Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system
We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensionalk-constrained KP-hierarchy, briefly called the 2d k-c-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2d k-...
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| Date: | 1999 |
|---|---|
| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4585 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510734406909952 |
|---|---|
| author | Samoilenko, A. M. Samoilenko, V. G. Sidorenko, Yu. M. Самойленко, А. М. Самойленко, В. Г. Сидоренко, Ю. М. |
| author_facet | Samoilenko, A. M. Samoilenko, V. G. Sidorenko, Yu. M. Самойленко, А. М. Самойленко, В. Г. Сидоренко, Ю. М. |
| author_sort | Samoilenko, A. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensionalk-constrained KP-hierarchy, briefly called the 2d k-c-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2d k-c KP-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for the construction of exact solutions of equations belonging to the 2d k-c KP-hierarchy is proposed. |
| first_indexed | 2026-03-24T03:01:42Z |
| format | Article |
| fulltext |
Y,~K 517.9
A. M. CaMogJ'lenXO (IH-T t, laTeMaa3~xe HAH YZpalHa, Irwin).
B. r . Ca~otU~eHKo, IO. M. CHRopeHKO (Hau. y,-T i M. T. RIenqeaxa. Kais)
I E P A P X I H PIBHHI-I~ K A ~ O M I I ~ B A - IIETBIAIII~I ,rII
3 H E J I O K A 2 I b H H M H B ' H 3 H M H : B A F A T O B H M I P H I
Y 3 A F A S I ~ H E H H H T A T O q H I P O 3 B ' H 3 K H
P E ~ Y K O B A H H X C H C T E M *
The spatially two-dimensional generalization of hierarchy of the Kadomtsev-Petviashvili equations
under nonlocai constraints or the so-cailed 2-dimensional k-constrained If, P-hierarchy is given (its
abrigded notation is 2 d k-c-hierarchy). As examples of (2 + 1)-dimensional nonlinear models
belonging to the 2 d k - c KP-hierarchy, both generalizations of already known systems and new
nonlinear systems are presented. A method for constructing exact solutions of equations belonging to
the 2d k-c KP-hierarohy is proposed.
./~aHO npOCTOpoBO-~aJ~omo4ipHe y3ara.qbHeHH~l iqmpxff pisH~lUb Ka~o~tteBa-l'leTBiamBiafi 3 He~oza.nb-
HH~H n ' a a ~ - - v a g 3saHa 2-dimensional k-constrained KP-hierarchy (czopoqeHo: 2 d k - c KP-
iepapxia). Ha~e~e.o npazaazta (2 + 1)-BHMipHHx HeJliHiRHHX Mo/~e.Jlcg, u~o e rlpe]lc-raBHl4gaMH 2 d
k -c KP-iepapxii; cepe~ aKttX nKa~aao, 3oKpeMa, aK yaara.nbHe, aa pauline Bi/toMI4X, Tag i Honi He-
afittil~Hi cac'reMa. 3anpononoeaHo MeTeor no6y/tosa TOqnHX po3n'~i3ZiB IX Jill pinaaHb ~ 2d k-c KP-
iepapxii.
BcTyn. B cy,-lacHitt Teopii HealiHiflHHX iHTeFpoBHHX CHCTeM MaTeMaTHqHOi Ta Teope-
THqHOi (~)i3HKH 3HaqHy po.rlh Bi~iFpaioTb ~nre6pa~Hi KOHCTpyKKi~ Bi~IOMO~ rpyuH
SnOHCbZa~X MaTe~avHZiB [1--3], ~Ki npHaH~TO Ha3Hsav~ Teopiem CuTe. IIi KOm
c r p ~ e np~o~HHM y3ara~sHeHHxM i no~zanbmHM p o 3 8 H ~ e ~ nioHepcbK~X Me-
TO~iB i pesy~Taris ~eJmKOi rpynH ~OCJfi~HHKiS, 3ogpcMa MOCgOBCSgOi Ta J~e~iH-
rpa~ch~oi mKiJ~ C. H. HosiKoaa i J-I. ]L q)a~)zr162 a Ta~o)z B. ~. 3axaposa,
A. B. HIa6aTa, I. M. FeJmqbaH~a, Y[. A. ]IiKoro, t0. I. MaHiHa, B. M. ]~piHc])eJ~s]Ia,
M. A~Jmpa, B. KOCTaHTa, P. XipovH, LbZ. Bi~s~o~a Ta 6arav~ox iHmax (~eTanSHHa
o r a ~ ~ e . s [4]). Hposi~Hy pore, S tmX ~ocHi~KeHHP, X 3att~ae Teopi~ Tax 3BaH0i
"c-~yHI~ii Ta Teopi~ iepapxii pieHmtb Ka~omtesa-l ' Iers iaxtmi~i , ix y3ara.r~HeHH~ Ta
aaCT0CySaHHa [1 - 4].
~o6pe si~o~o [1, 4], mo i~apxia pis~HS Ka~oM~esa-rIeTeiatusi~i ~onycKae
npocTOpOBi pe~yK~ii ~O (1 + 1 )-BHMipHHX iHTeFpOBHHX Mo~eylelL TaK 3BaHi k-
reduced KP-hierarchy, cepc~z ~ZHX ~ic'~T~CS 6araTO Bi~oMHx cHCTeM HeJIiHigHHX
pisH~Hb 3 qaCT~HHHMH noxi~HH~H. TaK, piBHaHHa KopTesera-~e<l)pi~a,
EycciHccKa -- Kayna Ta piSH~,HH~ HeniHiaHoi cTpyH~ ~ic-rsT~Ca s 2-pe~lyzuii Ta 3 -
pe~yg~ii iepapxii piBH~Hb K a ~ o m t e s a - H e ~ i a - m i ~ I i Bi/HIoBi~Ho. IHKOYg4 k-pe~yzt~ii
a iepapxii piBHSHb Ka~o~ t t c sa -1 - [ eTs i ams in i HasHsamTs TaKo3K pc~yz t t i a~H
F e a b c ~ a H ~ a - ~ i z o r o Ta A ~ e p a - - K o c T a H T a , sk i nepmH~H 3anponoHysa~H ~i
pe~yztfii Ta KOCJIiK3KyBaJItl iX c~op~a~bHo-a~retpai~Hi Ta raMiJIbTOHOBi aCHeKTH
si~oBi~o.
~crrpm~oq~c~ noma~e~m Ta Tep~/Ho~orff c ~ r i [1 ], iepapxim pisH~m K~o~-
l~eBa - Her~iaumi~i MO3KHa 3anacaTH y BHFJIYJ~i HecKiHqeHHO1 IIOCJIigOBHOCTi onepa-
TOpHllX pi~HYg4b CaTo -- BDmcoHa
w,. = - ( w ~ w - ' ) _ w , . = ~ , 2 . . . . . (~)
ge
W = 1 + co I D -I + ~2 ~ - 2 +... (2)
* BSzoHaHa npH ,mc'rKoei~ qbimmconi/~ ninTp~ai ~epx<a~Horo ~omly qbyn~aMem~LnhMxx ~oc~i-
~eHb rips Mi~icrepcrai YKpa~ y cnpa~ax Hayga i TvxnO~orig.
A. M. CAMO~:IYIEHKO, B. F. CAMO~fflEHKO, 10. M. CH~OPEHKO. 1999
78 !$$N 0041-6053. YKp. ~am. ~y. ps., 1999, m. 51, N e 1
IEPAPXI$I PIBH$1Hb KA/IOMI.[EBA-HETBIAILIBI~I 3 HEJIOKAJIbHHMH ... 79
r Mixpo}~HqbepeHtfiaribHaM onepaTopoM ( M ~ O ) 3 xoeqbiuieHTaMn
3arie~xaT~ BiA 3MiHHHX t = ( q , t 2 . . . . ), t I : = X, i
D D -~ = 1 , (3) D : = ~x,
a AHqbepeHI~tianbHa i iHTerpa~rbHa qaCTHHH MimpoAHcXDepeHI~iari~,Horo onepaTopa
W ~Z~' W- 1 nO3H~qaIOTt,CA BiAnoBi/XHO ( W D n W" I ) + Ta ( W D n W- 1 )_, npH tmOMy
( W ~ D n W - I ) _ := W D " W '~ - ( w D n w - I ) + . (4)
1-lpH qbiKcOBaHO~y 3HaaeHHi qncria n ~ N KO~XHe a piBHJ~Hb (1) piBHOCnm, He He-
CKiHqeHHi~ CHCTeMi HeriiHi~HHX piBHAHb 3 qaCTHHHHMrl rIOXiAHHMrl Aria KOeqbil2ieH-
"lib O~i(t ) MiKpo/IHd~epenlAia~bHOrO onepaTopa W BHrnAAy (2). 3a AonoMorom Mi-
KpoAHqbepeHttia~bHOrO onepavopa L, Mo BHaHayaerbcA aa qbopMyriom
L := W~DW - l = ~D + U ~ f-1 + r D - 2 + . . . . (5)
CHCTeMy (1) ~ o ~ H a noAaaa4 B qbopui ao6pa~eHHa ~ a x c a BHrriaAy
Ae
~ln ~ ~
(o i, i > 1, m o
~t L = [Bn, L ], (6)
B n = (Ln)+ = (WD"W-I)+ := E+, neN. (7)
Mac Mici~e TaKe TBepAbKeHHJt (AHB., HalIpHKria~, [4] ).
T~epa~xenna 1. Cnpase3nusi maKi msepa~enn~:
1 o. HacniOKozt pisnocmi (6) e cniesianoutenna
~tmBn- atnB m +[Bn, Bm] = O, m,n~ N. (8)
2 o. Bemnopni non.~ { 3t. }, eusna,leni pisnicmlo (6), Koztymytomb, mo6mo
Otto (~t. L) = bin (3t,. L).
OnepaTopHa qbopMa 3anHcy CHCTeMH (8) Ha3HBaeTI~CA 3o6pa~eRl-ta~t 3axapo~a-
l_lIa6ama [5] a6o pisn.~nna~t nynbosoi" KpusuHu[ 6, 7].
3ay6a~Keuna 1. KomHe piBHAHHA (6) npH qbiKCOBaHOMy n > 1 eKBiBa~eaTne
HecKiHqeHHh~ CHCTeMi AH(]3epeHl.ljaribaHX piBaJtHb Arla HeCKiltqeHnoi MHOJ~I4HH HeBi-
/~OMHX c13yHKHi~ U , U 2 . . . . . IiXO ABriAIOTI:~ CO6OIO AHC1)epeHl.ljaJlbHi IIoriiHOMH BiA
Koec1Di~ieHTiB 03i(t ) MiKpOAHqbepeHuiaribHOrO onepaTopa W BHrri~I/:~y (2) i
aarie~aTb BiA ~BOX.He3aJIeaCrIHX 3MiHHHX t I = X Ta t n. 3 inmoro 6oxy, X o ~ a e a
0nepaT0pHHX piBnAm, (8) rIpa c1DiXcoBarIHX m , n E N exBiBaneaTHe CKiHYeHHi~
CHCTeMi /Iklcl:)epeHIXia.rlbHHX piBHAHb, xi.rtbxicTb JtKHX cniBnaAae a tlHCeriBHiCTIO
HeBiAOMHX qbyHXUilt TpI,ox He3arie~r, HaX 3MirmHX fi : = X, t m, t n, TO6TO CHCTeMa (8)
- - aaMxHeHa, l l a CHCTeMa r aBHYaIaHO~O CtlCTeMOIO ~auqbepeHIfia.n_bHHX piBHAm, a
qaCTHHHHMH IIOXi/IHHMH.
l'lpH n = 2, m = 3 piBHSI~HS (8) piBHOCHm, He ~i~aoMouy pinHJmmO KaAorca~eBa-
rleTBiam-i~i [8] ~na qbyaxtdi U, rIpn t~OMy t 2 = y , t 3 = t i piBnzrraa Mac nHr.r t~
ISSN 0041-6053. Yxp. ~tam. ~. pn., 1999, m. 51, IV e I
80 A.M. CAMOI~IJIEHKO, B. r. CAMO~IYlEHKO, IO. M. CH~2OPEHKO
Yipot~e~ypa
MiKpo/~rlqbeperiIliaJ-tbHrll~ oriepaTop W BI4rJIally (2) TaKOi ]Io]IaTKOBOi yMOBrI:
WDkW-~ _- ( w D k w - ~ ) + ,
( 4 U t - g/~xx- 12~g/x)x = 3 U ~ . (9)
k-pe/lyKtlii aa FeJIbqbaH/IOM--JliKiM [1, 4] noJaarar n HaK.qa/IaHni Ha
~ 0 piBHHOCrlm~HO yMoBi
(10)
L t - B t a6o ( L t ) _ - - 0. (11)
KopeKTHiCTb o6Memeltb (10), (11) no:Iarar B ToMy, mo tti yMosa 36epiramT~cz
npH eBoJnomi MiKpo~nqbepeauia~bmlx onepaTopiB W i L BHac~i/IOK piBazat, (1) i
(6) Bi~noBi~ao. Ilpa t~OMy a (1) i (6) BnrizaBae
~t~W= ~tx.L= 0 (12)
i piBnanHa 3axapoBa-IIIa6aTa (8) pe~yKyeTbCa ~O piBnanHa Ylaxca Brir~a~y (npn
LI~OMy m = k)
~t,, Bk= [Bn, Bt], (13)
TO6TO ~O iHTerpoBrmX ( 1 + 1 )-BrI~ipHnX cncreM 3 arlqbepenuia~HrtMa onepaTopaMn
riapa Ylaxca Bnrzz;ay
Bk = (Lk)+ = Lk ' Bn = ( (B k),,/k)+ = (Ln)+. (14)
~ana po6owa ripHcaaqeHa ripo6~eMaM npoc-ropoBoro yaara.nbueuna ai;aOMriX ( 1 +
+ 1 )-BrIMipHrlX iHTeFpOBHHX CHCTeM, I_120 MaIOTb IIeBHI4i~I qbiaaqHHi~ 3MiCT, ra poapo6tfi
/~Jl~ TaKHX CHCTeM MeTo/~iB rio6y/~oBH IIIHpOKHX K~aciB TOqHHX pO3B'~I3KiB.
CTa-t-I'Jl no6y~oBaua TaKHM qHHOM. B riyriKTi I /~ar 03riaqeI-IH~ iepapxii piB-
nJm~, Ka~tomteBa-l'leTBiamBizi 3 nenoKa~n,nnMH B'aaaml (k-c KP-hierarchy), Ha-
BO/~JtTIaC~I riprmaa/Irl iHTerpoBnaX CrtCTeM, mo ~iicTJrr~ca e k-c KP-iepapxii. ~aJ~i
rio~aayr mo na~eiieai TyT KOMyraTopni ao6pa~enHa T~riy flaKca/1Ha ne~ini~tHriX
CnCTeM a TepMinax iriTerpo-zmqbepeatxia~baHx oriepaTopiB 6i~mtu ripncrOCOBaHi ZI~a
no6yaoaa TOqnaX pO3B'Z3KiB (aan. TeopeMy 4), Hi~ ni~oMi pa~ime. B nya~Ti 2 BBe-
~eao noaaTTa npocTOpOBO-~BOBaMipnoro yaara~baegHa iepapxii piBHaH~, Ka~oMtte-
Ba-l-leT~iamBi~i 3 He~o~a~bnn~rl B'ZaJ~Mn (2d k-c KP-hierarchy). Ty-r Ha~e~eao
n p n K ~ a (2 + 1 )-Bnuipanx neziHiitnnx Mo~eae~, ~ o e ripe/IcTa~nazaM~t 2d k
KP-icpapxii. 3o~pe~a, Bzaaaai zz yaa ra~uennz paHime BiaOMaX, Tax i aoBi ne-
ziHiflni CrlCTeMrL 3arrponoHoBano MeTO/I no6yaoBrt TOqHHX pOaB'~taKiB piBaJ~ab 2d
k-c KP-icpapxii (Teope~a 5). Ha aa~epmeHHa CTaTTi ~OpOT~O o6roBopmmTSCa
~eJ~Ki ripo6~eMH, mO TiCHO noB'aaaai 3/xario~ pO6OTO~O, z~i, Ha Ham nortH;a, CTa-
HOBd-I~ITb 3HaqHHi~ iHTepec ]~JI$1 crieIIiaJIiCTiB y 3B'Jt3Ky 3 MOXKJIHBriMri Ho~aJIblIIHMH
jaOCJlJ~KeHKgMri.
1. He~oKa~bHi pe~yKuii ~ iepapxii piBHanb Ka~o~ueea-IIeT~iamni~i. 5/~
BiaOMO, ~piM k-pe~y~tfi~t Fe~mqba~aa-~[iKoro (A/mepa-KOCTaHTa) BHFJI~I/Iy (10),
(11), ~aa iepapxii pinaaHb Ka~om~eBa-IIeTniaumiai (1), (6), (8) icayr irimma Ba~K-
mmriil i 5 i ~ m 3ara~sririit, Hi~ (10), (1 I), Trm o6Me~KeHb ;aaa MiKpo~dpepeatfia~,-
rinx onepaTopiB W i L, mo cy~icrraia 3 Bi;arioBi/xaOm ~_anaMi~om, sa3riaqeriom pi~-
riariaamt (1) i (6). Ue Tax aBaHi cn~erpi~tai pe/xyKttii, Briepme 3aripOnOHOBaHi
erarrax [9, 10], a raxO~ iX BeKTOpHi (6araToKo~moneaTHi) yaarara, rieHHa [11, 12].
Pe;ayKosaHa iepapxia piaHaH~ Ka~omteBa- IIeTaiamniai npri CHMexpiltai~ k-pe~tyK-
ttii aarmcyer~c~ ~a ~ono~orom qbop~yz
ISSN 0041-6053. Yrp. ~tam. ~.'vpn., 1999, m. 51,N ~ 1
I~PAPXI~I PIBH$IHb KA~OMIIEBA-HETB1AIIIBIJII 3 HEJ'IOKAIlbHHMH ... 81
~t. L = lB. , L] , (15a)
~t. qi =B.(q i ) , (15b)
~t, r/ = - B~, (r/), (15c)
]~e n ~ N, i = 1, l , a MiKpo~nqbepeHtlianbn.ta onepaTop L anrna~ay (5) 3a/~oBons-
rise yMoBy
l
L k = L~+ + ~ qi D-I r i . (16)
i=l
TyT qi = qi(t), ri = ri(t) ; B;~ ~/IrlqbepeHIXianbrlrU~ onepaTop, d@opMa.rlbHO cnpa-
~ e n n ~ (TpaHcnoHoaannfl) Z~o B. : = (L")+ anrna~ay (7). B (16) cn~nonoM D - ~ r i
noanaqeno xoMnoanuim onepaTopa D - l a n r n a ~ y (3) i onepaTopa MHo~enna Ha
d~ynKuiIO r i, T06TO
~ - I r i := ri59 -I +r ix D-2 - rixx D-3 - ... = ~ ( - I )J~(J)D - l - j , (17)
j=0
/Ie
r/y ) = ~OJr," i = 1,'-'l �9 j ~ N .
Ox j ,
Cnpaaez~mma TaKa TeopeMa [ 12].
TeopeMa 1. l-lpu dpiKcooano~o" k e N iepapxi~ pio..~nb Ka~zaceea-Hem~a-
moi.ai ouz.a.~y (15a) z ne.aos:a.abnu.~m o ' ~ t u (16), ( 15b, c) peaytg'embcs Oo inme-
epoottoi" ( I + 1 )-t3uzdpnoi" iepapxii" nenini#nux pisn.~nb, onepamopne zo@a~enn.~
Jla~ca nKoi',~tae ouen.qa
I ' ] B~ + ~ . q iD-Ir i , 0t. - B. = 0, n > l . (18)
i--I
3ay6a~enua 2. B aHrnOMOBH.X aH/IaHHsX HeCKiH'~eHHy nocni~oBnic'rb piBHSm.
(18) npH~n~'ro HaaUBaT. k-constra ined KP-hie ra rchy a6o cKopo~eHO k - c K P -
iepapxicm.
Ha~aa i ~tna cKopoqenaa iaTerpa.m.an~ /Io/IaIqoK B (16) ~ CMMBOJI onepaTopa
BozsTeppa I~ 3 B.pO~a~eHaM ziapoM V(x, y) Bnr~a~ay
1
V ( x , y ) = ~ qi(x)r i (y) := q (x ) r ( y ) , (19)
j = l
6y/~eMO noaaaqaTri 3a i lonoMoro~ cnMsozy q D-tr, zle q = (ql . . . . . ql), r T =
= (r~ . . . . . rl), T ~ anax TpaHcnonyBanna.
3ayBaT.crtMO TaKo~, mo TyT no 3aB~sc~rl ~ll~no SKaayeTt,Ca ~rteacaic ' r t , ~yrmttiti
q, r , co i a H r z s n y (2)Ta Uj a a r n ~ y (5) ai~ eso~or t i t lnnx napaMeTpin t 2, t 3 . . . . .
xoqa Taxa aanexuticTs ~aeT~Ca Ha yaaai, lXe erocyeTsCa TaKO~K i xoeqb iu i~Tis u i
onepaTopa L n aa raaz ty
ISSN 0041-6053. Y~p. ~tam, ~.~pu.. 1999, m. 5 !, hi e I
82 A.M. CAMOI~J'IEHKO, B. F. CAMOI~31EHKO, RD. M. CH~OPEHKO
~Zr L n := D n + u,~2 + ... + u o +
+ U_ I ~Z) -1 + U_2~) -2 + . . . . (20)
l ie
u i = ui(t ) = ui(x , t2, t 3 . . . . ), i < n - 2 ,
u,,_ 2 = n U , u,~-3 = n U 2 + n ( n - 1) ~/x . . . . .
2
3raliana Battle k-c KP-iepapxia Bl4raI#lliy (18) MiCTHTb 6araTo aK BiliOMrlX neJai-
HitlnHX (1 + I )-BH~ipHaX iaTerpOBHaX MOlieJzetl i ix BeKTopn~x l-KoMnonenTnaX
yaaraa~eab , TaK i 3Ha~Hy Kia~ziCT~ npmmHnoso HOBHX CnCTe~ HeniHitaHaX piBnanS
[9 -- 18]. 71r npHKJmli Tazoi iepapxii MO~ZHa BzazaTa I -c KP-iepapxi~, mo e sez-
TOpaa~ y a a r a ~ e a n a M siliOMOi iepapxii A6JmBitta - Kayna- HsmeJ~a - Cirypa [19],
axa aanncyea-bcz y BHrJ~alii
[ D + q D - ~ r , ~ t , - B , J = 0, (21)
l ie
B. : = (D + q D - t r ) ~ . .
FIepumM (HeTprlBia.rlbHrtM) npelicTaBHrlKOM ttiei iepapxii e CHCTeMa piBHJ~Ib
~t2q = qxx + 2 q r q ,
3t2r -- - r x x -- 2 r q r .
(22)
B 2-c KP-iepapxii nepmHM IlpelicTaBHHKOM e BeKTOpHC yaara~baeHHa cHerem~
~ n - OlaKa~n [ 19] B~rJazliy
~t2q = qxx + 2q . lq ,
~t2 U = (qr) x , (23)
~t2r = - rxx - 2 U r ,
npa tt~OMy L 2 = D 2 + u o + q D - l r , u o = 2 U (aaB. (20)).
Cepeli piBnaab 3 -c KP-iepapxii MiCTZTbCa yaara~baenna CHCTeM MeabHazoBa,
~Ipiaqbembaa- CozoaoBa, KyneptuMi~aTa Ta i~mnx [11-- 16]. 3ay~anmlMo TaKo>z, tU0
Bci k-peliyKona~i iepapxii piBamm KalioMuesa- rleTBiamBiJfi Bnraal~y (10), (11)
mCTJrrbcg s k -c KP-iepapxii Bnr~aliy (18), ;ae noTpi6no noKJmCTa q = r = 0.
3 a y 6 a ~ e n n ~ 3. KoMyTaropne ao6pa)zenHa J'Iazca (18) liJm k -c KP-iepapxii e
yMosom cy~icnoc-ri ~ a ]~osiJ~HOrO zoMnnezcHoro napaMcrpy 7~) niHi~m~X aaliaq
sarJ~uy
( B k + q D - l r ) ( f tt)) = ~,k f ( t ) ,
(24)
~)t,,f(t) = B n ( f ( t ) ) ,
/ I ~ s n~Ke siliOMHX HeJliniflUHX piBa$lHb, mO Mier~rhca ~,K qacr~oei BHIIa/~KH n
k- c KP-iepapxii, icnyIOTb i imui MO~K~mOeTi iX nolianH~i B ortepaToplti/t qbopMi
ISSN 0041-6053. Yrp, ~tam. ~Tpu.. 1999, m, 51,1~ 1
I~PAPXIfl PIBHflHb KA~[OMUEBA-IIETBIAIIIBIJII 3 HEJ1OKAJIbHHMH ... 83
J-laKca (a6o 3axapoBa-IIIa6aTa), mo dO3BOJIHYI0 B 6araT~ox BHrladKaX 3aCTOCyBaTM
dJE,q iX dOCJI1,D~KCHH~I C1~opMaYli3M MCTOdy 06CpHCHO~ 3a~a~i [6, 19-25]. ~r~q CHCTCM
(22), (23) ~ OdHOKOMnoHemaIoMy BHnaaKy (I = I ) ~axcosi napH onepaTopie
3a~a~TbC~ RHqbepeHIIia~bHHMH onepaTopaMH nepmoro nop~dKy a MaTpHILqMH
pOaMipHOCTi 2 X • 2 Ta 3 • 3 Bid~OBidHO. j[~.q CHCTeMH (22) -- BOKTOpHOi Bepcii
piSHaHH~ A6~noBiaa-Kayna- Hs~e~a - Cirypa -- cTaHdapame so6paaceHHa
Hy~moBoi KpHBHaHH 3anHcyeTbCa aa dOnoMoro~ MaTpHt~ posMipHOCTi (l + I ) X (I +
I) [6, 24].
Bzara~i Ka,'KyqH, dJLq 3aFaYlbHOFO BHIIa~Ky MO~KHa rlOKa3aTH, IllO KOMyTaTOpHe
ao6pa~eHH~ d~a k-c KP-iepapxii s H r ~ d y (18)piBH0CHYIbHe 306pa_,~KeHHIO HyJIbO-
BOi KpHBH3HH 3 0ncpaTopaMH, 3aHHCaHHMH 3a donoMoro~ MaTpmtb po3MipHOCTi (k +
+ l) • (k + l). ~ n z ttboro noTpi6no pO3mHpHTH CHCTCMy (24), Bn0d~'m /IOdaTKOBi
nOYm0Bi 3MiHHi, HaHpnKJIa~, TaKHM qHHOM:
f i : = ~xi-i , f i : = f , i = 1, k ; (25)
fk+j := D - l ( r j f ) , j = 1,--1; (26)
f : = ( f l , f 2 . . . . . fk+l) T
~ia onepaTopa D-1 Ha cl)yHKLdi BH3Haqa~I~Cg, BHXOdSlqH 3 KOHKI~TH3aI~ii d0-
c~id~Ky~aHnX nviTa~S dag KO)KHOFO cIDyHKI~iOHa/IbHOFO npocropy. TaK, dJL~ smla~-
Ky npocr0py L l (+o% s), e~eMeHTaMH aKOrO r qbyHKaii F(x), mo a6CO~THO in-
TeFpOBHiHanpoMimKy (--oo, S) a60 Ha ll'pOMbKKy (S, +oo), de s e : R ~ dOBi.rll, He
~HC~0, onepaTop D- ~ BHaHaqaeT~,Ca 3a dOnOMOrO~ qbop~y~m
X
D - I ( F ) := ~ F(y)dy, (27)
•
a ~zmo qbyHKRia r ~ Ll (R), TO onepaTop D-~ (dOdaTZOBO dO pianocTi (27))
MO~gHa BH3HaqHTH I/~ TaKHM qHHOM:
1 O(y)dy - r (28/ ~ -1 ( { ~ ) ' =
X
CHCTeMa k + I ziHiflHHX dHqbepcwaia~SHHX piBHY.Hb nepmoro n0p~dKy BHrYL~dy
~--- f = A f , (29)
~x
mo OTpH~y~n~C~ a nepmoro (iHTer~-dHqbep~H~iammoro) piBH~HH2 S ( 2 4 ) n i c ~
ni~c'l'aHOBOK (25), (26), Mac ~4aTpHn.rO A = (Aij), i= 1, k , j = 1,1, de
A~j = ~i/+~, i = 1, k - 1 ;
Akj = Xk ~ - uj_ t, j = 1, k - l ; (30)
A~(k+y) = - q j , A(k+j)! = rj, j = 1,1.
TyT 8~ --cmdso.rt KponeKepa, a sci ne,.~allHcaHi r HaTpmfi A pis~i SyJleBi.
B poaropHyTo~y BHrJ/dldi MaTDI41~ A ~ CHCTCMi pim~sm, (29) Mac B I 4 F ~
ISSN 0041-6053, Yap. ~tam, ~.'ypn., 1999 , m. 51, N ~ 1
84 A.M. CAMOI;IJ'IEHKO, B. F. CAMOFIJIEHKO, IO. M. CH2~OPEHKO
A=
0 1 0 ... 0 0 0 ... 0
0 0 I ... 0 0 0 ... 0
: : : : �9 : : : :
0 0 0 ... I 0 0 ... 0
0 0 0 ... 0 l 0 ... 0
)~k - - UO _ i l l _ U 2 . . . - - U k _ 2 0 - - q l " '" - q l
r~ 0 0 ... 0 0 0 ... 0
r 2 0 0 ... 0 0 0 ... 0
: : : : : * : : : :
r t 0 0 ... 0 0 0 ... 0
(31)
Koeqbiniem~a uj, j = 0, k - 2, ~anqbepemfia~Horo onepaTopa
k - 2
Bt = D k + ~ u j D j
i f 0
M O n a 3anHCaTH B BHrSI~]~i/~rlqbepeHlliaa-II,HHX noniHo~is CTOCOBHO BHXi~HHX Koed~i-
~r q./, s 7-/3 . . . . ~iKpo~nqbepeauian~aoro onepaTopa L Barna~ay (5).
3oKpe~a, cTaptai KoeqbiuiCHTn onepaTopa Bk MamT~ BHrm~a
Uk_ 2 = k 'U, uk-3 = k7.l z + k (k - 1__..~) ~ x " (32)
2
Inmn~ MeTO/~ 3Be/~eHH~I iI~erpo-~rlqbepeHtfiaJll, noi cneKTpaa'lba0i aa~aai
I ff)k + kf2 UJ DJ + q D _ l r ) ( f ) = ~,kf (33)
y=0
~o 3a~aai B ~nqbepeattiasmHi~ qbop~i nonarae y BrlKOpnCTaHrIi 3MirlHHX
Cp : = f ; ~ x := - - r ~ ( T O 6 T O ~t := - D - l ( r ~ p ) ) . (34)
ToAi 3~aqa (33) B HOBnX ~iHHHX 3anHcyea~ca S BHrsta~i cn~reMn
k + uj gJ + _ q
j=o = 0, (35)
r I D
~ae MaTpmfi MamTb pozMipnicTb (1 + 1)X (1 + 1). T y r I - - O ~ H H H q H a (1 • l)-~ta-
Tpm~g.
3ay6a.~eun~ 4. CneKTpam, ni 3axtaqi (29), (31) i (35) He e zaz~aqaml saza.abnOgO
noaoz, cenn~, ocKismzn Bi~rloBi/l~li ~qbcpemf ia~n i onepaTopn blalOTb CHa'I~HO pe~y-
Koruny erpyKrypy i i~ ~ a a e m ~ n~po~cemia erocosHo cneHwpasmHoro napaMerpa
~, e C. Y 3B'JI3Ky 3 I.!IIM 3ayBa~r IMO, II~O ~.Ii~ ~aoezfi~enna si~oBizBmX He3IiHiflHHX
i~rrerpOSHHX cnercM 3 k -c KP-iepapxii s n r s ~ a y (18) (HaCKi311bKH Re BiROMO gtBTO-
pabt) 3acTOeyBaltltJl Ka'laeHtlHOl'O BapialtTy M~'O/Iy 06epHeHOi 3a/~aqi, RIO l'pyHTy6TbC~I
Ha smcopHerarmi i r r rerpa~rmx pisnmm Feamqbant~a-3IesiTana-MapqeHHa [5, 6],
tX:m ~aoesfig~erm~ cHere~ sarz~gly (29), (31) i (35) He r ~o~ramH~ (s 6 y ~ - J m o ~
pazi - - na ~xamat ~ac).
ISSN 0041-6053, Y~p. ~ m . ~.'ypn., 1999, m. 51, N ~ 1
ICPAPXI~t PIBH~IHb KA/~OMI2EBA-I'IETBIAIIlBIYll 3 HF21OKAJIBHHMH ... 85
3ayBa~KHMO TaKO~K, mo BlIepuie YlaKCOBi 306pa~KeHH$1 3/~HdpepeHI/ja.rlbHHMH oIIe-
paTopaMH BHr~az~y (35) i ~Htue ~ I a BHIIa~Ky CKa2IJtpHI4X IlOJliB q, r (npH t~OMy 1 =
= 1 ) po3r~Jt/~a~lrtcb B. K. MOJIbHrlKOBrlM B [26, 27], de 6yJIo 3aNporlOHOBaHO MeTO/~
3HaXO/~KeHH$1 /~e~tKHX K~aciB TOqHHX pO3B'$I3KiB /~JI~l Bi/~noBi/~HHX He~liailtHr~X /~H-
qbepeHI/jaYlbHHX piBHJtHb. Y 3B'$I3Ky 3 RHM He~IiltiitHi k r KP-iepapxii Montana Ha-
3BaTh 6aFaTOKOMIIOHOHTHHMI, I y3ara,rlbHeHHJtMH CHCTeM MeJIbHHKOBa 3 iHIIIHM, Hick
(18), KoMyraTOpHHM 306pa)KerlH)l~I. B po6oTax [26, 27] TaKO)K 6y~o 3po6~eHo Bn-
CHOBOK IlpO HeMO~IJIHBiCTb 3aCTOCyBaHHJ/ BiZ~oMoro MeTO/~y O~azattt-t.~ (dressing
method) 3axapoBa- IIIa6aTa [5] /~o HeJIiHillHHX piBHJtHb T~ny Me~bHrlKOBa. /Ia~i Re
TBep/~KeHH~q cnpoCTOByeTbCa, OCKi~bKH 6y/~e JIBHO no6y/~oBaHo oz~ara~o~H~ onepa-
Top 3axapoBa- IiIa6aTa/~a CHCTeM piBH~IHb BKa3aHOFO T~ny.
1.1. Toqai po3B'SaKa ao~eaett 3 k-c KP-iepapxii'. Flic~a Bi~KpHTTa k-c
KP-iepapxii [9- 13, 17] BezaKa Ki~bKiCTb ~oc~i/DKeab 6y~a npr~cBa'~eHa pOaBHTKy
raMiZbTOHOBOI, TeOpeTago-rpynoBOi Ta Yli-a.are6paiar~oi Teopi~t mix piBHam, (daB.,
nanpnKza~, [9- 18, 28 - 42] Ta ttaTOBaHi TaM pO6OTa). ~Za no6y~aoBa TOqHaX poa-
B'Z3KiB piBHZHb, fRO BXO/~aT~ ~aO k -c KP-iepapxii, zacrocoByBa~ac~ 6i~iHi~iai Hero-
An XipoT~ [29 - 31, 34], pe~yKRii K~acr~aHOI x-~yHKRiI ~J~ iepapxii Ka~oM~eBa-
IIeTBiatuBizi [33], MeTO/~ iTepaRi~ ~zaca~rmx i 6iHapHaX nepeTBopem, ]Iap6y-BeK-
~yH~a [35, 36, 40, 41] Ta ill. HH~Kqe ~ a Z~oczi/I)KeHHJ~ k -c KP-icpapxii Barbary
(18) 8nKopacToSyeMO i~ei MeTO~y o~araHHa 3axapoBa- IIIa6aTa [5]. B r~o~a~bmo-
My HaM 3Ha~O6~ZT~Ca/~eaKi KOHCTpyKaii Ta peay~TaTa po6oTa [42] (~B. nyH~T 3,
TeopeMy 3).
Po3r~JIHeMO MiKpO~HqbepeHtlia.qI, HH~ onepaTop W [ f ] BHF-q-q/W
i
f / 0 ) . . . 1
1 i : : : , ( 3 6 ) W [ f ] =
/~e ma/~onoMorolo r no3HaqerIo BH3HaHHHK BpOHCbKOrO ,uiHillao-He3ane:~Knoi
CaCTeM~ qbyHKt~i~ f~ . . . . . flY, mO yTBOplOIOTb BeKTOp-pJt/~OK f = ( f l . . . . . fN), a
f(k '0 =- ~n f t ( x ' t 2 .... ) k = l , N n = N U { 0 } .
X n '
BHaHaqHrtK (36) rli/~paxony6~rbcJl IIIZJIXOM po3zna~y 3a e.rleMeaTaMH ocrarlnboro
CTOnnaa, npa ttboMy onepaTopa ~D j, j = 0, N , y ai~noai~aanx aHpaaax MaIOTb
aHaxoaaTaCb npaBopyq Bid ai~noBi~aanx Minopis.
Oqesaztno, tuo W [ f ] B ~aaqbepeat~iaJronait onepaTop N-to nopaz~Ky, TO6TO
Ord W [ f ] = N , npa UbOMy goedpit~icnT npa ~D N /iopiamor 1, a Iloro a~apo
Ker W [ f ] cniBna~ar 3 zini/~no~o o6ozori~oa) L ( f ) CnCTeMU qbyazrailt f l . . . . . fly,
TO6TO KerW[f ] = L ( f ) .
l'Io3HaqrlMO qepea cld/i,j Minop, I-1IO OTpaMyer~ca 3 arl3naqaHga ffl/[] e] Brlgpe-
c~IermaM i-parKa Ta j-cTonmr~. Poar~taHeMo MiKpoB,adpeperIaia2mHHlt ormpaTop K
Bar~y
K = h ~9-~f (t0, (37)
hj = (- I)A'+~-J'Wt~,j(Wb"]) -~. (38)
I$$N 0041-6053. Yxp. ~tam. ~.Hpn., 1999, m. 51, IV r 1
86
Mac ~ictte TcopeHa [42].
T e o p e M a 2. ] / ~ ~dxtJoau~epentdanbnozo onepamopa
I + K = 1 + h D - l f (:r
icnye odepnenuti onepamop, ur zanucyembc~ za 8ono~tozolo ~op~o,,au
(1 + K) - j --- 1 + ~ - 1 c o ~ + ... + D-~col =
N
= ~ D-iO~N+l_i = ( -1 )ND-NW*[ f ] , (39)
i=0
ae a,a.a ~tirtopi8 onepamopnozo 8usna~nur, a W[ f ] 8uKopucmano nosna~enn,a
co k := Wk, fN+l)[f], k = 1, N + I . (40)
TeopeMa 3. Onepamop 8uzn.aigy
( W [ f ] ) - t := W -~ = - f D - t h (41)
e o6epnenu~t Oo ~udpepen~ia.m,nozo onepamopa W[f] .
l'Ipu ~bO.~ty onepamop (41)e 8O.nbmeppi6cbKu.~t onepamopo.,',t (dpop.~uT, nbnu~t
cu.~,~o.ao~O z ~upoS~enu.,,,~ ~Spo~t.
]~oeeaenn,q,, BHKOpltCTOByloqH TeopeMy 2 Ta dpopHy•y (39), Moa~..la aalmcaTrl
TaXi pimxocri:
( W t f ] ) -1 = ( ( W * ) * ) - l = [ ( ( - 1 ) N ~ f l ( 1 + K ) - l ) ' ] - t =
= [(1 + K * ) - I ( - 1 ) 2 N ~ ] -1 = D-~'(1 - f ( N ) D - l h ) =
= D -N - D-Nf(N)D-Ih . (42)
3aeroco~yIo,-m rtpar~a.no Ylell6rdua ~.ri~ Brlna/~Ky HiKpoaHt~eperltda.rmHrlx onepa-
TOp~
f(N)D-I = D-If (/v) + D-2f (N+I) + ... = ~ D-if (N+i-l), (43)
i=I
orpmnycHo
i=I i=I
N
= ~ D-i f ( i -Oh - ~ D- i f ( i - l )h . (44)
/=1 i=1
B p a x o w ~ i a TOTOaCHier~
f(m)h = - ~m ~-1 , m = 0, N - 1, (45)
mo r Hac.ai/IxoM y~os (38), t~r dpymrafii hi, j = 1, N , r pO3S'xaKaMH crtercMa JdHil~-
rmx pisHarm (45), aKi HO~Ha no6y/Iy~ara n aSnOMy snraa~ti, nanpHr~eta aa ~aono~o-
ro~o Me'rony Kpa~epa, ~ y r t ~ ,ao~aHox n npasia ~ae'rnHi cni~ni~Homerma (44) aarm-
IIICMO "I~dKHM qm-lOM:
ISSN O041-605 3. Y~p. ,~am. ~'y. pn.,1999 , m. 51, N ~ 1
A. M. CAMOI~J'IEHKO, B. F. CAMO~EHKO, IO. M. CI~OPEHKO
ICPAPXDI PIBHI;IHB KA~OMI2EBA-HETBIAIIIBIJII 3 HEdlOKAflbHHMH ... 87
N
D - i f ti-I) h = - D - ~ (46)
i=l
a nepmnlt, ari~ano a pianicalo (43), Momna no~aTn y aarzazf i
~ D - i f f i - l ) h = f D - t h . (47)
i=1
TaKrIM qrtI-IOM, TeopeMy 3 ,/~OBe~eHO.
Hexa/4 Tcnep m = N U { 0 } i N = l + m. HO3HaqHMO 3a JlOnOMOrOm f l i fm
BeKTop-panKn
f t := ( f l , f 2 . . . . . ft), fm := (fl+l,fl+2 . . . . . fN)'
/Ie, aK i pardme,
f = (fl,f2 ..... fl,fl+l, fl+2 ..... fN) = (fl,fm)"
Oqesrt~HO, mo n p . m = 0 MacMo piBHiCTb f = f t , a n p a 1 = 0 ui/moai~mo
cnpaaaxyeTsCa cniBai~momeHHa f = fm-
Jliz onepaTopa W[f] . a cKa~apHy qbyHK1LilO g B143HaqaeTbCa 3a qbopMyJlOlO
W [ f ] ( g ) := q 4 ? [ f ; g ] =
= r , f2 . . . . . fN, g) q4?-! ( f l , f2 . . . . . fN), (48)
a/I•a BHrla~Ky aeKrOpHOi qbyHKllii g = (g I . . . . . gp) m BrlpaaoM
W [ f ] ( g ) := (r/d?[f; g l ] . . . . . ffl?[f; gp]) . (49)
Hexa~ MatNx M(C) ~ M m ~ Z~OBi~qbHa cTa~a KOMrl.rleKctta (N x m )-MaTpmt~, a
Ma tu (C ) ~ MOO --~aoBizbHa cTa.na Kaa~parna ( N x N)-MaTprma npn n = 2, 3 . . . . .
TyT n - - z~tirmH~t iH~teKC, mo ai/moBi~ar napaMeTpy t n, a N ~ qbiKcOBaHe
HaTypa.rlbHe qHC~O, ZKe a n He rlOB'Jt3aHe.
TeopeMa 4. MiKpoOudpepen.ia,~,nua onepamop L = W D W -1 e po36'asro~t
onepamopno? k -c KP-iepapxff eue,aaOy (15a) - (15c) , (16), aru~o cy~ticna matca
cucme~ta piananb:
f(k) = f Mm '
(50)
~t,,f = f ( " ) + fM(n)"
]loseaeuua. 3naR~aeMo cnoqaaxy i~rrerpazbny ClC~a/IoBy qacTrmy onepa ' ropa
L/. 3 raieIo MeroIo, apaxoay~Oqa (41), poar~aneMo pianocTi
(Li)_ = ( W D k W - ~ ) _ = [ ( W g i ) W - t ] _ =
= - [ ( w D k ) f ~ T ' i h ] = - [ ( W g k f ) D - l h ] _ =
= _[(wtDk) (/)] ~-lh = _W(f(k)) ~-I h =
= - W ( f / ~ ) ) D - l h l - W(f(m~))D-ihm, (51)
l~c W := W [ f ] , h ! : = ( h i , h 2 . . . . . ht), hm : = ( h i + l , hl+2 . . . . . hN).
ISSN 0041-6053. Yh'p. ~am. ~.'vpu., 1999. m. 51.1~ 1
88 A, M. CAMO~YlEHKO, B. F. CAMOI~IJ'IEHKO, IO. M. CI4~OPEHKO
Ha rli~cTaBi IleplllOFO piBH.,qHH.,q CHCTeMH (50), BH3HaqeHH,q (49) i dpopMyJBl (41)
lxe
X N = W [ f ] = W ( f t , f 2 . . . . . fN).
IIoKa.~eMo Tenep, mo onepaTop W(~t, ' - D") W- t ~ qrICTO ~tHdpepeHttia.m, HH~,
TO6TO, m0 ~oro iH'rerpa.nL,Ha cKna~oBa qaca"nHa z~opinH~Oe HyneBi. Cnpa~i , MaCMO
TaKi piBHOCTi:
w ( ~ , . - ~ ' ) w -~ = [ w ( ~ , . - ~ " ) w - ~ ] + + [ w ( ~ , . - ~ ' ) w - ~ ] _ =
= ~,,, - ( W D " W - ~ ) § + W ( W - ~ ) , , , - ( W D " W - t ) _ =
= ~,~ - B n - W ( f , , ) D - ~ h - W - t ( f ) D-~h , . + W ( f (n ) )D-~h =
= 8,,, - B n - W ( f ) D-~h,,, - W(f, , , _ f n ) ) D-~ h. (57)
Tpcri~ ~O~aHOK B npa~i~t ~aCTHHi cninni~omeHHa (57) ~opinH~ ayne~i ~ri/xHO 3
OaHa~eUHaM (48), (49), a acrncpTma - - nnacni~ox ~tpyroro piBUaHHa CHCTe~m (50) i
dpopMyn (48), (49). TaKH~ tumor, o~acpaty~to
ISSN 0041-6053.: Y~p. .~utm. ~..'vpn.. 1999. m. 51, N ~- 1
MaeMO f(m k) ~ Ker W[f ] , TaKHM qI4HOM, o~aep:,Kyea,,to
(Lk)_ = - W(f/(k))D -! h I := q D - I r , (52)
~e, ari~Ho 3 dpopMy.naMa (48) Ta (38), BHKOHylOTI~CJt pinnocTi
qi = + w ( f /k ~ ) := + W [ / ; f / k ) ] =
= - + ' W ( f l . . . . . f,v, f / ~ ' ) ) W - t ( . t i ..... f N ) , (53)
r i = +(- 1 ) l V - i W N i [ f ] W - ~ ( f t ..... fN) . (54)
3ayBamHMO, mo KoeclJilIi6HTH U j , j = 0, k - 2, ~ItqbepeHuia.nbHOrO onepaTopa
B k, aKnfl aanHcyeTt,Ca aa aonoMoro~o @opMy.rIH
B k := (Lk)+ = ( W D k W - I ) + = - ( W D k f D - Z h ) + :=
k-2
= Dk + ~_, u jD .i, (55)
j=o
e Bi/~HOtUemt~M ~Htloepe~.ittian~HHX no.niHOMiB Bi~ d,3yHzuifl f t , f2 . . . . . f/~ mo Brl-
n.naBae 3 ,qBHHX BHpaai" /I.na Koecl3iuieHriB onepaTopis W[f] BHrn.a~y (36) i
W - l = - f D - t r BrW.na~y (38). H.~HVlfl BrWna~ d~ynKttii.i Uj, j = 0 , k - 2 , MO*Ha
3HaflTH/1.rlJ~ KO:a<HOFO qbiKcOBaltOI'O 3aaqeaaJt k ~ N ariaHo 3 qbopMy.noto (32), rlpH-
'.tOMy ui qbyaKuii ~OrlycKatOTb KOMnaKTHy dpopMy ao6pax~eHHa B TepMiHax/m~epeH-
ttia.m,HHx no.niHOMiB Bi/I .norapH~Ma x-qbyHKttii Cawo [2, 4] ~a.na piBHaHb iepapxii
Ka~oMttena - HeTBiatuni.rii. HarlpHK.natt,
U = (ln'CN)xx, U 2 = l[(InXN)m2 - (ln'rN)xxx], ir.~a., (56)
z
ICPAPXI~I PIBHflHb KA]IOMHEBA-I'IETBIARIBLrll 3 HFAIOKAJ'IbHHMH ... 89
3,.- e.= ( 5 8 )
OrlepaTop L = W D W- l 3a/IOBOJ.,H~tr piBHm-ma
~t L= [Bn, L ] r [3,. - B , , L ] = O, (59)
mo ~an~msar 3 KOMyTaTHBH0CTi onepaTopin g) i 2, . - ~Z~' Ta qbopMyan (16) (~nn.
cIIiBBiBJ-IOIIIeHHSI (52)), TaK RK BHacJIiLIOK piBHOCTi (58) MaeMo
- = - e, w- ,w gw =
= W [ b t , ' - B,~L] W -1 = O.
TeopeMy ~o~e~eao.
Hac.dOoK. Hpu m = N onepamop W D W -I, no6yOo~anufl ~a cucme~to~o
pose' aztr pienanb (50), sa~o,~om, nae k-pe~yx~ito Fen~pataga-,~iKozo euzaaBy
(I0), (I I).
IAefl pe3y.qbTaT BrlrlslrtBar 3 piBHOCTi (51), OCKialbZrl f ( t ) = f(k) ~ Ker W [ f ]
npa m = N.
2. ~SOBHMipHi yaaradlbnenH~l k-c KP-iepapxiL Poar~a~eMo ~J~a cl~iKCOBa-
HnX k, n e N onepaTopae piBH~IHH$1
[ak~x, - (B t + q ~7-1r), ~,,~t. - A,J = 0,
k-2 n-2
Bt = g)k + Z uJ Dj" A,,= D n + ~, vi Di,
j=0 i=0
uj = uj(x, *k, tn), Vi = vi(x, Xk, t.), ak, ~ . e C.
PiBH~IIIHH (60) MOZKHa 3anncaTa y B n r ~ t i CHCTeMH TaKHM qHHOM:
~,,~t.Bk = ak~x,A,, + JAn, BIr ] + ([A n, q D - l r ] ) + ,
~ , ,~ t , q = A n ( q ) .
n3t.r = - A n (r) ,
( 6 0 )
(61)
(62)
(63)
de CI~MBO.rlOM P+ no3HaqeHo, aK i paHime, ~ttcloepeHuia.rlbHy qaCTHHy MiKpo~HCl0e-
pcHRia.nbHOrO oncpaTopa P, a 3a ~onoMorom A,~ no3HaqeHo onepaTop, mo c
QbopMaJlbnO cHpfl~KeHHM ]~O An i Mae TaKHfl aMrnaa:
n-2
A,~ = (-1)riD" + Z (-1)iDivi �9 (64)
i=0
OnepaTopHe piBHmma (61) pisHocH.m, He CHCTeMi n + k - 2 ~MqbepeHRia.nbHHX
piBHSIHb/Ia-l~l ( n + k - 2 ) + 2l IIOJIbOBHX 3MiHHItX Uj, 1~, qs, rs, l~e j = 0, k - 2 ; i =
= 0 , n - 2 ; s = l,--l, a KOaCae 3 BeKTOpnHX pisn~nb (62), (63) exnieasteHTne
8i~anoni~tHita CHCTeMi ~aHqbepeHuia~bHHX pinHJmb, aKa n noKoMno.ewrnila qbopMi
3armcy MaC rarlr~#l~
I SSN 0041-6053. YKp. Ham. ~.'vpu., 1999, m. 51, N e I
90 A.M. CAMO~tJIEHKO, B. F. CAMOflJIEHKO~ IO. M. Cld~IOPEHKO
~J,,Ot. qs =An(qs ), s = 1,"l, (62 ' )
[$,,b, r s = - A,](rs), s = 1,--1. (63 ' )
TaKn~ qnnoM, cncTer~a (61)--(63) ~na nesi~oMnx ~bynKttilt uj, v i, qs, rs, ~e j =
= 0 , k - 2 ; i = 0, n - 2 ; s = 1, l , r aaMKHeHOIO, ocKi~KH KiSlbrica'b ii piBHaeb cnia-
na~ae a KDII, KiCTIO HeBiaOMHX no~iB, i Jmn~e co6om 3BHqat~Hy esoanouiltny ( ~ a a M i -
qHy) cHc'reMy 3 qaC'rHHHHMH IIOXi~HHMH, ~e g ~ t n w eBOJIIO11i~Hn~ napaMeTp, a
R 2 ~ (x, x k) w npocmopooi 3MiHHi.
OaHaqeHnm llocMOoonicmb onepamopnux pion~nb (60), 0e n = 2, 3 . . . . . npu
dpirco6ano~ty k e N na3u~amu~w~to npocmopoeo Oooou~dpnu~ y3aea,~bnenn.~t k -
c KP-iepapxii" a6o ctcopo~eno 2 d k -c KP-iepap~eu~ (noxiOne oiO mep~dny an~-
Ai~cbroto ~woOtO 2-dimensional k-constrained KP-hierarchy).
Ha IIi~ITBep~7~eHH~ 3MiCTOBHOCTi Cq~opMyJ~b0Ban0r00aHaqeHHg, /Iablo ~eaKi
npnr..na~a CHCTeM HcJIiHifiHHX/IHq~epeHRiadIbHaX piBH~tltb, I.IJ, O ltaYieT~aTb ,110 03Haqe-
Hoi ~ m e iepapxii pi~HaH~.
1. Hexait k = I , n = 2. Tozfi onepaTopHe piBH~IHH~I (60) Ha6yBar Bnr~a;Xy
[tXl0xt -- D + q D - l r , ~ 2 O t 2 - D 2 - v0] = 0, (64)
npHqoMy OCTaHHe piBH~IHH$I MO)KHa aanncaTa elgBiBa~eHTHHM HHHOM y aHrY[~/~i
CHc'rcMI, I piBHaltb
~2Otzq = qxx + Voq,
~2~t2 r = - r x x - vor, (65)
0~1 ~x~Vo = VOx - 2(qr)x.
PoaFJI~HeMO HOai 3MiHHi
z~ : = x + a ? ~x~, z2 : = x - a ~ ~x~ (66)
Ta BiAHoBiRHi ~i ortepaTopa Raqbcpe~ i loaamta CTOCOBHO U~4X 3MiHHHX, m o 3 a n a c y ~ -
TbCH T a K H M HHHOM:
= + =
Hpt4u~oMy, oqesrlaHO, O x = 3z, + 0z2, a l O ~ = Oz~ - 3z2.
Hexalt ~2 = i~ e iR . HaK~a/ieMo a 'g3i r = g~. r , zte R ~ kt ~ / l e J~Ka c r a ~ a
as 'zaKy. To/Ii CrICTeMy /~nqbepeHuiam, mix pi~HaHb (65) MO~KHa 3anrlcarrl TaKHM
HHHOM:
i~3t2 q = (o~z, + Oz2)2q + glql2q + Sq;
(67)
~e
! !
S=vo-r t lq l z, iql = ,lqil 2.
iffil i - - I
ISSN 0041-6053. Yxp. ~tam. ~'vpH., 1999 , m. 5 I , N e I
I@PAPXI.q PIBH,qHS KA~OMI.IEBA-IIETBIAI//BIJ'II 3 HF.JIOKAJIbHHMH ... 91
CHCTeMy (67) nprl l = 1 iHo/Ii HaaHBalOTb "rpe'r~o~o MoRe.n.mo ~es i - C'noap/IcoHa
(DS-II I ) [ 4 3 - 45]. CTOCOBHO Mo~e.neA D S - I i D S - I I Rrmric~, HanpH~.aa~t, Taxo~x
[45 -- 53].
3 (67) npH a ~ = 0, l = 1 MO:~,Ha o ' rpn~a 'm He.aiHillHe pinmaHHa MpeRiHrepa, y
3B'R3Ky 3 qHM crlcTeMa piBttRHb (65) Ha3HBa~rbcA npocmopoao Oaoau~tipnu~t l-
t~o~monenmnu~ ),saea~,nenn~t~t pi~ttnnn,~ HlpeOinzepa.
2. Hexatt k = 1, n = 3. T o ~ onepaTopne pimt.anaa (60) na6yr~ae nHr.natty
[ a I Oct - 9 - q ~ r ' l r , ~!3 0t3 - / )1 9 - D0] -- 0, (68)
NpH KbOMy OCTaHHr piBH~IHH~I Mo~a-la 3artHcaTH eKBiBa.rleHTHHM HHHOM y BHF.rDIRi
CHCTCbIH piBH~lHb
~30t3 q = qxxx + Vl qx + voq,
~30t~r = rxx x + ( v l r ) x - vor,
(69)
a l O~Vo = POx- 3(qxr)x,
a l 0 x t v l = V l x - 3(qr)x.
Hexatt [33 : = [3 ~ R . HaK~a/IeMo n 'a3i r = Ix~ T, Re R ~ It - - ~aea~a KOH-
cTarrra 3B'Jt3zy. ToRi CHCTeMa (69) B 3MiHHHX (66) 3aIIHCy~TbC$I TaKHM qHHOM:
3
~Ot3q = qxxx + "-~lql2qx + (q~ r )q + .~Sq x
O~2S = tt0zl I~l 2,
oz2P = ~tOz ~(q~r) ,
3
+ ~ P q ,
(70)
~ e
2 2
S = ~v l - lxlql 2, P = ~ v 0 - I x ( q x q r ) ,
i Rna cKopoqeHH~I 3anncy BHKOpHCTa.HO.rloaHaqeltttA 3 (66) BHr . IL~
Oi q = (Oz, + Oz2)iq Ox'
CHereMH ~taqbepeHuia~bnax piBnaab (69), (70) e sHmH~n cnMexpiaMH cncreM
(65) Ta (67) Bi~anosiRHo. CaereMa (70) r HOSOm (neKTOpHOS3) sepcieao npoc ' roposoro
fiBOBHMipaoro MoR~qbizoBaaoro pisHaaHa K o v r e s e r a - ~ e Opiaa, mo n CKa~apHoMy
Bnnazmy (npH l = 1 ) si~MiH~a s i r aanponoHoBawax pauitue [54 - 56].
2. Hexa~ k = 2, n = 2. Toni onepaTopKe piBHarma (60) Ha6ynar BHrna~Xy
[~20~2 - D 2 - u 0 - q D - ' l r , ~ 2 O t 2 - D 2 - v 0 ] = 0,
nprtqoMy oc'raHHe piB~aHH.,q MO:~,,.Ha aamlcaTH eKSiBa.neHTHHM qHHOM y nm-~i CHC-
TCmt piaHaHr~
~20t2 u ---- ~2U,c2 + 2 ( q r ) x,
~20t2 r = qxx + ( U + C ) q , (71)
ISSN 0041-6053. Yxp. ~aam, ~.'vpn., 1999, m. 51, N ~ 1
92 A.M. CAMOI;IXIEHKO, B. F. CAMOI;b'IEHKO, IO. M. CH~2OPEHKO
~2 bt2r = -rxx - (u + c)r ,
/~e U : = UO, V 0 = U+C, C~ C.
Hexala a 2 : = i a e iR, ~2 :-- i ~ e iR. HaKSla/IeMOB'~I3i r = i[.t~ T, ~e R 9
~t ~ / x e z ~ a cTana aa'zaKy, i nexai~ c ~ R , u = ft. To~ai CnCTeMa (71) pe~y~yeT~-
Ca/~O CHCTeMM piBH$IHb
I~,~u = aux~ + 21xlql 2,
(72)
i~3t2q = qxx + ( u + c ) q .
Cncre~ta (72) BnKopnCTOByeTbCa n qbi3ntti naaataa ~aa onncy B3aetaojaii naBKO-
aoaayKoanx neartalOpOBCbKtlX XBnJIb i e l-KoMnOHeHTrlnta npocTOpOBO ttBOBHMipHHM
y3al'a.rlbaeHHJtta piBI-Dtnb H/~3HMn--O~KaBn [57, 58]. CKa.rmpHy Bepci~ CHCTeta piB-
Ham, (71), (72) (npn l = 1 ) 6yno orpntaano B pO6oTax [26, 27], a ~emo ni3Hime ix
6yno nepeBirtKpnTo B [59]. 3aana,antao, mo s [60] rrpoBe~eHo anasria IlermeBe cncre-
tan (71) rtsm srma~Ky l = 1. 3 a y ~ a , ~ t a o Ta~O~, mo qac-moBn~ Bnna~o~ cr~cTetarx
piBHam, (65), tUO BianoBi~aar anaqenmo l = 2, o r p r ~ a n o s [61], z~e Ta~oa~ 3anHcaHo
ii sxaxco~y napy onepaTopiB, ~i~tr, fiHny Bi~a aanponoHoBaHOi s /~axai~ cTarvi. I2e~
Bma~OK TaKo~ Tieno noB'z3ann~ 3 piBnannata .rI. II. H a ~ m m a [62, 63] (~nB, Ta-
~ 0 ~ [61]).
Imui tliKaBi npnKna~aa CnCTeM neJIiHithtHX piBHJ~HI,, mo HaJIeY~aTb /10 2 d k -c
KP-iepapxii, taOaCHa OTpntaaTn npn k = 2, n = 3 ; k = 3, n = 2 i k = 3, n = 3, SKi
aa 6paxoM Micu~I B/lanita CTaTTi He BrlnncyeMo JtBH0.
2.1. Toqni pO3B'$1aKH p ianan~ , m o na~e~caTb 2d k ~ KP- iepapxff . Hexa~
f = f ( x , "C k, t 2, t 3 . . . . ) := ( f l , f m )
r cni.ahrInM (cyMiCHnM) pOaB'aaKOM nocni~aoaHocTi aiHiRaHX gnqbepermia.nbHHX
piBHaHb (]II4B, TaKO~K (50)) BHrJI~IIy
(Xk~t fm = f(k) + f Mm,
(73)
~n~t.f = f n + fM(n),
/Ie n = 2, 3 . . . . ; k --/Ie~tKe qbiKcoaarle rlaTypanbHe qncno, a W : = W [ f ] - - / l n -
dpepenuia~bHnfl onepaTop, mo BH3naqeHa~ 3rittno 3 qbop~tyaom (36).
Mae taictle Taxa Teopetaa.
TeopeMa 5. Onepamopu
L k : = W ( a k O x ~ - D k) W -t , (74)
L. := w ( I L , ~ , . - 9 " ) w -~ (75)
3aOoaonbttalomb pisnnnnn (60).
]/oeeSen.~. OCKiabKH ~zaqbcpcmfiam,ld oncpaTop~ (74), (75) i~oMyrymTb,
TO6TO
[Lk, Ln ] _ W [ ~ k ~ . c k _ ~ k , [~n~t, - ~ , ] W-I = 0,
TO ~aoeraam~ noxa3aa'H, mo onepa'ropH L k i L n MaKrrb Ta.KHI~ BHFJIJI~:
Lt = c t k ~ t - (Bk + q D - t r ) , (76)
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I~PAPXIH PIBH$IHB KA~OMIIEBA-rlETBIAIIIBIIII 3 HEJ'IOKAJISHHMH ... 93
Ln = ~,,~t. - An" (77)
~OBe/~eMo cno,~aTKy pimdcaa, (76). BHKOpHCTOByIOqH qbop~yny (74), MOgKHa 3a-
IIHCaTH cIIiBBi~HoUIeHH~
L~ = w [ a ~ - ~ ' ] w -~ =
= a ~ x ~ + a k W ( W - l ) ~ -- W D k W - l . (78)
HecK.rla/IHO rIOKaaaTH, triO orIepaTop W ( W- 1 ) xt ~ qrlCTO iHTerpaamaHIt oneparop.
~e Bl4n.rlI4Bae 3 nopiBHaHHJt n0pJt]~KiB Bil/JIOBi]~HHX onepaTopiB, a came 3 TaKHX CHiB-
Bi~HOILIeHb;
Oral W -- N, TOMy mo W = D N + ... BHacninOK qbop~y.rm (36) ;
O r d W -~ = - N , TOMymO W - l
O r d ( W - l ) x ~ < - N - 1 .
= D-N+...
3a TeopeMoIO 3 i qbopMyno~o (45) ;
ToMy MO:~KHa 3arn, lcaTrt piBHiCTr~ L k = ( L k )+ + ( L k )_, /Ie 3 ypaxyBaHH~IM C13opMy.rl
(41) i (73) BHKOHymaa, CJ~ pi~aHOCTi
(Lk)+ = a k ~ - ( W D k W - l ) + := a k ~ - 8k,
(Lk)_ = a k W ( W-~)~, - ( W D k W - ~ ) - =
= - a k W f ~ k ~ 7 " l h - a k W f D - l h x k + W f i O k f D - l h =
= - O ~ k W ( f x ~ ) D - l h - a k W ( f ) D - l h x t + W ( f (k)) D - l h =
= - W ( a k O ~ k f - f ( k ) ) D - I h =
= - W ( a k ~'k fm - f (mt) )D-lhm : W ( ~ t ~xk ~ - "f /k))D-I ht =
= - W(akO~ k f t - f t ( k ) )D- lh l := q ~D-lr, (79)
a6o B nO-KOMIIOHeHTHOMy BHF.rIJt/~i 0:[HB. (53), (54) ) :
qi = ++'q41(fl ..... f N , ( a k ~ x t f i _ f/(k))) q4]-I ( f , . . . . . JR), (80)
r i = +_ (- 1 )N-if f l2Ni[f]q42-1 ( f l ..... fN)" (81)
3a ~onoMoro~o 6canocepc/~ix O6qHCJIeHb HeBa3KKO IlepeKOHaTHCb, II~O /~.rDI /~0-
ninbHHX auqbepem4ia~bnUX oncpaTopin A l , fil 2 , A 3 c n p a s e ~ m i TaKi TOT0~KHOCTi:
A l f ~ D - l h = ( A l f ~ D - ~ h ) + + A 1 ( f ) D - l h , (82)
fD-l~q~h = ( f ~ ' l ~ h ) + +fD"l.~z(h), (83)
f ~ ' l h . , ~ 3 ffi ( f D - I A ~ h ) + + f D - I . , ~ f h ) . (84)
ISSN 0041.6053, Ygp. ~lam. ~Tpa., 1999, m. 51, bl ~ 1
94 A.M. CAMO~J'IEHKO, B. I'. CAMOPd]EHKO, IO. M. CHj3OPEHKO
Ilpn rlepeTBopeHH~X B qbopMyaax (79) BHKOpliCTaH0 dpopMyay (82). PianicTb
(84) nHKOpnCTaHa HeaBHO npu nepexo6i Bi~ onepaT0pHHx pisnJ~H~ k -c KP- i 2d
k-c KP-icpapxilt ~o si6n0ni6aHx CHCTeM He~iHiflHHX 6HqbepeHllia~baHx piBH~qHl,
6Za Zoe~bit~ieHTiS MiKpo/mdpepeHuia~mHOrO onepaTopa B k + .q D- I r .
PiBHiCTb (77) ~tOn06wrbca aHzaoriaHo Srmm~eHOMy BmI~e B nyHKTi 1 ~OBe6eHH~
TeOpeMH 4.
TaKrrM tmaoM, TeOpeMy 5 ~aoBe6eHo.
3aKalotmi 3ayBameHHa, O/mUM i3 ocaoBnnx pe3y~bTaTiB ttiei pO6OTH r y3a-
ra.m, HeHHa HeJIOKaJIbH0 pe/~yKOBaHOi iepapxii piBnmtb KaaoMIiena- 1-leTBiamBi~i ~ I a
npocTOp0B0 ~aBOBHMipH0r0 BHna~Ky. HOBa iepapxia (2 + 1 )-BHMipHHX He~iHillHaX
iHTerponnHx CHCTeM, aKy HaaBa~la 2d k -c KP-iepapxie~, MiCamTb B c06i ZK yaa-
ra~maerma 6o6pe Bi6oMHx paHime HeziHiilHHX CHCTeM, TaK i cyrTr HOBi He~iHi~Hi
CHCTeMH piBH$lHb. IIpH 606aTKOBHX (JIOKaJIbHHX) pe6yKttiax THNy cpMiTOB0rO
cnpzmeHHZ (/roB., narrpnKaa~, qbopMyaa (67), (70), (72)) ai CnCTeMn MmOTS 60-
CHTb npo30pal~ cl3i314qHHfl 3MiCT, OCKiJ-ISKH MO:~KyTb BI4KOpHCTOByBaTHCF /~JI~l Mo/Ie-
~OBaaHa aeaiHii~HOi BaaeMo~ii 6OBrHX XBHa~ (a npoqbiza~n u j , j = O, k - 2 ) 3
naKeTaMrl K0p0TKrlX XBrI~I, ([qi[, i = 1 ~ ) B naomrIHi R 2 B (x, xt) . MO~KHa
TaXOm no~aaaTa, mo nic~z 6eaxoi Mo~nqbiKauii TeopeT~o-rpynoBa cxeMa 6ByMe-
pHzat~ii He~iaiiaH~X piBnaHb ne~iHiiaHnX pinaaas T~ny KopTenera- /xe r [64]
Mome 6yTH 3aCTOdOBaHa i ZX~a pinHaHb, mo BX0~aZT~ 60 2 d k -c KP-iepapxii.
Taxn~ r MocHa, 3oKpeMa, OTpH~aT~ iX ra~i~sTOnOBy iHTepnpeTattiux I_[e
mrramaa n~aayeT~Ca BHCBiTJH4TH B OKpeMil~ CTaTri.
O/IHa 30CHOBHHX Bi/IKpHTHX npo6JIeM, MO TiCrlO nOB'~13aHa 3 6aHo~ CTaTTe~, no-
~arar B nacTynaoMy. Teope~H 4, 5 BCTaHOBJIIOIOTb B gBHOMy BHr~a/li 3B'~I30K Mi~
pO3B'.CI3KaMH CHCTeM JIiHii~tHaX/IHCl0epCHIiia~IbHtlX pil~li~IHb Ta pOaB'~I3KaMH HeYliHiI~-
HaX pisnm4~, mo BXO~taaa, ~o k -c Ta 2d k -c KP-icpapxii~ Bi6noBi6HO. Ane Bi/mo-
Bi6~ Ha nwraHaa npo Te, an nora Jingo IIpOBe~IeHH}I B III4X KYlacax pO3B'~3KiB ~O~aT-
KOBHX KOMIIJICKCHHX pe~xyxuifl BHFBg6y r = l~qr, r 6a~eKO He OqeBH~HOIO HaBiTb B
HafinpocTimOMy mma6~y npH l = I . Ha ~aaunll qac t~a npo6~e~a He pO3B'~13aHa y 3a-
raYlbHOMy srlIIa~Ky HaBiTb 6JIJt C KP-iepapxii, HO 3BamalOqH Ha qHCYleHHi 60czi-
//)KCHH~I B ~ M y HanpJIMKy B OCTaHHi 6ecffrb pOKiB (6a~eKo HenOBHH~I CnHCOK ny-
6JfiKaujA Ha~e6eao HHmqe). Hama rinoTeza nonarar ~ HacTynHozy: 3aMicTb o0a-
eatottoeo onepaTopa W BHr~La6y (36) noTpi6HO BHK0pHCTOByBaTH onepaTop Bo~b-
Teppa 3 BHpO6~KeHHM Jt/ip0M 3ara:lbHOrO n0JlO~KettH~, ~K lie 6yJ~o pea2fisOBaHO rlprl
no6y6oBi 6iHapnoi x-qbyHKttii B [42]. B aaHatt ~ac tt~ i/~ea aKTHBBO a~TOpaMU onpa-
u~onyerbc~.
HapemTi, ocrarme 3ayBameHH~. JTtK 3a3Haqa~Iocb BHMC, 6eJ~Ki CHCTeMH pisHmt~,,
mo Bxoa~rn, 6o c KP- Ta 2d k ~ KP-iepapxii, cepe6 aKHX, HaxlpnK~a~, MiCTJ~T~CZ
CHCTeMH, mo CTaHOB~IATI, IICBaI4fl irrrepec A~a ix 3aCTOCyBarIb B pi3HHX pos6DIax dpi-
zrma, ZmOT~ O/mo~acHO 6CKia~za KOMyTaTOpHHx 3o6pameHb .rlaKca [26, 27, 59, 61],
cepe6 JIKrm r TaXi, ~t:L~ ~mHX, ~ 3a3Haqa~oc~.me B [5, 26, 27], HeMO~nHBO aanHcaT~
piBnJm~ r e ~ a r t a a - MapqeHXa-- JIeaiTa~a. Cno]~iBacMocb, mo npo]~e~oncTp0Ba-
Ha B ~aHii~l CTaTri MO:)KJIHBiCTb 3aCTOCyBaHH~I MeTOAy 06~IFaHH~I 3axapoBa-ll la6aTa
[5] ~ piBHJIHb 2d k -c KP-iepapxi~ y dpop~i (60) CTr~y~Baam~r irrrepec cnctti-
azlicTin s ranyzi HedliHiflHHX 6HHaMiqHHX CHCTeM 60 n0eTaH0SKa Ta ~oc.rli~IX~eHHJt
06r :~a6a~I ~ YliHi~HHX HeCTatIi0HapHHX iHTerpo-6H~bCpeHRiaYlbHHX piBH$1HI~
nHrJL~y
o, = + o +
ae /~ ~ irrrerpaymHHR onepaTop BoJl~T~ppa 3 BHp06X~eHHM P~pOM BHrJ'm6y
ISSN 19041-6053. Ysp. ~tam. ~.'vpu., 1999, m. 51, IW 1
ICPAPXI~I PIBHJ:IHS KA~OMI.IEBA-HETBIAI//BIJ 'II 3 HE2IOKAJ'ISHHMH ... 95
K ( x , y ; t) =
m e ~tie 3ri~HO a qbopMyao~o
( t?O) (x , t) =
1.
2.
3.
4.
5.
6.
7.
8.
9.
l
qi (X; t) r / ( y ; t ) ,
i=1
x
f
+**
K (x, y ; t) ~ ( y , t) dy .
Date E., Jimbo M., Kashiwara M., Miwa T. Nonlinear integrable systems: classical theory and
quantum theory / Ed. M. Jimbo and T. Miwa. - Singapore: World Scien., 1983. - P. 39-119 .
Ohta Y., Satsuma J., Takahashi D., Tokihiro 7". An elementary introduction to Sate theory //
Progress Theoret. Phys. Supp l . - 1988. - 9 4 . - P. 210-241.
Came M., ]lsu~tSo M., Muoa M. FO.nOIIOMH~e xlmlrron~e noJia. - M.: Mxp, 1983. - 304 c.
Dickey L. A. Soliton equations and Hamiltonian systems/ /Adv. Ser. in Math. Phys. - 1991. - 12. -
310p.
3axapo6 B. E., llJaSam A. 1~. CxeMa H14TerpHponalma im~mwe~ln~xypan14enH~l MaTeMaT14qecKO~t
~H3HK14 MeTO/]OM o6pa'rriolt 3a/~aqa pacceaHrla / /~ynguaoH, aHallH3 14 ero npH.a. - 1974. - 8,
Ng 3. - C. 43-53.
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| id | umjimathkievua-article-4585 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:01:42Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b0/85f64bd81ea64beb356888939ddd2bb0.pdf |
| spelling | umjimathkievua-article-45852020-03-18T21:09:14Z Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system Ієрархія рівнянь кадомцева-петвіашвілі з нелокальними в'язями: багатовимірні узагальнення та точні розв'язки редукованих систем Samoilenko, A. M. Samoilenko, V. G. Sidorenko, Yu. M. Самойленко, А. М. Самойленко, В. Г. Сидоренко, Ю. М. We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensionalk-constrained KP-hierarchy, briefly called the 2d k-c-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2d k-c KP-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for the construction of exact solutions of equations belonging to the 2d k-c KP-hierarchy is proposed. Дано просторово-двовимірне узагальнення ієрархії рівнянь Кадомцева-Петвіашвілі з нелокаль-ними в'язями — так звана 2-dimensional $k$-constrained $KP$-hierarchy (скорочено: $ 2d k-cKP$-ієрархія). Наведено приклади $(2+l)$-вимірних нелінійних моделей, що є представниками $2dk−cKP $-ієрархії; серед яких вказано, зокрема, як узагальнення раніше відомих, так і нові нелінійні системи. Запропоновано метод побудови точних розв'язків для рівнянь з $2dk−cKP$-ієрархії. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4585 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 78–97 Український математичний журнал; Том 51 № 1 (1999); 78–97 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4585/5874 https://umj.imath.kiev.ua/index.php/umj/article/view/4585/5875 Copyright (c) 1999 Samoilenko A. M.; Samoilenko V. G.; Sidorenko Yu. M. |
| spellingShingle | Samoilenko, A. M. Samoilenko, V. G. Sidorenko, Yu. M. Самойленко, А. М. Самойленко, В. Г. Сидоренко, Ю. М. Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system |
| title | Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system |
| title_alt | Ієрархія рівнянь кадомцева-петвіашвілі з нелокальними в'язями: багатовимірні узагальнення та точні розв'язки редукованих систем |
| title_full | Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system |
| title_fullStr | Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system |
| title_full_unstemmed | Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system |
| title_short | Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system |
| title_sort | hierarchy of the kadomtsev-petviashvili equations under nonlocal constraints: many-dimensional generalizations and exact solutions of reduced system |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4585 |
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