϕ = ( ) /λ ϕ− b1 2 . Krome toho, yspol\zuq (9),
pry n → ∞ naxodym
M i tn[ sin ( , )]− 2 ξϕ ξ = ˙ ( , ) sin ( , )f t i f t e in
0 0ξ ξ ξ ξ−( ) − –
– ˙ ( , ) sin ( , ) ( )f t i f t e oin
0 0 1− + −( ) +ξ ξ ξ ξ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1148 Ah. X. XANMAMEDOV
Odnako uravnenye Lψ = λψ ymeet edynstvennoe reßenye s takoj asymptoty-
koj.
Sledovatel\no,
M i tn[ sin ( , )]− 2 ξϕ ξ = ˙ ( , ) sin ( , ) ( , )f t i f t f tn0 0ξ ξ ξ ξ−[ ] − –
– ˙ ( , ) sin ( , ) ( , )f t i f t f tn0 0− + −[ ]ξ ξ ξ ξ .
Sopostavlqq πto toΩdestvo s (9) y ravenstvom
M tn[ ]( , )ϕ ξ =
λ ϕ ξ− b
tn
1
2
( , ), λ = 2 cos ξ ,
poluçaem uravnenyq
˙ ( , ) sin ( , )f t i f t0 0ξ ξ ξ− =
λ ξ− b
f t1
02
( , ),
˙ ( , ) sin ( , )f t i f t0 0− + −ξ ξ ξ =
λ ξ− −b
f t1
02
( , ) ,
yz kotor¥x sleduet pervoe yz sootnoßenyj (15). Krome toho, v sylu predpo-
sledneho uravnenyq ymeem
f z t0( , ) = f z i z z t b d
t
0
0
10
1
2
( , ) exp ( sin cos ) ( )+ −
∫ τ τ , Im z ≥ 0.
Sohlasno poslednemu toΩdestvu nuly z tk ( ) funkcyy f z t0( , ) ot t ne zavysqt:
z tk ( ) = zk ( )0 .
Rassmotrym teper\ normyrovannug sobstvennug funkcyg u tk( )( ) =
= { }( , )u z tn k n=
∞
1 operatora L. Poskol\ku sobstvenn¥e znaçenyq πtoho operatora
qvlqgtsq prost¥my, to
˙ ( ) ( )( ) ( )u t Au tk k− = Cu tk( )( ) .
UmnoΩyv obe çasty posledneho ravenstva skalqrno v l2 1( , )∞ na u tk( )( ) y vos-
pol\zovavßys\ tem, çto u tk( )( ) — normyrovannaq sobstvennaq funkcyq, a A —
kososymmetryçeskyj operator, poluçym C = 0. Sledovatel\no, Mu tk( )( ) = 0.
S druhoj storon¥, pry n → ∞
Mu tk
n
( )( )( ) = ˙ ( ) sin ( ) ( )M t i z M t e o ek k k
inz inzk k+[ ] + .
Poπtomu
˙ ( ) sin ( )M t i z M tk k k+ = 0, otkuda y sleduet spravedlyvost\ tret\eho
toΩdestva yz (15).
Lemma dokazana.
Formul¥ (15) pozvolqgt najty b¥stroub¥vagwee reßenye zadaçy (1) – (3).
Dlq πtoho nuΩno najty dann¥e rasseqnyq pry t = 0, zatem postroyt\ funk-
cyg F tn( ) po formule (11), v kotoroj vmesto S ( ξ, 0 ) , Mk
2 0( ) sleduet yspol\-
zovat\ (15). Dalee sleduet reßyt\ uravnenye (11) s parametrom t otnosytel\no
A tnm( ) y najty an ( t ), bn ( t ) po formulam (7).
V opysannoj sxeme postroenyq b¥stroub¥vagweho reßenyq zadaçy (1) – (3)
predpolahaetsq, çto takoe reßenye suwestvuet. Ot posledneho predpoloΩenyq
moΩno yzbavyt\sq, ubedyvßys\, çto postroenn¥e ukazann¥m v¥ße sposobom
funkcyy an ( t ), bn ( t ) dejstvytel\no udovletvorqgt sootnoßenyqm (1), (3).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
BÁSTROUBÁVAGWEE REÍENYE NAÇAL|NO-KRAEVOJ ZADAÇY … 1149
Kak otmeçalos\ v¥ße, esly an ( t ), bn ( t ) — b¥stroub¥vagwee reßenye zada-
çy (1) – (3), to operator M perevodyt reßenyq uravnenyq Lψ = λψ v reßenyq
πtoho Ωe uravnennyq. Naoborot, esly an ( t ), bn ( t ) y yx perv¥e proyzvodn¥e po
t b¥stro ub¥vagt y operator M perevodyt lgboe reßenye ψ uravnenyq Lψ =
= λψ v kakoe-to reßenye πtoho Ωe uravnenyq, to ( )L̇ AL L A− + ψ = 0. Tohda
v sylu (10) poluçym (16). Sohlasno (10) b¥stroub¥vagwye funkcyy an ( t ),
bn ( t ) s b¥stroub¥vagwymy proyzvodn¥my ȧn, ḃn qvlqgtsq reßenyem zadaçy
(1), (3) v tom y tol\ko v tom sluçae, kohda M u tn[ ( , )]ξ , M u z tn j[ ( , )], j = 1, … , N,
qvlqgtsq reßenyqmy uravnenyq Lψ = λψ . Poskol\ku u0 ( ξ, t ) ≡ 0, u0 ( zj, t ) ≡
≡ 0, to
M u tn[ ]( , )ξ = ˆ ( , )[ ]M u tn ξ , M u z tn j[ ]( , ) = ˆ ( , )[ ]M u z tn j ,
hde operator M̂ dejstvuet po formule
( )M̂y n = ẏ a y a yn n n n n+ −+ − −
1
2
1
21 1 1, n ≥ 1.
S druhoj storon¥, v¥raΩenyq
ˆ [ ( , )]M u tn ξ ,
ˆ [ ( , )]M u z tn j , j = 1, … , N , za-
vedomo qvlqgtsq reßenyqmy uravnenyq Lψ = λψ , esly
ˆ [ ( , )]M f z tn pry Im z ≥
≥ 0 sluΩyt reßenyem uravnenyq (4), koπffycyentamy kotoroho qvlqgtsq
an ( t ), bn ( t ) , n ≥ 1.
Zameçaq, çto
ˆ ( , )[ ]M f z tn pry n → ∞ ymeet asymptotyku
ˆ ( , )[ ]M f z tn = 2i z e o einz inzsin ( )+ , Im z ≥ 0,
poluçaem
ˆ ( , )[ ]M f z tn = 2i z f z tnsin ( , ), (17)
tak kak reßenye uravnenyq (4) s takoj asymptotykoj edynstvenno.
Takym obrazom, esly funkcyy an ( t ), bn ( t ) y yx perv¥e proyzvodn¥e po t
b¥stro ub¥vagt y verno ravenstvo (17), to an ( t ), bn ( t ) — b¥stroub¥vagwee re-
ßenye zadaçy (1), (3).
Rassmotrym teper\ sootnoßenyq (11), (15). Yz πtyx sootnoßenyj sleduet,
çto funkcyq F tn( ) neperer¥vno dyfferencyruema po t y
˙ ( )F tn = F t F tn n− +−1 1( ) ( ) . (18)
V prostranstve l1 1( , )∞ opredelqem operator F( )( )n t , polahaq
F( )( )n m
t y( ) =
k
n m k kF t y
=
∞
+ +∑
1
2 ( ) .
Norma operatora F( )( )n t v prostranstve l1 1( , )∞ ocenyvaetsq neravenstvamy
F( )( )n t ≤ sup ( )
m k
n m kF t
≥ =
∞
+ +∑
1 1
2 ≤
s n
sF t
= +
∞
∑
2 2
( ) . (19)
Esly Q3 ( 0 ) < 0, to funkcyq S ( ξ, 0 ) na otrezke [ – π, π ] ymeet neprer¥vnug
proyzvodnug vtoroho porqdka. Tohda, podstavlqq (15) v formulu (11), ynteh-
ryruq po çastqm y yspol\zuq (19), naxodym, çto operator F( )( )n t neprer¥ven po
norme na kaΩdom koneçnom otrezke [ 0, T ] . V toj Ωe formule yntehryruq
dvaΩd¥ po çastqm, ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1150 Ah. X. XANMAMEDOV
sup ( ) ( )
0 1
2
≤ ≤ =
∞
+∑ −
t T n
n nF t F t < ∞ . (20)
Odnako pry uslovyy (20) dlq lgboho n operator I tn+ F( )( ) ymeet pry vsex
t ∈ [ 0, T ] ohranyçenn¥j obratn¥j ( ( ))( )I tn+ −F 1. Poskol\ku operator I tn+ F( )( )
neprer¥ven po norme na [ 0, T ] , πtot obratn¥j takΩe neprer¥ven [7, c. 12, 13]
po norme na [ 0, T ] y, v çastnosty, ravnomerno ohranyçen na [ 0, T ] . Krome toho,
v sylu (18), (20) operator I tn+ F( )( ) syl\no neprer¥vno dyfferencyruem.
Tohda [7, c. 12, 13] operator ( )( )( )I tn+ −F 1
takΩe syl\no neprer¥vno dyffe-
rencyruem y spravedlyva formula
[ ]( ( ))( )I tn t+ ′−F 1 = – ( ( )) ( ) ( ( ))( ) ( ) ( )[ ]I t t I tn n t n+ ′ +− −F F F1 1. (21)
Zametym, çto obratn¥j operator ( ( ))( )I tn+ −F 1
ohranyçen v l1 1( , )∞ po nor-
me ravnomerno otnosytel\no n, n ≥ 0, y t, t ∈ [ 0, T ] . Dejstvytel\no, v sylu
(19), (20) pry n > n0 ymeem
sup ( )( )
0≤ ≤t T
n tF ≤
sup ( )
0 2 2≤ ≤ = +
∞
∑
t T s n
s tF <
1
2
.
Tohda
( )( )( )I tn+ −F 1 ≤ 1
1−( )−F( )( )n t < 2 pry n > n0 .
Ostaetsq koneçnoe çyslo operatorov ( )( )( )I tn+ −F 1, 0 ≤ n ≤ n0 , kaΩd¥j yz ko-
tor¥x ravnomerno ohranyçen na [ 0, T ] .
S druhoj storon¥, pry v¥polnenyy neravenstva (20) uravnenye (12) pry
kaΩdom n odnoznaçno razreßymo v l1 1( , )+ ∞ . Pry πtom reßenye A tnm( ) y, so-
hlasno (21), eho proyzvodnaq po t udovletvorqgt neravenstvam, analohyçn¥m
(14), a samy ravenstva (12), (13) moΩno dyfferencyrovat\ po t.
3. Dokazatel\stvo teorem¥. Pust\ Q3 ( 0 ) < ∞ . Opredelym funkcyy an =
= an ( t ) , bn = bn ( t ) po formulam
bn = A An n1 1 1− − , , an
2 = 1 2 1 2 1+ − −−A A b An n n n, ,
hde Anm — reßenye uravnenyq (12). Yz rezul\tatov pred¥duweho punkta sle-
duet, çto funkcyy an , bn , ȧn, ḃn qvlqgtsq b¥stroub¥vagwymy. Dlq dokaza-
tel\stva teorem¥ pokaΩem, çto esly Q3 ( 0 ) < ∞ , to reßenye A tnm( ) uravne-
nyq (12) udovletvorqet ravenstvu
˙ ( ), ,A A A b Anm n m n m n nm+ − −+ − +1 1 1 = 0, m = 1, 2, … , (22)
a velyçyna αn , opredelqemaq po formule (13), — sootnoßenyg
α̇n = –
1
2
bn nα . (23)
Sohlasno rezul\tatam p.T2, k obeym çastqm ravenstva (12) moΩno prymenyt\
operator M̂1, dejstvugwyj po formule
( )M̂ h nm1 = ˙ ( ), ,h h h b hnm n m n m n nm+ − −+ − +1 1 1 .
Tohda poluçym
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
BÁSTROUBÁVAGWEE REÍENYE NAÇAL|NO-KRAEVOJ ZADAÇY … 1151
˙ ( )( ) ( )F F F b F M An m n m n m n n m nm2 2 1 2 1 1 2 2+ + + − + + ++ − − + +
+
k
nk n m k
k
nk n m k
k
nk n m kA F A F A F
=
∞
+ +
=
∞
+ +
=
∞
+ + +∑ ∑ ∑+ +
1
2
1
2
1
2 1
˙ ˙ –
–
k
n k n m k n
k
nk n m kA F b A F
=
∞
− − + + +
=
∞
+ +∑ ∑−
1
1 2 1 1
1
2, ( ) = 0.
Yz (18) sleduet
˙ ( )( )F F Fn m n m n m2 2 1 2 1 1+ + + − + ++ − = 0,
k
nk n m k
k
nk n m k
k
n k n m kA F A F A F
=
∞
+ +
=
∞
+ + +
=
∞
− − + + +∑ ∑ ∑+ −
1
2
1
2 1
1
1 2 1 1
˙
, ( ) =
=
k
nk n k n m kA A F
=
∞
− + + −∑ −
1
1 2 1( ), = ( ),A A Fn n n m1 1 1 2− − + +
+
k
n k n k n m kA A F
=
∞
+ − + + +∑ −
1
1 1 1 2( ), , =
= b F A A Fn n m
k
n k n k n m k2
1
1 1 1 2+
=
∞
+ − + + ++ −∑ ( ), , .
Sledovatel\no,
( ) ( )ˆ ˆM A M A Fnm
k
nk n m k1
1
1 2+
=
∞
+ +∑ = 0.
Poskol\ku πto uravnenye dlq ( )M̂ A nm1 ymeet lyß\ tryvyal\noe reßenye, to
( )M̂ A nm1 ≡ 0, çto dokaz¥vaet spravedlyvost\ ravenstva (22).
DokaΩem teper\ ravenstvo (23). Yzvestno, çto pravaq çast\ ravenstva (13)
poloΩytel\na (sm. [5]). Tohda, dyfferencyruq ravenstvo (13) po t, poluçaem
α̇n = –
1
2
3( )αn np ,
hde
pn = ˙ ˙ ˙F A F A Fn
k
nk n k
k
nk n k2
1
2
1
2+ +
=
∞
+
=
∞
+∑ ∑ .
Yspol\zuq (12), (18) y (22), ymeem
pn = F A F F A Fn
k
n k n k n
k
n k n k2 1 1
1
1 2 1 1 2 1
1
2 1( ) , ( ) ,− +
=
∞
− − + + +
=
∞
+ ++
− +
∑ ∑ +
+
k
nk n k n k
k
nk n kA A F A F
=
∞
− + −
=
∞
+∑ ∑− +
1
1 2 1
1
2( ) ˙
, =
= – A A A A Fn n n n n− −+ + −1 1 1 1 1 1 2, ,( ) +
+
k
n k n k n k
k
nk nkA A F A F
=
∞
+ − + +
=
∞
∑ ∑− +
1
1 1 1 2
1
2( ) ˙
, , =
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1152 Ah. X. XANMAMEDOV
= b b F b A Fn n n n
k
nk n k+ +
=
∞
+∑2
1
2 = bn n( )α −2 ,
çto y trebovalos\ dokazat\.
Zaverßym dokazatel\stvo teorem¥. V sylu (8), (22) ymeem
˙ ( ),A A A b An n n n n1 2 1 2 1
1
2
1
2
+ − −− =
1
2
1 2( )− an ,
˙
, ,A b A a A Anm n nm n n m n m− + −+ − − +
1
2
1
2
1
2
2
1 1 1 1 =
1
2
1
21 1A An m n m, ,− +− .
Zapys¥vaq teper\ v uravnenyy (17) vmesto f zn( ) eho predstavlenye (6), vydym,
çto poslednye ravenstva vmeste s (23) πkvyvalentn¥ uravnenyg (17). Tem sam¥m
suwestvovanye b¥stroub¥vagweho reßenyq dokazano. Edynstvennost\ reße-
nyq sleduet yz odnoznaçnoj razreßymosty obratnoj zadaçy
*
.
Teorema dokazana.
1. Berezanskyj G. M. Yntehryrovanye nelynejn¥x raznostn¥x uravnenyj metodom obratnoj
spektral\noj zadaçy // Dokl. AN SSSR. – 1985. – 281, #T1. – S.T16 – 19.
2. Flashka H. On the Toda lattice. Inverse scattering solutions // Progr. Theor. Phys. – 1974. – 51,
# 3. – P. 703 – 716.
3. Manakov S. V. O polnoj yntehryruemosty y stoxastyzacyy v dyskretn¥x dynamyçeskyx
systemax // Ûurn. πksperym. y teor. fyzyky. – 1974. – 67, #T2. – S.T543 – 555.
4. Kac M.,van Moerbeke P. On the explicitly soluble system of nonlinear differential equation related
to certain Toda lattices // Adv. Math. – 1975. – 16, # 2. – P. 160 – 169.
5. Xabybulyn Y. T. Uravnenye KDF na poluosy s nulev¥m kraev¥m uslovyem // Teor. y mat.
fyzyka. – 1999. – 119, #T3. – S.T397 – 404.
6. Husejnov H. Í. Opredelenye beskoneçnoj matryc¥ Qkoby po dann¥m rasseqnyq // Dokl.
AN SSSR. – 1976. – 227, #T6. – S.T1289 – 1292.
7. Krejn S. H. Lynejn¥e dyfferencyal\n¥e uravnenyq v banaxovom prostranstve. – M.:
Nauka, 1967.
Poluçeno 10.06.2004
*
Edynstvennost\ b¥stroub¥vagweho reßenyq sleduet takΩe yz [1].
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
|
| id | umjimathkievua-article-3673 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:46:51Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/31/c9531b2be24d58c6fd5433680f611131.pdf |
| spelling | umjimathkievua-article-36732020-03-18T20:01:36Z Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice Быстроубывающее решение начально-краевой задачи для цепочки Тоды Khanmamedov, A. Kh. Ханмамедов, А. Х. Ханмамедов, А. Х. Using the inverse scattering transform, we investigate an initial boundary-value problem with zero boundary condition for the Toda lattice. We prove the existence and uniqueness of a rapidly decreasing solution and determine a class of initial data that guarantees the existence of a rapidly decreasing solution. Методом оберненої задачі розсіяння досліджується початково-крайова задача з нульовою крайовою умовою для ланцюжка Тоди. Доведено існування та єдиність швидкоспадного розв'язку. Вказано клас початкових даних, який забезпечує існування швидкоспадного розв'язку. Institute of Mathematics, NAS of Ukraine 2005-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3673 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 8 (2005); 1144 – 1152 Український математичний журнал; Том 57 № 8 (2005); 1144 – 1152 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3673/4074 https://umj.imath.kiev.ua/index.php/umj/article/view/3673/4075 Copyright (c) 2005 Khanmamedov A. Kh. |
| spellingShingle | Khanmamedov, A. Kh. Ханмамедов, А. Х. Ханмамедов, А. Х. Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice |
| title | Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice |
| title_alt | Быстроубывающее решение начально-краевой задачи для цепочки Тоды |
| title_full | Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice |
| title_fullStr | Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice |
| title_full_unstemmed | Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice |
| title_short | Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice |
| title_sort | rapidly decreasing solution of the initial boundary-value problem for the toda lattice |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3673 |
| work_keys_str_mv | AT khanmamedovakh rapidlydecreasingsolutionoftheinitialboundaryvalueproblemforthetodalattice AT hanmamedovah rapidlydecreasingsolutionoftheinitialboundaryvalueproblemforthetodalattice AT hanmamedovah rapidlydecreasingsolutionoftheinitialboundaryvalueproblemforthetodalattice AT khanmamedovakh bystroubyvaûŝeerešenienačalʹnokraevojzadačidlâcepočkitody AT hanmamedovah bystroubyvaûŝeerešenienačalʹnokraevojzadačidlâcepočkitody AT hanmamedovah bystroubyvaûŝeerešenienačalʹnokraevojzadačidlâcepočkitody |