On group classification of differential equations: algebraic approach
Using the classical Lie theorem on realizations of Lie algebras by vector fields on the line, we substantially simplify the proof of the known results on the group classification of the classes of (1+1)-dimensional nonlinear evolution equations ut=H(uxx) and ut+uux=H(uxx).
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Інститут математики НАН України
2025
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| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Репозитарії
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872553043801669632 |
|---|---|
| author | Huraka, Sofiia Lokaziuk, Oleksandra Гурака, Софія Локазюк, Олександра |
| author_facet | Huraka, Sofiia Lokaziuk, Oleksandra Гурака, Софія Локазюк, Олександра |
| author_institution_txt_mv | [
{
"author": "Софія Гурака",
"institution": "Кафедра математичних та статистичних наук, Університет Альберти, Едмонтон, Альберта, T6G 2G1, Канада"
},
{
"author": "Олександра Локазюк",
"institution": "Інститут математики НАН України"
}
] |
| author_sort | Huraka, Sofiia |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2026-02-24T12:22:41Z |
| description | Using the classical Lie theorem on realizations of Lie algebras by vector fields on the line, we substantially simplify the proof of the known results on the group classification of the classes of (1+1)-dimensional nonlinear evolution equations ut=H(uxx) and ut+uux=H(uxx). |
| doi_str_mv | 10.3842/trim.v21n1.539 |
| first_indexed | 2026-08-04T01:09:03Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Ií-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè (2024) ò. 21, �1, 35�58
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ
ðiâíÿíü: àëãåáðà¨÷íèé ïiäõiä
Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Abstract. Using the classical Lie theorem on realizations of Lie algebras
by vector �elds on the line, we substantially simplify the proof of the known
results on the group classi�cation of the classes of (1+1)-dimensional non-
linear evolution equations ut “ Hpuxxq and ut ` uux “ Hpuxxq.
Àíîòàöiÿ. Âèêîðèñòîâóþ÷è êëàñè÷íó òåîðåìó Ëi ïðî ðåàëiçàöiþ àë-
ãåáð Ëi âåêòîðíèìè ïîëÿìè íà ïðÿìié, ñóòò¹âî ñïðîùåíî äîâåäåííÿ
âiäîìèõ ðåçóëüòàòiâ ïðî êëàñèôiêàöiþ êëàñiâ (1+1)-âèìiðíèõ íåëiíié-
íèõ åâîëþöiéíèõ ðiâíÿíü âèãëÿäó ut “ Hpuxxq òà ut ` uux “ Hpuxxq.
1. Âñòóï
Äèôåðåíöiàëüíi ðiâíÿííÿ âiäiãðàþòü âàæëèâó ðîëü ó ñó÷àñíié íàó-
öi, àäæå çíàõîäÿòü ñâî¹ çàñòîñóâàííÿ ó áàãàòüîõ ñôåðàõ. Îäèí iç âàæ-
ëèâèõ íàïðÿìêiâ öi¹¨ ãàëóçi ìàòåìàòèêè � ãðóïîâèé àíàëiç äèôåðåí-
öiàëüíèõ ðiâíÿíü. Ôóíäàìåíòàëüíèì ïîíÿòòÿì ãðóïîâîãî àíàëiçó äè-
ôåðåíöiàëüíèõ ðiâíÿíü ¹ ïîíÿòòÿ ñèìåòðié � ïåðåòâîðåíü, ÿêi ïåðåâî-
äÿòü ðîçâ'ÿçêè ðiâíÿííÿ (àáî ñèñòåìè ðiâíÿíü) ó ðîçâ'ÿçêè öüîãî ñàìîãî
ðiâíÿííÿ (àáî ñèñòåìè ðiâíÿíü). Ðîçðiçíÿþòü ðiçíi òèïè ñèìåòðié: ëî-
êàëüíi, íåïåðåðâíi, äèñêðåòíi, êîíòàêòíi, ïîòåíöiàëüíi, óçàãàëüíåíi òî-
ùî. Ñèìåòði¨ øèðîêî âèêîðèñòîâóþòüñÿ, îñêiëüêè âîíè äàþòü ìîæëè-
âiñòü áóäóâàòè òî÷íi ðîçâ'ÿçêè çâè÷àéíèõ äèôåðåíöiàëüíèõ ðiâíÿíü òà
ðiâíÿíü iç ÷àñòèííèìè ïîõiäíèìè, iç äîâiëüíîãî òðèâiàëüíîãî ðîçâ'ÿçêó
îòðèìóâàòè iíøèé, ñêëàäíiøèé ðîçâ'ÿçîê, iíòåãðóâàòè çâè÷àéíi äèôå-
ðåíöiàëüíi ðiâíÿííÿ, ïîíèæóâàòè ïîðÿäîê ðiâíÿííÿ, âèêîíóâàòè ðåäó-
êöiþ, îïèñóâàòè çàêîíè çáåðåæåííÿ, áóäóâàòè iíòåãðàëè ðóõó, áóäóâàòè
ïàðàìåòðè÷íi ñõåìè â ÷èñåëüíèõ ìåòîäàõ òîùî. Ïåðåëi÷åíi âèùå çàäà÷i
çâîäÿòüñÿ äî òàê çâàíî¨ ïðÿìî¨ çàäà÷i ñèìåòðiéíîãî àíàëiçó, ÿêà ïîëÿãà¹
â îïèñi ñèìåòðié äëÿ çàäàíîãî êëàñó äèôåðåíöiàëüíèõ ðiâíÿíü.
This work was supported by grants from the Simons Foundation (1290607, O.V.L.)
2020 Mathematics Subject Classi�cation: 01A72, 16G50, 18G80
Êëþ÷îâi ñëîâà: åâîëþöiéíå ðiâíÿííÿ, ãðóïîâà êëàñèôiêàöiÿ, ïåðåòâîðåííÿ åêâiâà-
ëåíòíîñòi, ñèìåòðiÿ, àëãåáðà Ëi, àëãåáðà¨÷íèé ïiäõiä, ðåàëiçàöi¨ àëãåáð Ëi íà ïðÿìié
DOI : https://doi.org/10.3842/trim.v21n1.539
35
36 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Ñîôóñ Ëi çâiâ çàäà÷ó ïîøóêó ñèìåòðié äî ëiíiéíî¨ çàäà÷i ðîçâ'ÿçàííÿ
ïåðåâèçíà÷åíèõ ñèñòåì äèôåðåíöiàëüíèõ ðiâíÿíü, òàêèì ÷èíîì çíà÷íî
ñïðîñòèâøè ïî÷àòêîâó çàäà÷ó. Îäíàê ÷åðåç áðàê åôåêòèâíèõ iíñòðó-
ìåíòiâ äëÿ ðîçâ'ÿçêó ñêëàäíèõ ïåðåâèçíà÷åíèõ ñèñòåì äèôåðåíöiàëü-
íèõ ðiâíÿíü íàâiòü iíôiíiòåçèìàëüíèé ìåòîä ìîæå iíîäi ïðèâîäèòè äî
äóæå ãðîìiçäêèõ îá÷èñëåíü, ÿêùî ïî÷àòêîâà çàäà÷à ìiñòèòü äîâiëüíi
ïàðàìåòðè àáî æ ¨¨ ðîçìiðíiñòü äóæå âåëèêà. Ùå áiëüøå ñïðîñòèòè
ðîçâ'ÿçîê ìîæóòü ðiçíi àëãåáðà¨÷íi ìåòîäè, ÿêi íà äàíèé ÷àñ ðîçðîáëÿ-
þòüñÿ â òîìó ÷èñëi ïðåäñòàâíèêàìè êè¨âñüêî¨ íàóêîâî¨ øêîëè ñèìåòðié-
íîãî àíàëiçó.
Îñíîâíèì ïðåäìåòîì öi¹¨ ðîáîòè ¹ åâîëþöiéíi ðiâíÿííÿ
ut “ Hpt, x, u, u1, u2, . . . , urq, (1.1)
äå t, x � íåçàëåæíi çìiííi (÷àñîâà òà ïðîñòîðîâà çìiííi, âiäïîâiäíî);
u “ upt, xq � çàëåæíà çìiííà; ut “ Bu
Bt � ÷àñòèííà ïîõiäíà çà çìiííîþ t;
uk “ Bku
Bxk � k-òà ÷àñòèííà ïîõiäíà çà çìiííîþ x, k “ 1, . . . , r, r P N,
r ě 2, äå r � ïîðÿäîê ðiâíÿííÿ; H � äîâiëüíà ãëàäêà ôóíêöiÿ çìiííèõ
t, x, u, u1, u2, . . . , ur, Hur ‰ 0.
Ñèìåòði¨ åâîëþöiéíèõ ðiâíÿíü òà ¨õ âëàñòèâîñòi � ïîïóëÿðíi òåìè áà-
ãàòüîõ íàóêîâèõ äîñëiäæåíü òà ïðàöü. Îêðiì òîãî, äóæå ÷àñòî ðiâíÿííÿ
öüîãî êëàñó ñëóãóþòü áàçîâèìè ïðèêëàäàìè â ñèìåòðiéíîìó àíàëiçi äè-
ôåðåíöiàëüíèõ ðiâíÿíü (äèâ., íàïðèêëàä, [3, 8�10, 17]). Çàçíà÷èìî, ùî,
çãiäíî ç ðåçóëüòàòàìè Ñîêîëîâà [20] òà Ìàãàä¹¹âà [16], êîíòàêòíi ïå-
ðåòâîðåííÿ çáåðiãàþòü âèãëÿä ðiâíÿíü iç êëàñó (1.1) òîäi é ëèøå òîäi,
êîëè âîíè ìàþòü íàñòóïíèé âèãëÿä:
t̃ “ κptq, x̃ “ ϕpt, x, u, uxq, ũ “ ψpt, x, u, uxq,
ïðè ÷îìó íà ôóíêöi¨ ϕ òà ψ íàêëàäåíà óìîâà êîíòàêòíîñòi [21]
ϕuxpuxψu ` ψxq “ ψuxpuxϕu ` ϕxq.
Ó ðîáîòi [16] Ìàãàä¹¹â âñòàíîâèâ, ùî ó âèïàäêó ñêií÷åííîâèìiðíî¨
àëãåáðè êîíòàêòíèõ ñèìåòðié (Cont) p1 ` 1q-âèìiðíèõ åâîëþöiéíèõ ðiâ-
íÿíü iç êëàñó (1.1) ¨¨ ðîçìiðíiñòü ñòàíîâèòü ùîíàéáiëüøå r ` 5. Íà-
òîìiñòü, ó âèïàäêó íåñêií÷åííîâèìiðíî¨ àëãåáðè êîíòàêòíèõ ñèìåòðié
ðiâíÿííÿ çàâæäè ìîæíà çâåñòè äî ëiíiéíîãî çà äîïîìîãîþ äåÿêèõ êîí-
òàêòíèõ ïåðåòâîðåíü. Ó öié ñòàòòi àâòîð òàêîæ íàâiâ ïîâíèé ïåðåëiê
àëãåáð ñêií÷åííîâèìiðíèõ êîíòàêòíèõ ñèìåòðié ðiâíÿíü iç êëàñó (1.1) i
âiäïîâiäíi åâîëþöiéíi ðiâíÿííÿ, ùî ¨õ äîïóñêàþòü.
Äîáðå âiäîìà (äèâ., íàïðèêëàä, [11, Theorem 1] i [22, ëåìà 4.5, ñ. 266])
íàñòóïíà ëåìà:
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 37
Ëåìà 1.1. Äëÿ äîâiëüíîãî åâîëþöiéíîãî ðiâíÿííÿ âèãëÿäó (1.1) t-êîì-
ïîíåíòà áóäü-ÿêîãî iíôiíiòåçèìàëüíîãî îïåðàòîðà, ùî ïîðîäæó¹ îäíî-
ïàðàìåòðè÷íó ãðóïó ëîêàëüíèõ ïåðåòâîðåíü ñèìåòði¨ öüîãî ðiâíÿííÿ,
íå çàëåæèòü âiä x òà u.
Òàêèì ÷èíîì, âåêòîðíi ïîëÿ, ùî íàëåæàòü ìàêñèìàëüíié àëãåáði ëi-
¨âñüêî¨ iíâàðiàíòíîñòi gH åâîëþöiéíèõ ðiâíÿíü ç êëàñó (1.1), ìîæíà øó-
êàòè ó âèãëÿäi:
Q “ ξ0ptqBt ` ξ1pt, x, uqBx ` ηpt, x, uqBu, (1.2)
äå ξ0ptq, ξ1pt, x, uq i ηpt, x, uq ïðîáiãàþòü ìíîæèíó ãëàäêèõ ôóíêöié ñâî¨õ
àðãóìåíòiâ.
Âiäîìîþ ¹ êëàñè÷íà òåîðåìà Ëi ïðî ëi¨âñüêi àëãåáðè âåêòîðíèõ ïî-
ëiâ íà äiéñíié ïðÿìié [15, Satz 6, S. 455] (òàêîæ [18, Theorem 2.70], [5,
Theorem 1] i [24, òåîðåìà 1.1, ñ. 26]).
Òåîðåìà 1.1. Íååêâiâàëåíòíi ðåàëiçàöi¨ ñêií÷åííîâèìiðíèõ ëi¨âñüêèõ
àëãåáð âåêòîðíèìè ïîëÿìè íà t-ïðÿìié âè÷åðïóþòü íàñòóïíi àëãåáðè:
t0u, xBty, xBt, tBty, xBt, tBt, t2Bty.
Ïîçíà÷èìî ÷åðåç π ïðî¹êöiþ ç Rt ˆ Rx íà Rt i íåõàé k :“ dimπ˚gH .
Îñêiëüêè t-êîìïîíåíòà âåêòîðíèõ ïîëiâ âèãëÿäó (1.2) çàëåæèòü ëèøå
âiä çìiííî¨ t, òî, ç îãëÿäó íà òåîðåìó 1.1, k ď 3.
Ðàíiøå òåîðåìà Ëi âæå óñïiøíî çàñòîñîâóâàëàñÿ äëÿ ãðóïîâî¨ êëàñè-
ôiêàöi¨ êëàñiâ åâîëþöiéíèõ ðiâíÿíü òà ðiâíÿííÿØðåäiíãåðà [2,13,14,19],
à òàêîæ êëàñó ëiíiéíèõ çâè÷àéíèõ äèôåðåíöiàëüíèõ ðiâíÿíü äîâiëüíî-
ãî ôiêñîâàíîãî ïîðÿäêó r ě 2 [7, �3], äå iñíó¹ àíàëîãi÷íà ïðî¹êòîâ-
íiñòü íà îäíîâèìiðíèé ïðîñòið ÷àñîâî¨ àáî íåçàëåæíî¨ çìiííî¨, âiäïî-
âiäíî. Ó ðîáîòi [5] òåîðåìó 1.1 âèêîðèñòàíî ó çàäà÷i ãðóïîâî¨ êëàñèôi-
êàöi¨ ðiâíÿííÿ Êëåéíà�Ãîðäîíà, ïðè÷îìó îäíî÷àñíî ðîçãëÿäàëèñÿ ïðî-
¹êöi¨ íà îáèäâi íåçàëåæíi çìiííi t i x (äèâ. òàêîæ äèñåðòàöiþ [24, ðîç-
äië 1]). Íàòîìiñòü, ó ðîáîòi [6] ïðî¹êòîâíiñòü âåêòîðíèõ ïîëiâ íà íåçàëå-
æíó çìiííó t ä๠ìîæëèâiñòü åôåêòèâíî âèêîðèñòîâóâàòè òåîðåìó 1.1
äëÿ ãðóïîâî¨ êëàñèôiêàöi¨ ëiíiéíèõ ñèñòåì çâè÷àéíèõ äèôåðåíöiàëü-
íèõ ðiâíÿíü äðóãîãî ïîðÿäêó ç äîâiëüíîþ êiëüêiñòþ çàëåæíèõ çìií-
íèõ.
2. Ãðóïîâà êëàñèôiêàöiÿ íåëiíiéíîãî ðiâíÿííÿ
òåïëîïðîâiäíîñòi
2.1. Ïîïåðåäíi âiäîìîñòi. Ìåòîþ öüîãî ïàðàãðàôó ¹ óòî÷íåííÿ òà
ñïðîùåííÿ äîâåäåííÿ ðåçóëüòàòiâ Àõàòîâà, Ãàçiçîâà òà Iáðàãiìîâà [1, �4]
38 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
ïðî ãðóïîâó êëàñèôiêàöiþ êëàñó åâîëþöiéíèõ ðiâíÿíü âèãëÿäó
ut “ Hpuxxq, (2.1)
äå ut “ Bu
Bt , uxx “ B2u
Bx2 , H � äîâiëüíà ãëàäêà ôóíêöiÿ àðãóìåíòó uxx.
ßêùî H ëiíiéíà, òî ðiâíÿííÿ (2.1) âiäîìå ÿê ëiíiéíå ðiâíÿííÿ òåïëî-
ïðîâiäíîñòi, i òîìó âiäïîâiäíèé êëàñ (2.1) iíîäi íàçèâàþòü êëàñîì íåëi-
íiéíèõ ðiâíÿíü òåïëîïðîâiäíîñòi.
Äàëi âèêîðèñòîâó¹ìî ïîçíà÷åííÿ: u0 :“ ut, u1 :“ ux, u11 :“ uxx. Òîäi
ïî÷àòêîâå ðiâíÿííÿ ìîæíà ïåðåïèñàòè ÿê
u0 “ Hpu11q. (2.2)
Íèæ÷å ââàæà¹ìî, ùî iíäåêñè i, j, . . . “ 0, 1, à çà ïîâòîðþâàíèìè iíäå-
êñàìè çäiéñíþ¹òüñÿ ïiäñóìîâóâàííÿ, íèæíi iíäåêñè � äèôåðåíöiþâàííÿ
çà âiäïîâiäíèìè çìiííèìè. Âèêëþ÷àþ÷è ç ðîçãëÿäó âèïàäîê ëiíiéíî-
ãî ðiâíÿííÿ òåïëîïðîâiäíîñòi u0 “ u11, ïðèïóñòèìî, ùî H � íåëiíiéíà
ôóíêöiÿ çìiííî¨ u11. (Ñëiä çâåðíóòè óâàãó íà ðîáîòó Êîâàëÿ òà Ïîïîâè-
÷à [12], ó ÿêié âèïðàâëåíî äåÿêi íåäîëiêè â ãðóïîâîìó àíàëiçi ëiíiéíîãî
ðiâíÿííÿ òåïëîïðîâiäíîñòi.)
Íèæ÷å âèêîðèñòàíî íàñòóïíi ïîçíà÷åííÿ äëÿ âåêòîðíèõ ïîëiâ àëãåáð
iíâàðiàíòíîñòi ðiâíÿíü iç êëàñó (2.2):
Q1 “ Bt, Q2 “ Bx, Q3 “ Bu, Q4 “ 2tBt ` xBx ` 2uBu,
Q5 “ xBu, Q6a “ 2tBt ´ x2Bu, Q6b “ xBx ´ 2tBu,
Q6c “ p1 ´ kqtBt ` uBu, k “ 0,˘1
3 , 1, Q6c1 “ 2tBt ` 3uBu,
Q6c2 “ 4tBt ` 3uBu, Q7c1 “ uBx, Q7c2 “ x2Bx ` xuBu,
äå Q6c1 òà Q6c2 � ÷àñòèííi âèïàäêè âåêòîðíîãî ïîëÿ Q6c äëÿ k “ 1
3 òà
k “ ´1
3 , âiäïîâiäíî.
Òåîðåìà 2.1 (ãðóïà åêâiâàëåíòíîñòi, äèâ. [1, �3.3]). Ãðóïà åêâiâàëåí-
òíîñòi G„ êëàñó (2.2) ïîðîäæåíà ïåðåòâîðåííÿìè âèãëÿäó
t̃ “ µ0t` µ1, x̃ “ ν0x` ν1,
ũ “ κ0u` κ1x2 ` κ2x` κ3t` κ4, H̃ “ κ0
µ0
H ` κ3,
äå µ0, µ1, ν0, ν1, κ0, . . . , κ4 � äîâiëüíi êîíñòàíòè, òàêi, ùî µ0ν0κ0 “ 0.
Âåêòîðíå ïîëå (1.2) íàëåæèòü ìàêñèìàëüíié àëãåáði ëi¨âñüêî¨ iíâàði-
àíòíîñòi gH ðiâíÿííÿ ç êëàñó (2.2) äëÿ áóäü-ÿêîãî H òîäi é ëèøå òîäi,
êîëè âîíî çàäîâîëüíÿ¹ íàñòóïíå âèçíà÷àëüíå ðiâíÿííÿ:
pζ0 ´ ζ11H 1qˇ̌
u0“Hpu11q “ 0, (2.3)
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 39
äå ζ0 òà ζ11 � âiäïîâiäíi êîåôiöi¹íòè ïåðøîãî òà äðóãîãî ïðîäîâæåíü
âåêòîðíîãî ïîëÿ (1.2), H 1 :“ BHpu11q{Bu11.
Ðîçïèñóþ÷è ðiâíÿííÿ (2.3), îòðèìó¹ìî
η0 ` ηuH ´Hξ00 ´ u1pξ10 ` ξ1uHq
´H 1“η11 ` 2η1uu1 ` ηuuu
2
1 ` ηuu11 ´ 2u11pξ11 ` ξ1uu1q
´ u1pξ111 ` 2ξ11uu1 ` ξ1uuu
2
1 ` ξ1uu11q‰ “ 0.
Ïiñëÿ ðîçùåïëåííÿ öüîãî ðiâíÿííÿ çà ñòåïåíÿìè çìiííî¨ u1 ïðèõîäèìî
äî íàñòóïíî¨ ñèñòåìè:
ξ1uu “ 0, ηuu ´ 2ξ11u “ 0, (2.4)
ξ10 ` ξ1uH ` p2η1u ´ ξ111 ´ 3ξ1uu11qH 1 “ 0, (2.5)
η0 ` pηu ´ ξ00qH ´ pη11 ` pηu ´ 2ξ11qu11qH 1 “ 0. (2.6)
Çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè (2.4) ì๠âèãëÿä
ξ1 “ αpt, xqu` βpt, xq, η “ α1u
2 ` γpt, xqu` δpt, xq,
äå αpt, xq, βpt, xq, γpt, xq i δpt, xq ïðîáiãàþòü ìíîæèíó ãëàäêèõ ôóíêöié
ñâî¨õ àðãóìåíòiâ.
Ïiäñòàíîâêà öèõ âèðàçiâ ó ðiâíÿííÿ (2.5) i (2.6) ïðèâîäèòü äî íàñòóï-
íî¨ ñèñòåìè:
α0u` β0 ` αH ` p3α11u` p2γ1 ´ β11q ´ 3αu11qH 1 “ 0,
α01u
2 ` γ0u` δ0 ` p2α1u` γ ´ ξ00qH ´ rα111u
2 ` γ11u
` δ11 ` p2α1u` γ ´ 2pα1u` β1qqu11qsH 1 “ 0.
Ðîçäiëÿþ÷è âèùåíàâåäåíó ñèñòåìó çà ñòåïåíÿìè u, ïðèõîäèìî äî ñèñ-
òåìè
α0 ` 3α11H
1 “ 0, (2.7)
β0 ` αH ` p2γ1 ´ β11 ´ 3αu11qH 1 “ 0, (2.8)
α01 ´ α111H
1 “ 0, (2.9)
γ0 ` 2α1H ´ γ11H
1 “ 0, (2.10)
δ0 ` pγ ´ ξ00qH ´ pδ11 ` pγ ´ 2β1qu11qH 1 “ 0. (2.11)
Áåðó÷è äî óâàãè, ùî H2 ‰ 0, ç ðiâíÿííÿ (2.7) îäåðæó¹ìî
α0 “ 0, α11 “ 0.
Ó òîé æå ÷àñ ðiâíÿííÿ (2.9) ñò๠íåçíà÷óùèì. Äàëi, ïiñëÿ äèôåðåíöiþ-
âàííÿ ðiâíÿííÿ (2.8) çà çìiííîþ u11, îòðèìó¹ìî
´2αH 1 ` p2γ1 ´ β11 ´ 3αu11qH2 “ 0.
40 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Ïiñëÿ ùå îäíîãî äèôåðåíöiþâàííÿ ïîïåðåäíüîãî ðiâíÿííÿ çà çìiííîþ x
ìà¹ìî
´2α1H
1 ` p2γ11 ´ β111 ´ 3α1u11qH2 “ 0. (2.12)
Äèôåðåíöiþþ÷è ðiâíÿííÿ (2.10) çà çìiííîþ u11, îòðèìó¹ìî
2α1H
1 ´ γ11H
2 “ 0. (2.13)
Äîäàþ÷è ðiâíÿííÿ (2.12) i (2.13), ìà¹ìî
pγ11 ´ β111 ´ 3α1u11qH2 “ 0.
Iç óìîâè H2 ‰ 0 îäåðæó¹ìî
α1 “ 0, γ11 “ 0, β111 “ 0.
Ç iíøîãî áîêó, ç ðiâíÿííÿ (2.10) ñëiäó¹, ùî γ0 “ 0. Òîäi
α “ C1 “ const, β “ β2ptqx2 ` β1ptqx` β0ptq, γ “ C2x` C3,
äå ôóíêöi¨ β2ptq, β1ptq òà β0ptq ïðîáiãàþòü ìíîæèíó ãëàäêèõ ôóíêöié
çìiííî¨ t, i C1, C2, . . . � äîâiëüíi êîíñòàíòè. Ïiäñòàíîâêà öèõ âèðàçiâ ó
ðiâíÿííÿ (2.8) äà¹
β20x
2 ` β10x` β00 ` C1H ` p2C2 ´ 2β2 ´ 3C1u11qH 1 “ 0.
Ðîçùåïëþþ÷è çà çìiííîþ x, ìà¹ìî
β20 “ 0, β10 “ 0, β000 “ 0.
Òîäi
β2 “ C4, β1 “ C5, β0 “ C6t` C7,
C6 ` C1H ` p2C2 ´ 2C4 ´ 3C1u11qH 1 “ 0. (2.14)
Îòæå, êîåôiöi¹íòè âåêòîðíîãî ïîëÿ Q íàáóâàþòü òàêî¨ ôîðìè:
ξ0 “ ξ0ptq, (2.15)
ξ1 “ C1u` C4x2 ` C5x` C6t` C7, (2.16)
η “ pC2x` C3qu` δpt, xq. (2.17)
Ïiñëÿ ïiäñòàíîâêè (2.16) i (2.17) â ðiâíÿííÿ (2.11) ïðèõîäèìî äî íàñòóï-
íî¨ êëàñèôiêàöiéíî¨ óìîâè:
δ0 ` pC2x` C3 ´ ξ00qH´
´ pδ11 ` pC2x` C3 ´ 2p2C4x` C5qqu11qH 1 “ 0.
(2.18)
ßêùî H � äîâiëüíà ôóíêöiÿ, òî iç (2.14) i (2.18) çíàõîäèìî
C6 “ 0, C1 “ 0, C2 “ C4, δ0 “ 0, C2x` C3 ´ ξ00 “ 0,
δ11 “ 0, pC2 ´ 4C4qx` C3 ´ 2C5 “ 0.
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 41
Âèõîäèòü, ùî
δ “ C8x` C9, ´ 3C4 “ 0, C3 “ 2C5, C2 “ 0,
i êîåôiöi¹íòè âåêòîðíèõ ïîëiâ (1.2) íàáóâàþòü íàñòóïíîãî âèãëÿäó:
ξ0 “ C3t` C10 “ 2C5t` C10, ξ1 “ C5x` C7, η “ 2C5u` C8x` C9.
Òàêèì ÷èíîì, ðiâíÿííÿ (2.2) ç äîâiëüíîþ ôóíêöi¹þ H ó ïðàâié ÷àñòèíi
äîïóñê๠5-âèìiðíó àëãåáðó Ëi g0, ïîðîäæåíó âåêòîðíèìè ïîëÿìè
Q1 “ Bt, Q2 “ Bx, Q3 “ Bu, Q4 “ 2tBt ` xBx ` 2uBu, Q5 “ xBu.
Ïðèõîäèìî äî íàñòóïíîãî òâåðäæåííÿ.
Òåîðåìà 2.2. g0 “ xBt, Bx, Bu, 2tBt ` xBx ` 2uBu, xBuy � ÿäðî ëi¨âñüêèõ
àëãåáð iíâàðiàíòíîñòi ðiâíÿíü iç êëàñó (2.2).
2.2. Ãðóïîâà êëàñèôiêàöiÿ. Íåõàé dimπ˚gH “ k. Òàê ÿê dimπ˚g0 “
2, òî äëÿ áóäü-ÿêîãî ðiâíÿííÿ ç êëàñó (2.2) ìà¹ìî k “ 2 àáî k “ 3, à
t-êîìïîíåíòà ¹ êâàäðàòè÷íîþ ôóíêöi¹þ çìiííîþ t, òîáòî,
π˚gH “ xBt, tBty àáî π˚gH “ xBt, tBt, t2Bty.
Íèæ÷å ðîçãëÿíóòî êîæåí iç öèõ äâîõ âèïàäêiâ îêðåìî.
Ñïî÷àòêó, äèôåðåíöiþþ÷è (2.18) äâi÷i âiäíîñíî çìiííî¨ x, îòðèìó¹ìî
δ011 ´ δ1111H
1 “ 0,
òîìó δ011 “ 0, δ1111 “ 0, îòæå,
δ “ C20x3 ` C21x2 ` ρ1ptqx` ρ0ptq, (2.19)
äå ôóíêöi¨ ρ1ptq òà ρ0ptq ïðîáiãàþòü ìíîæèíó ãëàäêèõ ôóíêöié çìií-
íî¨ t.
k “““ 3. Ó öüîìó âèïàäêó
ξ0 “ λ2t2 ` λ1t` λ0, (2.20)
äå λ2, λ1 i λ0 � äîâiëüíi êîíñòàíòè.
Ïiñëÿ ïiäñòàíîâêè (2.19) i (2.20) â (2.18) îòðèìó¹ìî
ρ10x` ρ00 ` pC2x` C3 ´ 2λ2t´ λ1qH´
´ p6C20x` 2C21 ` pC2x` C3 ´ 4C4x´ 2C5qu11qH 1 “ 0.
Ðîçùåïëåííÿ öüîãî ðiâíÿííÿ çà çìiííîþ x ïðèâîäèòü äî íàñòóïíî¨ ñè-
ñòåìè:
ρ10 ` C2H ´ p6C20 ` pC2 ´ 4C4qu11qH 1 “ 0,
ρ00 ` pC3 ´ 2λ2t´ λ1qH ´ p2C21 ` pC3 ´ 2C5qu11qH 1 “ 0.
42 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Äèôåðåíöiþþ÷è äðóãå ðiâíÿííÿ âèùåçãàäàíî¨ ñèñòåìè âiäíîñíî çìií-
íî¨ t, îäåðæó¹ìî
ρ000 ´ 2λ2H “ 0,
i òîìó λ2 “ 0, îñêiëüêè H � íåëiíiéíà ôóíêöiÿ çìiííî¨ u11. Ìà¹ìî,
ùî ôóíêöiÿ ξ0 ìîæå áóòè ùîíàéáiëüøå ëiíiéíîþ. Öå ñóïåðå÷èòü óìîâi
k “ 3.
k “““ 2. Ó öüîìó âèïàäêó ôóíêöiÿ ξ0 ëiíiéíà, òîáòî,
ξ0 “ λ1t` λ0, (2.21)
äå λ1 i λ0 � äîâiëüíi êîíñòàíòè.
Iç ïiäñòàíîâêè (2.19) i (2.21) â (2.18) ñëiäó¹
ρ10x` ρ00 ` pC2x` C3 ´ λ1qH´
´ p6C20x` 2C21 ` pC2x` C3 ´ 4C4x´ 2C5qu11qH 1 “ 0.
Ðîçùåïëåííÿ öüîãî ðiâíÿííÿ çà çìiííîþ x ïðèâîäèòü äî íàñòóïíî¨ ñèñ-
òåìè:
ρ10 ` C2H ´ p6C20 ` pC2 ´ 4C4qu11qH 1 “ 0, (2.22)
ρ00 ` pC3 ´ λ1qH ´ p2C21 ` pC3 ´ 2C5qu11qH 1 “ 0. (2.23)
Ïiñëÿ äèôåðåíöiþâàííÿ (2.22) i (2.23) çà çìiííîþ t îòðèìó¹ìî, ùî ρ100 “
0 i ρ000 “ 0, âiäïîâiäíî. Òîìó ρ1ptq “ C22t ` C23 i ρ0ptq “ C24t ` C25.
Îòæå,
δpt, xq “ C20x3 ` C21x2 ` pC22t` C23qx` pC24t` C25q.
Iç ðiâíÿíü (2.15)�(2.17) îäåðæó¹ìî íàñòóïíi âèðàçè äëÿ êîìïîíåíò âåê-
òîðíîãî ïîëÿ Q:
ξ0 “ λ1t` λ0, ξ1 “ C1u` C4x2 ` C5x` C6t` C7,
η “ pC2x` C3qu` C20x3 ` C21x2 ` pC22t` C23qx` pC24t` C25q,
ó òîé ÷àñ, ÿê (2.14), (2.22) i (2.23) äàþòü ñèñòåìó âèçíà÷àëüíèõ ðiâíÿíü
äëÿ ôóíêöi¨ H ó âèãëÿäi
C6 ` C1H ` p2C2 ´ 2C4 ´ 3C1u11qH 1 “ 0, (2.24)
C22 ` C2H ´ p6C20 ` pC2 ´ 4C4qu11qH 1 “ 0, (2.25)
C24 ` pC3 ´ λ1qH ´ p2C21 ` pC3 ´ 2C5qu11qH 1 “ 0. (2.26)
Áà÷èìî, ùî âñi öi ðiâíÿííÿ ìàþòü çàãàëüíèé âèãëÿä
a` bH ` cH 1 ` du11H
1 “ 0 (2.27)
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 43
iç äåÿêèìè ñòàëèìè ïàðàìåòðàìè a, b, c i d. Ç òî÷íiñòþ äî åêâiâàëåíòíî-
ñòi, âèçíà÷åíî¨ â òåîðåìi 2.1, iñíó¹ òiëüêè òðè ìîæëèâîñòi äëÿ ðîçâ'ÿçêiâ
ðiâíÿííÿ òèïó (2.27), à ñàìå, eu11 , lnpu11q i up11, äå p ‰ 0, 1.
ßêùî H “ eu11 , òî ìà¹ìî
ξ0 “ 2pC5 ´ C21qt` λ0, ξ1 “ C5x` C7
η “ 2C5u` C21x2 ` C23x` C25.
Òàêèì ÷èíîì, çà óìîâè H “ eu11 áàçèñ ìàêñèìàëüíî¨ àëãåáðè ëi¨âñüêî¨
iíâàðiàíòíîñòi ì๠íàñòóïíèé âèãëÿä:
Q1 “ Bt, Q2 “ Bx, Q3 “ Bu, Q4 “ 2tBt ` xBx ` 2uBu,
Q5 “ xBu, Q6 “ 2tBt ´ x2Bu.
ßêùî H “ lnpu11q, òî
ξ0 “ C3t` λ0, ξ1 “ C5x` C7, η “ C3u` C23x` pC3 ´ 2C5qt` C25.
Ó âèïàäêó H “ lnpu11q îäåðæó¹ìî íàñòóïíèé áàçèñ ìàêñèìàëüíî¨ àëãå-
áðè ëi¨âñüêî¨ iíâàðiàíòíîñòi:
Q1 “ Bt, Q2 “ Bx, Q3 “ Bu, Q4 “ 2tBt ` xBx ` 2uBu,
Q5 “ xBu, Q6 “ xBx ´ 2tBu.
ßêùî H “ up11, òî ç ñèñòåìè (2.24)�(2.26) îòðèìó¹ìî
C6 ` C1up11 ` 2pC2 ´ C4qpup´1
11 ´ 3C1pup11 “ 0,
C22 ` C2up11 ´ 6C20pup´1
11 ´ pC2 ´ 4C4qpup11 “ 0,
C24 ` pC3 ´ λ1qup11 ´ 2C21pup´1
11 ´ pC3 ´ 2C5qpup11 “ 0.
Ðîçùåïëþþ÷è öþ ñèñòåìó, îäåðæó¹ìî
C1 “ 3pC1, C2 “ C4, C2 “ ppC2 ´ 4C4q,
λ1 “ p1 ´ pqC3 ` 2pC5, C6 “ C20 “ C21 “ C22 “ C24 “ 0.
ßêùî p ‰ ˘1
3 , òî
λ1 “ p1 ´ pqC3 ` 2pC5,
C1 “ C2 “ C6 “ C20 “ C21 “ C22 “ C24 “ 0,
òîáòî êîìïîíåíòè âåêòîðíîãî ïîëÿ Q ìàþòü âèãëÿä
ξ0 “ pp1 ´ pqC3 ` 2pC5qt` λ0, ξ1 “ C5x` C7,
η “ C3u` C23x` C25.
44 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Îòæå, êîëè H “ up11, p ‰ ˘1
3 , òî àëãåáðà ìàêñèìàëüíî¨ ëi¨âñüêî¨ iíâàði-
àíòíîñòi ïîðîäæåíà íàñòóïíèìè âåêòîðíèìè ïîëÿìè:
Q1 “ Bt, Q2 “ Bx, Q3 “ Bu, Q4 “ 2tBt ` xBx ` 2uBu,
Q5 “ xBu, Q6 “ p1 ´ pqtBt ` uBu.
Äëÿ p “ 1
3 :
λ1 “ 2
3
pC3 ` C5q, C2 “ C6 “ C20 “ C21 “ C22 “ C24 “ 0,
òîáòî êîìïîíåíòè âåêòîðíîãî ïîëÿ Q ìàþòü íàñòóïíèé âèãëÿä:
ξ0 “ 2
3
pC3 ` C5qt` λ0, ξ1 “ C1u` C5x` C7,
η “ C3u` C23x` C25.
Òàêèì ÷èíîì, ó âèïàäêó H “ u
1{3
11 îòðèìó¹ìî íàñòóïíèé âèãëÿä àëãåáðè
ìàêñèìàëüíî¨ ëi¨âñüêî¨ iíâàðiàíòíîñòi:
Q1 “ Bt, Q2 “ Bx, Q3 “ Bu, Q4 “ 2tBt ` xBx ` 2uBu,
Q5 “ xBu, Q6 “ 2tBt ` 3uBu, Q7 “ uBx.
Äëÿ p “ ´1
3 :
λ1 “ 2
3
p2C3 ´ C5q, C1 “ C6 “ C20 “ C21 “ C22 “ C24 “ 0,
i êîìïîíåíòè âåêòîðíîãî ïîëÿ Q òàêi:
ξ0 “ 2
3
p2C3 ´ C5qt` λ0, ξ1 “ C2x2 ` C5x` C7,
η “ pC2x` C3qu` C23x` C25.
Îòæå, ÿêùî H “ u
´1{3
11 , òî àëãåáðà ìàêñèìàëüíî¨ ëi¨âñüêî¨ iíâàðiàíòíî-
ñòi ïîðîäæó¹òüñÿ íàñòóïíèìè áàçèñíèìè âåêòîðíèìè ïîëÿìè:
Q1 “ Bt, Q2 “ Bx, Q3 “ Bu, Q4 “ 2tBt ` xBx ` 2uBu,
Q5 “ xBu, Q6 “ 4tBt ` 3uBu, Q7 “ x2Bx ` xuBu.
Îòðèìàíi ðåçóëüòàòè ïiäñóìîâàíî â íàñòóïíîìó òâåðäæåííi.
Òåîðåìà 2.3 (ðåçóëüòàò ãðóïîâî¨ êëàñèôiêàöi¨, [1, �4]). Ïîâíèé ñïèñîê
G„-íååêâiâàëåíòíèõ pìàêñèìàëüíèõq ðîçøèðåíü ëi¨âñüêèõ ñèìåòðié ó
êëàñi (2.2) âè÷åðïóþòü òàêi âèïàäêè, ÿêi íàâåäåíî ó òàáëèöi 2.1.
Ó âèïàäêó ëiíiéíî¨ (íåñòàëî¨) ôóíêöi¨ H (îñòàííié ðÿäîê òàáëèöi,
ÿêèé âêëþ÷åíî äëÿ ïîâíîòè ðîçãëÿäó êëàñó (2.2)) àëãåáðà ëi¨âñüêî¨ iíâà-
ðiàíòíîñòi íåñêií÷åííîâèìiðíà. Òóò ïàðàìåòðè÷íà ôóíêöiÿ f çàëåæèòü
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 45
Òàáëèöÿ 2.1. Ðåçóëüòàòè ãðóïîâî¨ êëàñèôiêàöi¨ êëàñó (2.2).
Hpuxxq Ëi¨âñüêà àëãåáðà iíâàðiàíòíîñòi
@ xBt, Bx, Bu, 2tBt ` xBx ` 2uBu, xBuy
expuxx xBt, Bx, Bu, 2tBt ` xBx ` 2uBu, xBu, 2tBt ´ x2Buy
lnuxx xBt, Bx, Bu, 2tBt ` xBx ` 2uBu, xBu, xBx ´ 2tBuy
upxx, p “ 0,˘1
3 , 1 xBt, Bx, Bu, 2tBt ` xBx ` 2uBu, xBu, p1 ´ pqtBt ` uBuy
u
1{3
xx xBt, Bx, Bu, 2tBt ` xBx ` 2uBu, xBu, 2tBt ` 3uBu, uBxy
u
´1{3
xx xBt, Bx, Bu, 2tBt ` xBx ` 2uBu, xBu,
4tBt ` 3uBu, x2Bx ` xuBuy
uxx xBt, Bx, uBu, 2tBt ` xBx, tBx ´ 1
2xuBu,
t2Bt ` txBx ´ 1
4px2 ` 2tquBu, fpt, xqBuy
âiä çìiííèõ pt, xq i ïðîáiã๠ìíîæèíó ðîçâ'ÿçêiâ ëiíiéíîãî ðiâíÿííÿ òå-
ïëîïðîâiäíîñòi ut “ uxx.
3. Ãðóïîâà êëàñèôiêàöiÿ íåëiíiéíîãî ðiâíÿííÿ Áþðãåðñà
3.1. Ïîïåðåäíi âiäîìîñòi. Ìåòîþ öüîãî ïàðàãðàôà ¹ ñïðîùåííÿ äî-
âåäåííÿ ðåçóëüòàòiâ [4,25] ïðî ãðóïîâó êëàñèôiêàöiÿ êëàñó åâîëþöiéíèõ
ðiâíÿíü âèãëÿäó
u0 ` uu1 “ Hpu11q, (3.1)
äå u0 :“ ut “ Bu
Bt , u1 :“ ux “ Bu
Bx , u11 :“ uxx “ B2u
Bx2 , H � äîâiëüíà ãëàäêà
ôóíêöiÿ àðãóìåíòó uxx. Êîëè H � ëiíiéíà ôóíêöiÿ, òî ðiâíÿííÿ (3.1) ¹
äîáðå âiäîìèì ðiâíÿííÿì Áþðãåðñà. Çãiäíî ç ëåìîþ 1.1, âåêòîðíi ïîëÿ,
ùî íàëåæàòü ìàêñèìàëüíié àëãåáði ëi¨âñüêî¨ iíâàðiàíòíîñòi gH åâîëþ-
öiéíèõ ðiâíÿíü iç êëàñó (3.1), ìîæíà øóêàòè â âèãëÿäi (1.2).
Òåîðåìà 3.1 (ãðóïà åêâiâàëåíòíîñòi êëàñó (3.1), äèâ. [23]). Ãðóïà åêâi-
âàëåíòíîñòi G„ êëàñó (3.1) ïîðîäæåíà îïåðàòîðàìè Bt, Bx, tBx ` Bu,
tBt ` xBx ´HBH , xBx ` uBu `HBH , t2Bx ` 2tBu ` 2BH , à òàêîæ äèñêðå-
òíèì ïåðåòâîðåííÿì åêâiâàëåíòíîñòi pt, x, u,Hq Ñ p´t,´x, u,´Hq.
Äiÿ áóäü-ÿêîãî ïåðåòâîðåííÿ åêâiâàëåíòíîñòi íà ôóíêöiþ H ì๠âèã-
ëÿä
H̃pu11q “ δ1Hpδ2u11q ` δ0,
äå δ0, δ1, δ2 P R, δ1δ2 ‰ 0.
46 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Âåêòîðíå ïîëå (1.2) íàëåæèòü ìàêñèìàëüíié àëãåáði ëi¨âñüêî¨ iíâà-
ðiàíòíîñòi gH ðiâíÿííÿ ç êëàñó (3.1) äëÿ áóäü-ÿêî¨ ôóíêöi¨ H òîäi é
ëèøå òîäi, êîëè âîíî çàäîâîëüíÿ¹ íàñòóïíå âèçíà÷àëüíå ðiâíÿííÿ:
pηu1 ` ζ0 ` ζ1u´ ζ11H 1qˇ̌
u0`uu1“Hpu11q “ 0, (3.2)
äå ζ0, ζ1 òà ζ11 � âiäïîâiäíi êîåôiöi¹íòè ïåðøîãî òà äðóãîãî ïðîäîâæåíü
âåêòîðíîãî ïîëÿ (1.2), H 1 :“ BHpu11q{Bu11.
Ðîçïèñóþ÷è ðiâíÿííÿ (3.2), îòðèìó¹ìî
ηu1 ` η0 `Hηu ` uu1ξ
0
0 ´Hξ00 ´ u1ξ
1
0 ´ u1ξ
1
uH ` uη1 ´ uu1ξ
1
1
´H 1“η11 ` 2η1uu1 ` ηuuu
2
1 ` ηuu11 ´ 2u11pξ11 ` ξ1uu1q
´ u1pξ111 ` 2ξ11uu1 ` ξ1uuu
2
1 ` ξ1uu11q‰ “ 0.
Ïiñëÿ ðîçùåïëåííÿ öüîãî ðiâíÿííÿ çà ñòåïåíÿìè çìiííî¨ u1 ïðèõîäè-
ìî äî íàñòóïíî¨ ñèñòåìè:
ξ1uu “ 0, ηuu ´ 2ξ11u “ 0, (3.3)
η ´ ξ10 ` upξ00 ´ ξ11q ´ ξ1uH ´ p2η1u ´ ξ111 ´ 3ξ1uu11qH 1 “ 0, (3.4)
η0 ` uη1 ` pηu ´ ξ00qH ´ pη11 ` pηu ´ 2ξ11qu11qH 1 “ 0. (3.5)
Çàãàëüíèé ðîçâ'ÿçîê ñèñòåìè (3.3) ì๠âèãëÿä
ξ1 “ apt, xqu` bpt, xq, η “ a1u
2 ` cpt, xqu` dpt, xq,
äå apt, xq, bpt, xq, cpt, xq i dpt, xq ïðîáiãàþòü ìíîæèíó ãëàäêèõ ôóíêöié
ñâî¨õ àðãóìåíòiâ.
Ïiäñòàíîâêà öèõ âèðàçiâ ó ðiâíÿííÿ (3.4) i (3.5) ïðèâîäèòü äî íàñòóï-
íî¨ ñèñòåìè:
pc´ a0 ` ξ00 ´ b1qu` pd´ b0q ´ aH
´ p3a11u` 2c1 ´ b11 ´ 3au11qH 1 “ 0,
a11u
3 ` pa01 ` c1qu2 ` pc0 ` d1qu` d0 ` p2a1u` c´ ξ00qH
´ pa111u2 ` c11u` d11 ` pc´ 2b1qu11qH 1 “ 0.
Ðîçùåïëþþ÷è âèùåíàâåäåíó ñèñòåìó çà ñòåïåíÿìè u, ïðèõîäèìî äî
ñèñòåìè
c´ a0 ` ξ00 ´ b1 ´ 3a11H
1 “ 0, (3.6)
d´ b0 ´ aH ´ p2c1 ´ b11 ´ 3au11qH 1 “ 0, (3.7)
a11 “ 0, (3.8)
a01 ` c1 ´ a111H
1 “ 0, (3.9)
c0 ` d1 ` 2a1H ´ c11H
1 “ 0, (3.10)
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 47
d0 ` pc´ ξ00qH ´ pd11 ` pc´ 2b1qu11qH 1 “ 0. (3.11)
Iç ðiâíÿííÿ (3.8) âèïëèâà¹, ùî
a “ λ1ptqx` λ2ptq.
Òóò i íèæ÷å λ1ptq, λ2ptq, . . . ïðîáiãàþòü ìíîæèíó ãëàäêèõ ôóíêöié çìií-
íî¨ t. Ç óðàõóâàííÿì ðiâíÿíü (3.8) i (3.9) ìà¹ìî a01 ` c1 “ 0, òîáòî
λ10 ` c1 “ 0, çâiäêè ìîæíà çíàéòè ôóíêöiþ cpt, xq:
c “ ´λ10ptqx` λ3ptq.
Äàëi ïiäñòàâèìî ôóíêöi¨ a i c â ðiâíÿííÿ (3.6):
´λ10x` λ3 ´ λ10x´ λ20 ` ξ00 ´ b1 “ 0.
Çâiäñè çíàõîäèìî ôóíêöiþ bpt, xq:
b “ ´λ10x2 ` pξ00 ` λ3 ´ λ20qx` λ4ptq.
Îñêiëüêè H � íåëiíiéíà ôóíêöiÿ, òî ç ðiâíÿííÿ (3.10) âèïëèâà¹, ùî
c0 ` d1 “ 0, a1 “ 0.
Òîäi ç ðiâíÿííÿ a1 “ 0 îäåðæó¹ìî, ùî λ1 “ 0, òîáòî
a “ λ2ptq, b “ pξ00 ` λ3 ´ λ20qx` λ4ptq, c “ λ3ptq.
Íàòîìiñòü, iç ðiâíÿííÿ c0 ` d1 “ 0 ìà¹ìî, ùî λ30 ` d1 “ 0, à îòæå,
d “ ´λ30x` λ5ptq.
Ïiäñòàíîâêà ùîéíî óòî÷íåíèõ âèðàçiâ äëÿ ôóíêöié a, b, c i d â ðiâ-
íÿííÿ (3.7) ïðèâîäèòü äî ñïiââiäíîøåííÿ
pλ5 ´ λ40q ´ p2λ30 ` ξ000 ´ λ200qx´ λ2H ` 3λ2u11H
1 “ 0.
Ïðèðiâíþþ÷è êîåôiöi¹íò ïðè x äî íóëÿ, îòðèìà¹ìî íàñòóïíå ðiâíÿííÿ:
2λ30 ` ξ000 ´ λ200 “ 0,
çâiäêè ñëiäó¹, ùî
ξ00 “ λ20 ´ 2λ3 `m1.
Òóò i íèæ÷å m1,m2, . . . � äîâiëüíi äiéñíi ñòàëi.
Îòæå, ïðèõîäèìî äî ïåðøî¨ êëàñèôiêàöiéíî¨ óìîâè:
pλ5 ´ λ40q ´ λ2H ` 3λ2u11H
1 “ 0.
Íàòîìiñòü, iç ïiäñòàíîâêè íàâåäåíèõ âèùå âèðàçiâ äëÿ ôóíêöié a, b,
c, d i ξ00 â ðiâíÿííÿ (3.11) îäåðæèìî
´λ300x` λ50 ` p3λ3 ´ λ20 ´m1qH ` pλ3 ` 2ξ00 ´ 2λ20qu11H 1 “ 0.
Ïðèðiâíþþ÷è êîåôiöi¹íò ïðè x äî íóëÿ, îòðèìà¹ìî íàñòóïíå ðiâíÿííÿ:
λ300 “ 0.
48 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Çâiäñè ðîáèìî âèñíîâîê, ùî λ3 “ m2t`m3 i, îòæå,
a “ λ2ptq, b “ p´m2t`m1 ´m3qx` λ4ptq, c “ m2t`m3,
d “ ´m2x` λ5ptq, ξ00 “ ´2m2t` λ20 ´ 2m3 `m1.
Äðóãà êëàñèôiêàöiéíà óìîâà íàáóâ๠âèãëÿäó
λ50 ` p3λ3 ´ λ20 ´m1qH ` pλ3 ` 2ξ00 ´ 2λ20qu11H 1 “ 0.
Òàêèì ÷èíîì, ïðèõîäèìî äî íàñòóïíî¨ ñèñòåìè:
pλ5 ´ λ40q ´ λ2H ` 3λ2u11H
1 “ 0,
λ50 ` p3m2t` 3m3 ´ λ20 ´m1qH ` pm2t`m3 ` 2ξ00 ´ 2λ20qu11H 1 “ 0,
ξ00 “ ´2m2t` λ20 ´ 2m3 `m1. (3.12)
ßêùî H � äîâiëüíà ôóíêöiÿ, òî ç (3.12) ïiñëÿ ðîçùåïëåííÿ çíàõîäèìî:
λ5 “ λ40, λ2 “ 0, λ50 “ 0, 3m2t` 3m3 ´ λ20 ´m1 “ 0,
´3m2t` 2m1 ´ 3m3 “ 0.
Çâiäñè îòðèìà¹ìî, ùî
λ5 “ m4, λ4 “ m4t`m5, m2 “ 0, 3m3 ´m1 “ 0, 2m1 ´ 3m3 “ 0,
òîìó m1 “ 0, m3 “ 0 i ôóíêöi¨ a, b, c, d i ξ00 íàáóâàþòü íàñòóïíîãî
âèãëÿäó:
a “ 0, b “ m4t`m5, c “ 0, d “ m4, ξ00 “ 0.
Îòæå, êîåôiöi¹íòè âåêòîðíèõ ïîëiâ (1.2) ìàþòü ôîðìó:
ξ0 “ m6, ξ1 “ m4t`m5, η “ m4.
Òàêèì ÷èíîì, ðiâíÿííÿ (3.1) ç äîâiëüíîþ ôóíêöi¹þ H ó ïðàâié ÷àñòèíi
äîïóñê๠3-âèìiðíó àëãåáðó Ëi g0, ïîðîäæåíó âåêòîðíèìè ïîëÿìè
Q1 “ Bt, Q2 “ Bx, Q3 “ tBx ` Bu.
Ïðèõîäèìî äî íàñòóïíîãî òâåðäæåííÿ.
Òåîðåìà 3.2 (äèâ. [23]). g0 “ xBt, Bx, tBx ` Buy � ÿäðî ëi¨âñüêèõ àëãåáð
iíâàðiàíòíîñòi ðiâíÿíü iç êëàñó (3.1).
3.2. Ãðóïîâà êëàñèôiêàöiÿ. Íåõàé çíîâó dimπ˚gH “ k. Îñêiëüêè
dimπ˚g0 “ 1, òî äëÿ áóäü-ÿêîãî ðiâíÿííÿ ç êëàñó (3.1) ìà¹ìî, ùî k “ 1,
k “ 2 àáî k “ 3. Äîäàòêîâî ïðèïóñêà¹ìî, ùî t-êîìïîíåíòà ¹ êâàäðàòè-
÷íîþ ôóíêöi¹þ çìiííîþ t. Îòæå,
π˚gH “ xBty, π˚gH “ xBt, tBty àáî π˚gH “ xBt, tBt, t2Bty.
Íèæ÷å ðîçãëÿíóòî êîæåí iç öèõ òðüîõ âèïàäêiâ îêðåìî.
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 49
k “““ 3. Ó öüîìó âèïàäêó
ξ0 “ µ2t2 ` µ1t` µ0, (3.13)
äå µ2, µ1 i µ0 � äîâiëüíi êîíñòàíòè. Ïîðiâíþþ÷è ç òðåòiì ðiâíÿííÿì
iç (3.12), ìà¹ìî
2µ2t` µ1 “ ´2m2t` λ20 ´ 2m3 `m1, (3.14)
çâiäêè ïiñëÿ äèôåðåíöiþâàííÿ çà çìiííîþ t îòðèìà¹ìî λ200 “ 2µ2 `2m2,
iíøèìè ñëîâàìè,
λ2 “ pµ2 `m2qt2 `m7t`m8.
Ïiäñòàâëÿþ÷è ùîéíî íàâåäåíèé âèðàç ó (3.14), îäåðæèìî
2µ2t` µ1 “ ´2m2t` p2µ2 ` 2m2qt`m7 ´ 2m3 `m1,
çâiäêè ñëiäó¹, ùî m7 “ µ1 ` 2m3 ´m1, òîáòî
λ2 “ pµ2 `m2qt2 ` pµ1 ` 2m3 ´m1qt`m8. (3.15)
Ïiñëÿ ïiäñòàíîâêè (3.15) â äðóãå ðiâíÿííÿ ñèñòåìè (3.12) îòðèìó¹ìî
λ50 ` ppm2 ´ 2µ2qt` pm3 ´ µ1qqH
` p´3m2t` p2m1 ´ 3m3qqu11H 1 “ 0. (3.16)
Ïiñëÿ äèôåðåíöiþâàííÿ äâi÷i çà çìiííîþ t îäåðæèìî
λ5000 “ 0,
ùî ñâiä÷èòü ïðî òå, ùî ôóíêöiÿ λ5 � êâàäðàòè÷íà, ñêàæiìî,
λ5 “ m9t
2 `m10t`m11.
Ïiäñòàâèìî öåé âèðàç íàçàä ó ðiâíÿííÿ (3.16), îòðèìà¹ìî
2m9t`m10 ` ppm2 ´ 2µ2qt` pm3 ´ µ1qqH
` p´3m2t` p2m1 ´ 3m3qqu11H 1 “ 0.
Ðîçùåïëþþ÷è çà çìiííîþ t, ïðèõîäèìî äî ñèñòåìè:
2m9 ` pm2 ´ 2µ2qH ´ 3m2u11H
1 “ 0,
m10 ` pm3 ´ µ1qH ` p2m1 ´ 3m3qu11H 1 “ 0. (3.17)
Ïiäñòàâèìî òåïåð ôóíêöi¨ λ5 i λ2 ó ïåðøå ðiâíÿííÿ (3.12):
pm9t
2 `m10t`m11 ´ λ40q
´ ppµ2 `m2qt2 ` pµ1 ` 2m3 ´m1qt`m8qH
` 3ppµ2 `m2qt2 ` pµ1 ` 2m3 ´m1qt`m8qu11H 1 “ 0. (3.18)
50 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Äèôåðåíöiþâàííÿ òðè÷i çà çìiííîþ t ïîêàçó¹, ùî λ40000 “ 0, ùî ñâiä-
÷èòü ïðî òå, ùî ôóíêöiÿ λ4 � êóái÷íà. Îòæå,
λ4 “ m12t
3 `m13t
2 `m14t`m15.
Ïiäñòàâëÿþ÷è ôóíêöiþ λ4 íàçàä ó ðiâíÿííÿ (3.18), ìà¹ìî
pm9t
2 `m10t`m11 ´ 3m12t
2 ´ 2m13t´m14q
´ ppµ2 `m2qt2 ` pµ1 ` 2m3 ´m1qt`m8qH
` 3ppµ2 `m2qt2 ` pµ1 ` 2m3 ´m1qt`m8qu11H 1 “ 0.
Ðîçùåïëþþ÷è çà çìiííîþ t, îäåðæó¹ìî ñèñòåìó:
pm9 ´ 3m12q ´ pµ2 `m2qH ` 3pµ2 `m2qu11H 1 “ 0,
pm10 ´ 2m13q ´ pµ1 ` 2m3 ´m1qH ` 3pµ1 ` 2m3 ´m1qu11H 1 “ 0,
pm11 ´m14q ´m8H ` 3m8u11H
1 “ 0. (3.19)
Î÷åâèäíî, ùî ðiâíÿííÿ ó ñèñòåìi (3.19) ìàþòü çàãàëüíèé âèãëÿä (2.27)
iç äåÿêèìè ñòàëèìè ïàðàìåòðàìè a, b, c i d. Îòæå, ç òî÷íiñòþ äî åêâi-
âàëåíòíîñòi, âèçíà÷åíî¨ â òåîðåìi 3.1, iñíó¹ òiëüêè òðè ìîæëèâîñòi äëÿ
ðîçâ'ÿçêiâ ðiâíÿííÿ (2.27), à ñàìå, eu11 , lnpu11q i up11, äå p ‰ 0, 1.
ßêùî H “ up11, p ‰ 0, 1, òî iç ñèñòåì (3.17) i (3.19) ìà¹ìî
2m9 ` pm2 ´ 2µ2qup11 ´ 3m2pu
p
11 “ 0,
m10 ` pm3 ´ µ1qup11 ` p2m1 ´ 3m3qpup11 “ 0,
pm9 ´ 3m12q ´ pµ2 `m2qup11 ` 3pµ2 `m2qpup11 “ 0,
pm10 ´ 2m13q ´ pµ1 ` 2m3 ´m1qup11 ` 3pµ1 ` 2m3 ´m1qpup11 “ 0,
pm11 ´m14q ´m8u
p
11 ` 3m8pu
p
11 “ 0. (3.20)
Ïðè ðîçùåïëåííi öi¹¨ ñèñòåìè îòðèìó¹ìî íàñòóïíi îáìåæåííÿ íà ñòàëi:
m9 “ 0, m10 “ 0, m12 “ 0, m13 “ 0, m11 “ m14.
Äàëi, ÿêùî p ‰ 1
3 , òî ìà¹ìî ùå íàñòóïíi îáìåæåííÿ:
m2 “ 0, µ2 “ 0, m8 “ 0
µ1 “ m1 ´ 2m3, p3 ´ 3pqm3 ` p2p´ 1qm1 “ 0.
Çà óìîâè p “ 1
3 ìà¹ìî ëèøå íàñòóïíi îáìåæåííÿ:
µ2 “ 0, µ1 “ 2
3
m1.
Îòæå, çà áóäü-ÿêîãî çíà÷åííÿ p ‰ 0, 1 (iíøèìè ñëîâàìè, êîëè ôóíê-
öiÿ H íå ¹ ëiíiéíîþ ôóíêöi¹þ ñâîãî àðãóìåíòà) µ2 “ 0, òîáòî k ‰ 3 i
ìà¹ìî ñóïåðå÷íiñòü.
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 51
k “““ 2. Ó öüîìó âèïàäêó
ξ0 “ µ1t` µ0, (3.21)
äå µ1, µ0 � äîâiëüíi êîíñòàíòè. Ïîðiâíþþ÷è ç òðåòiì ðiâíÿííÿì iç (3.12),
ìà¹ìî
µ1 “ ´2m2t` λ20 ´ 2m3 `m1, (3.22)
çâiäêè ïiñëÿ äèôåðåíöiþâàííÿ çà çìiííîþ t îòðèìà¹ìî
λ200 “ 2m2,
iíøèìè ñëîâàìè,
λ2 “ m2t
2 `m15t`m16.
Ïiäñòàâëÿþ÷è ùîéíî íàâåäåíèé âèðàç ó (3.22), îäåðæèìî
µ1 “ ´2m2t` 2m2t`m15 ´ 2m3 `m1,
çâiäêè ñëiäó¹, ùî m15 “ µ1 ` 2m3 ´m1, òîáòî
λ2 “ m2t
2 ` pµ1 ` 2m3 ´m1qt`m16. (3.23)
Ïiäñòàíîâêà (3.23) â äðóãå ðiâíÿííÿ ñèñòåìè (3.12) ïðèâîäèòü äî ðiâ-
íÿííÿ
λ50 ` pm2t` pm3 ´ µ1qqH ` p´3m2t` p2m1 ´ 3m3qqu11H 1 “ 0. (3.24)
Ïiñëÿ äèôåðåíöiþâàííÿ äâi÷i çà çìiííîþ t îòðèìà¹ìî λ5000 “ 0, ùî ñâiä-
÷èòü ïðî òå, ùî ôóíêöiÿ λ5 � êâàäðàòè÷íà, òîáòî,
λ5 “ m17t
2 `m18t`m19.
Ïiäñòàâèìî öåé âèðàç íàçàä ó ðiâíÿííÿ (3.24), îäåðæèìî
2m17t`m18 ` pm2t` pm3 ´ µ1qqH
` p´3m2t` p2m1 ´ 3m3qqu11H 1 “ 0.
Ðîçùåïëþþ÷è çà çìiííîþ t, ïðèõîäèìî äî ñèñòåìè:
2m17 `m2H ´ 3m2u11H
1 “ 0,
m18 ` pm3 ´ µ1qH ` p2m1 ´ 3m3qu11H 1 “ 0. (3.25)
Ïiäñòàâèìî òåïåð ôóíêöi¨ λ5 i λ2 â ïåðøå ðiâíÿííÿ (3.12):
pm17t
2 `m18t`m19 ´ λ40q ´ pm2t
2 ` pµ1 ` 2m3 ´m1qt`m16qH
` 3pm2t
2 ` pµ1 ` 2m3 ´m1qt`m16qu11H 1 “ 0. (3.26)
Äèôåðåíöiþâàííÿ òðè÷i çà çìiííîþ t ïîêàçó¹, ùî λ40000 “ 0, ùî ñâiä-
÷èòü ïðî òå, ùî ôóíêöiÿ λ4 � êóái÷íà. Îòæå,
λ4 “ m20t
3 `m21t
2 `m22t`m23.
52 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Ïiäñòàâëÿþ÷è ôóíêöiþ λ4 íàçàä ó ðiâíÿííÿ (3.26), ìà¹ìî
pm17t
2 `m18t`m19 ´ 3m20t
2 ´ 2m21t´m22q
´ pm2t
2 ` pµ1 ` 2m3 ´m1qt`m16qH
` 3pm2t
2 ` pµ1 ` 2m3 ´m1qt`m16qu11H 1 “ 0.
Ðîçùåïëþþ÷è çà çìiííîþ t, îäåðæó¹ìî òàêó ñèñòåìó:
pm17 ´ 3m20q ´m2H ` 3m2u11H
1 “ 0,
pm18 ´ 2m21q ´ pµ1 ` 2m3 ´m1qH ` 3pµ1 ` 2m3 ´m1qu11H 1 “ 0,
pm19 ´m22q ´m16H ` 3m16u11H
1 “ 0. (3.27)
Áà÷èìî, ùî âñi öi ðiâíÿííÿ ìàþòü çàãàëüíèé âèãëÿä (2.27) iç äåÿêèìè
ñòàëèìè ïàðàìåòðàìè a, b, c i d. Ç òî÷íiñòþ äî åêâiâàëåíòíîñòi, âèçíà-
÷åíî¨ â òåîðåìi 3.1, iñíó¹ òiëüêè òðè ìîæëèâîñòi äëÿ ðîçâ'ÿçêiâ ðiâíÿí-
íÿ (2.27), à ñàìå, eu11 , lnpu11q i up11, äå p ‰ 0, 1.
ßêùî H “ eu11 , òî ìà¹ìî
ξ0 “ µ0, ξ1 “ m19t`m23, η “ m19.
Òàêèì ÷èíîì, ó âèïàäêó ôóíêöi¨ H “ eu11 îòðèìó¹ìî ìàêñèìàëüíó
àëãåáðó ëi¨âñüêî¨ iíâàðiàíòíîñòi ç òåîðåìè 3.2.
ßêùî H “ lnpu11q, òî
ξ0 “ µ1t` µ0, ξ1 “ 2µ1x´ 3
2µ
1t2 `m19t`m23,
η “ µ1u´ 3µ1t`m19.
Çà óìîâè H “ lnpu11q áàçèñ ìàêñèìàëüíî¨ àëãåáðè ëi¨âñüêî¨ iíâàðiàí-
òíîñòi çàäà¹òüñÿ íàñòóïíèìè âåêòîðíèìè ïîëÿìè:
Q1 “ Bt, Q2 “ Bx, Q3 “ tBx ` Bu,
Q4 “ tBt ` `
2x´ 3
2 t
2
˘ Bx ` pu´ 3tqBu.
ßêùî H “ up11, p ‰ 0, 1, òî iç ñèñòåì (3.25) i (3.27) îòðèìó¹ìî
2m17 `m2u
p
11 ´ 3m2pu
p
11 “ 0,
m18 ` pm3 ´ µ1qup11 ` p2m1 ´ 3m3qpup11 “ 0,
pm17 ´ 3m20q ´m2u
p
11 ` 3m2pu
p
11 “ 0,
pm18 ´ 2m21q ´ pµ1 ` 2m3 ´m1qup11 ` 3pµ1 ` 2m3 ´m1qpup11 “ 0,
pm19 ´m22q ´m16u
p
11 ` 3m16pu
p
11 “ 0. (3.28)
Ïðè ðîçùåïëåííi öi¹¨ ñèñòåìè çà u11 îòðèìó¹ìî òàêi îáìåæåííÿ íà ñòàëi:
m17 “ 0, m18 “ 0, m20 “ 0, m21 “ 0, m19 “ m22.
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 53
Äàëi, ÿêùî p ‰ 1
3 , òî ìà¹ìî ùå íàñòóïíi îáìåæåííÿ:
m2 “ 0, m16 “ 0, µ1 “ m1 ´ 2m3, p3 ´ 3pqm3 ` p2p´ 1qm1 “ 0.
Îñòàííi äâi ðiâíîñòi ìîæíà ïåðåïèñàòè òàêèì ÷èíîì:
m1 “ p3p´ 3qm3
2p´ 1
, µ1 “ p` 1
1 ´ 2p
m3.
Ó öüîìó âèïàäêó êîìïîíåíòè âåêòîðíîãî ïîëÿ Q ìàþòü íàñòóïíèé âè-
ãëÿä:
ξ0 “ p` 1
1 ´ 2p
m3t` µ0, ξ1 “ 2 ´ p
1 ´ 2p
m3x`m19t`m23, η “ m3u`m19.
Íèæ÷å íàâåäåíèé áàçèñ ìàêñèìàëüíî¨ àëãåáðè ëi¨âñüêî¨ iíâàðiàíòíîñòi
äëÿ H “ up11, p “ 0, 13 , 1:
Q1 “ Bt, Q2 “ Bx, Q3 “ tBx ` Bu,
Q4 “ pp` 1qtBt ` p2 ´ pqxBx ` p1 ´ 2pquBu.
Çàóâàæåííÿ 3.1. ßêùî p “ ´1, òî â öüîìó âèïàäêó ïðî¹êöiÿ àëãåáðè
âèùåíàâåäåíèõ âåêòîðíèõ ïîëiâ íà t-êîìïîíåíòó ìàòèìå ðîçìiðíiñòü 1,
àëå öåé âèïàäîê âêëþ÷åíî òóò äëÿ ïîâíîòè ðîçãëÿäó ñòåïåíåâî¨ íåëi-
íiéíîñòi.
Çà óìîâè p “ 1
3 ìà¹ìî ëèøå íàñòóïíå îáìåæåííÿ:
µ1 “ 2
3
m1.
Îòæå, ó âèïàäêó H “ u
1{3
11 êîìïîíåíòè âåêòîðíîãî ïîëÿ Q òàêi:
ξ0 “ 2
3
m1t` µ0,
ξ1 “ `
m2t
2 ` `
2m3 ´ 1
3m1
˘
t`m16
˘
u
` pp´m2t`m1 ´m3qx`m19t`m23q,
η “ pm2t`m3qu` p´m2x`m19q.
Ç òî÷íiñòþ äî ëiíiéíèõ çàìií îäåðæó¹ìî íàñòóïíèé áàçèñ ìàêñèìàëüíî¨
àëãåáðè ëi¨âñüêî¨ iíâàðiàíòíîñòi:
Q1 “ Bt, Q2 “ Bx, Q3 “ tBx ` Bu, Q4 “ 4tBt ` 5xBx ` uBu,
Q5 “ uBx, Q6 “ p2tu´ xqBx ` uBu, Q7 “ ptu´ xqptBx ` Buq.
k “““ 1. Ó öüîìó âèïàäêó
ξ0 “ µ0, (3.29)
54 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
äå µ0 � äîâiëüíà êîíñòàíòà. Ïîðiâíþþ÷è ç òðåòiì ðiâíÿííÿì iç (3.12),
ìà¹ìî
0 “ ´2m2t` λ20 ´ 2m3 `m1, (3.30)
çâiäêè ïiñëÿ äèôåðåíöiþâàííÿ çà çìiííîþ t îòðèìà¹ìî λ200 “ 2m2, ií-
øèìè ñëîâàìè,
λ2 “ m2t
2 `m23t`m24.
Ïiäñòàâëÿþ÷è ùîéíî íàâåäåíèé âèðàç ó (3.30), îäåðæèìî
0 “ ´2m2t` 2m2t`m23 ´ 2m3 `m1,
çâiäêè ñëiäó¹, ùî m23 “ 2m3 ´m1, òîáòî
λ2 “ m2t
2 ` p2m3 ´m1qt`m24. (3.31)
Ïiñëÿ ïiäñòàíîâêè (3.31) ó äðóãå ðiâíÿííÿ ñèñòåìè (3.12) ìà¹ìî
λ50 ` pm2t`m3qH ` p´3m2t` p2m1 ´ 3m3qqu11H 1 “ 0. (3.32)
Ïiñëÿ äèôåðåíöiþâàííÿ äâi÷i çà çìiííîþ t îòðèìà¹ìî λ5000 “ 0, ùî ñâiä-
÷èòü ïðî òå, ùî ôóíêöiÿ λ5 � êâàäðàòè÷íà, òîáòî
λ5 “ m25t
2 `m26t`m27.
Ïiäñòàâèìî öåé âèðàç íàçàä ó ðiâíÿííÿ (3.32), îäåðæèìî
2m25t`m26 ` pm2t`m3qH ` p´3m2t` p2m1 ´ 3m3qqu11H 1 “ 0.
Ðîçùåïëþþ÷è çà çìiííîþ t, ïðèõîäèìî äî ñèñòåìè:
2m25 `m2H ´ 3m2u11H
1 “ 0,
m26 `m3H ` p2m1 ´ 3m3qu11H 1 “ 0. (3.33)
Ïiäñòàâèìî òåïåð ôóíêöi¨ λ5 i λ2 ó ïåðøå ðiâíÿííÿ ñèñòåìè (3.12):
pm25t
2 `m26t`m27 ´ λ40q ´ pm2t
2 ` p2m3 ´m1qt`m24qH
` 3pm2t
2 ` p2m3 ´m1qt`m24qu11H 1 “ 0. (3.34)
Äèôåðåíöiþâàííÿ òðè÷i çà çìiííîþ t ïîêàçó¹, ùî λ40000 “ 0, ùî ñâiä-
÷èòü ïðî òå, ùî ôóíêöiÿ λ4 � êóái÷íà. Îòæå,
λ4 “ m28t
3 `m29t
2 `m30t`m31.
Ïiäñòàâëÿþ÷è ôóíêöiþ λ4 íàçàä ó ðiâíÿííÿ (3.34), ìà¹ìî
pm25t
2 `m26t`m27 ´ 3m28t
2 ´ 2m29t´m30q
´ pm2t
2 ` p2m3 ´m1qt`m24qH
` 3pm2t
2 ` p2m3 ´m1qt`m24qu11H 1 “ 0.
Ðîçùåïëþþ÷è çà çìiííîþ t, îäåðæó¹ìî òàêó ñèñòåìó:
pm25 ´ 3m28q ´m2H ` 3m2u11H
1 “ 0,
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 55
pm26 ´ 2m29q ´ p2m3 ´m1qH ` 3p2m3 ´m1qu11H 1 “ 0,
pm27 ´m30q ´m24H ` 3m24u11H
1 “ 0. (3.35)
Áà÷èìî, ùî âñi öi ðiâíÿííÿ ìàþòü çàãàëüíèé âèãëÿä (2.27) iç äåÿêèìè
ñòàëèìè ïàðàìåòðàìè a, b, c i d. Ç òî÷íiñòþ äî åêâiâàëåíòíîñòi, âèçíà-
÷åíî¨ â òåîðåìi 3.1, iñíó¹ òiëüêè òðè ìîæëèâîñòi äëÿ ðîçâ'ÿçêiâ ðiâíÿí-
íÿ (2.27), à ñàìå, eu11 , lnpu11q i up11, äå p ‰ 0, 1.
ßêùî H “ eu11 , òî iç ñèñòåì (3.33) i (3.35) îäåðæó¹ìî
ξ0 “ µ0, ξ1 “ m27t`m31, η “ m27.
Îòæå, ó âèïàäêó ôóíêöi¨ H “ eu11 îòðèìó¹ìî ìàêñèìàëüíó àëãåáðó
ëi¨âñüêî¨ iíâàðiàíòíîñòi ç òåîðåìè 3.2.
ßêùî H “ lnpu11q, òî iç ñèñòåì (3.33) i (3.35) îòðèìó¹ìî
ξ0 “ µ0, ξ1 “ m27t`m31, η “ m27.
Òàêèì ÷èíîì, çà óìîâè H “ lnpu11q áàçèñ ìàêñèìàëüíî¨ àëãåáðè ëi¨â-
ñüêî¨ iíâàðiàíòíîñòi âèçíà÷à¹òüñÿ âèïàäêîì k “ 2.
ßêùî H “ up11, p ‰ 0, 1, òî iç ñèñòåì (3.33) i (3.35) îòðèìó¹ìî
2m25 `m2u
p
11 ´ 3m2pu
p
11 “ 0,
m26 `m3u
p
11 ` p2m1 ´ 3m3qpup11 “ 0,
pm25 ´ 3m28q ´m2u
p
11 ` 3m2pu
p
11 “ 0,
pm26 ´ 2m29q ´ p2m3 ´m1qup11 ` 3p2m3 ´m1qpup11 “ 0,
pm27 ´m30q ´m24u
p
11 ` 3m24pu
p
11 “ 0. (3.36)
Ïiñëÿ ðîçùåïëåííÿ öi¹¨ ñèñòåìè çà u11 îòðèìó¹ìî íàñòóïíi îáìåæåííÿ
íà ñòàëi: m25 “ 0, m26 “ 0, m28 “ 0, m29 “ 0, m27 “ m30. Äàëi, ÿêùî
p ‰ 1
3 , òî ç ïåðøîãî é ï'ÿòîãî ðiâíÿíü ñèñòåìè (3.36) ìà¹ìî ùå íàñòóïíi
îáìåæåííÿ: m2 “ 0, m24 “ 0. ßêùî p ‰ ´1, òî ç äðóãîãî é ÷åòâåðòîãî
ðiâíÿíü ñèñòåìè (3.36) ìà¹ìî ùå íàñòóïíi îáìåæåííÿ: m1 “ 0, m3 “ 0.
Òàêèì ÷èíîì, ïðè p ‰ 1
3 ,´1 êîìïîíåíòè âåêòîðíîãî ïîëÿ Q ìàþòü
íàñòóïíèé âèãëÿä:
ξ0 “ µ0, ξ1 “ m27t`m31, η “ m27.
Îòæå, áàçèñ ìàêñèìàëüíî¨ àëãåáðè ëi¨âñüêî¨ iíâàðiàíòíîñòi âèçíà÷à¹-
òüñÿ âèïàäêîì k “ 2.
ßêùî p “ 1
3 , òî áàçèñ ìàêñèìàëüíî¨ àëãåáðè ëi¨âñüêî¨ iíâàðiàíòíîñòi
òàêîæ âèçíà÷à¹òüñÿ âèïàäêîì k “ 2.
ßêùî p “ ´1, òî m1 “ 2m3, é îòðèìó¹ìî ÷àñòèííèé âèïàäîê äëÿ
ñòåïåíåâî¨ íåëiíiéíîñòi (äèâ. çàóâàæåííÿ 3.1).
Îòðèìàíi ðåçóëüòàòè ïiäñóìîâàíî â íàñòóïíîìó òâåðäæåííi.
56 Ñ. Ò. Ãóðàêà, Î. Â. Ëîêàçþê
Òåîðåìà 3.3 (ðåçóëüòàò ãðóïîâî¨ êëàñèôiêàöi¨, [25]). Ïîâíèé ñïèñîê
G„-íååêâiâàëåíòíèõ pìàêñèìàëüíèõq ðîçøèðåíü ëi¨âñüêèõ ñèìåòðié ó
êëàñi (3.1) âè÷åðïóþòü òàêi âèïàäêè, ÿêi íàâåäåíî â òàáëèöi 3.1.
Òàáëèöÿ 3.1. Ðåçóëüòàòè ãðóïîâî¨ êëàñèôiêàöi¨ êëàñó (3.1).
Hpuxxq Ëi¨âñüêà àëãåáðà iíâàðiàíòíîñòi
@ xBt, Bx, tBx ` Buy
lnuxx xBt, Bx, tBx ` Bu, tBt ` p2x´ 3
2 t
2qBx ` pu´ 3tqBuy
uxx xBt, Bx, tBx ` Bu, 2tBt ` xBx ´ uBu,
t2Bt ` txBx ` px´ tuqBuy
upxx, p “ 0, 13 , 1 xBt, Bx, tBx ` Bu, pp` 1qtBt ` p2 ´ pqxBx ` p1 ´ 2pquBuy
u
1{3
xx xBt, Bx, tBx ` Bu, 4tBt ` 5xBx ` uBu,
uBx, p2tu´ xqBx ` uBu, ptu´ xqptBx ` Buqy
Òðåòié âèïàäîê òàáëèöi âiäïîâiä๠äîáðå âiäîìîìó ðiâíÿííþ Áþð-
ãåðñà ut ` uux “ uxx, ÿêå äîïóñê๠ï'ÿòèâèìiðíó àëãåáðó ìàêñèìàëüíî¨
ëi¨âñüêî¨ iíâàðiàíòíîñòi.
4. Âèñíîâêè
Âïåðøå çàäà÷à ãðóïîâî¨ êëàñèôiêàöi¨ öèõ ðiâíÿíü áóëà ðîçâ'ÿçàíà
ó ðàìêàõ êëàñè÷íîãî iíôiíiòåçèìàëüíîãî ïiäõîäó ó ðîáîòàõ Àõàòîâà�
Ãàçiçîâà�Iáðàãiìîâà òà Áîéêî�Ôóùè÷à âiäïîâiäíî. Öi äîâåäåííÿ áóëè
äîñèòü ãðîìiçäêèìè. Çàâäÿêè çàñòîñóâàííþ äåÿêèõ àëãåáðà¨÷íèõ òåõíiê
âäàëîñÿ íå ëèøå ïåðåâiðèòè òà ïiäòâåðäèòè äîñòîâiðíiñòü ðåçóëüòàòiâ,
îòðèìàíèõ ðàíiøå â ðàìêàõ êëàñè÷íîãî iíôiíiòåçèìàëüíîãî ïiäõîäó, àëå
é çíà÷íî ñïðîñòèòè òåõíi÷íi âèêëàäêè.  îáîõ âèïàäêàõ ïðèíöèïîâèì
êðîêîì áóëî ñóòò¹âå ñïðîùåííÿ äîâåäåííÿ iç çàñòîñóâàííÿì àëãåáðà¨-
÷íîãî ïiäõîäó, ÿêèé ïîëÿãàâ â àíàëiçi ïðèäàòíèõ àëãåáð ëi¨âñüêî¨ iíâàði-
àíòíîñòi. Ñïî÷àòêó áóâ âèêîðèñòàíèé âiäîìèé ðåçóëüòàò, çãiäíî ç ÿêèì
äëÿ äîâiëüíîãî âåêòîðíîãî ïîëÿ, ÿêå äîïóñêà¹òüñÿ åâîëþöiéíèì ðiâíÿ-
ííÿì, éîãî t-êîìïîíåíòà çàëåæèòü ëèøå âiä çìiííî¨ t. Ïiñëÿ öüîãî áóâ
ïðîâåäåíèé àíàëiç ðîçìiðíîñòi ïðî¹êöi¨ àëãåáðè íà öþ t-êîìïîíåíòó ç
âèêîðèñòàííÿì êëàñè÷íî¨ òåîðåìè Ëi ïðî ðåàëiçàöi¨ àëãåáð Ëi âåêòîð-
íèìè ïîëÿìè íà ïðÿìié.
Ãðóïîâà êëàñèôiêàöiÿ äèôåðåíöiàëüíèõ ðiâíÿíü 57
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Ñ. Ò. Ãóðàêà
Department of Mathematical and Statistical Sciences, University of Alberta,
Edmonton, Alberta, T6G 2G1 Canada
Email: sophiahuraka@gmail.com
ORCID: 0009-0002-8655-0478
Î. Â. Ëîêàçþê
Iíñòèòóò ìàòåìàòèêè ÍÀÍ Óêðà¨íè, ì. Êè¨â
Email: sasha.lokazuik@gmail.com
ORCID: 0000-0001-9663-251X
|
| id | oai:trim.imath.kiev.ua:article-539 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:09:03Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| resource_txt_mv | trimimathkievua/12/51371609f373b19dce9ae2764f12a212.pdf |
| spelling | oai:trim.imath.kiev.ua:article-5392026-02-24T12:22:41Z On group classification of differential equations: algebraic approach Групова класифікація диференціальних рівнянь: алгебраїчний підхід Huraka, Sofiia Lokaziuk, Oleksandra Гурака, Софія Локазюк, Олександра Using the classical Lie theorem on realizations of Lie algebras by vector fields on the line, we substantially simplify the proof of the known results on the group classification of the classes of (1+1)-dimensional nonlinear evolution equations ut=H(uxx) and ut+uux=H(uxx). Використовуючи класичну теорему Лі про реалізацію алгебр Лі век\-торними полями на прямій, суттєво спрощено доведення відомих результатів про класифікацію класів (1+1)-вимірних нелінійних еволюційних рівнянь вигляду ut=H(uxx) та ut+uux=H(uxx). Інститут математики НАН України 2025-08-18 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/539 10.3842/trim.v21n1.539 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 21 No. 1 (2024): Special Issue "Actual problems of modern mathematics: methods, models and applications"; 35-58 Сборник Трудов Института математики НАН Украины; Том 21 № 1 (2024): Спеціальний номер "Актуальні проблеми сучасної математики: методи, моделі та застосування" ; 35-58 Збірник Праць Інституту математики НАН України; Том 21 № 1 (2024): Спеціальний номер "Актуальні проблеми сучасної математики: методи, моделі та застосування" ; 35-58 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/539/522 Авторське право (c) 2024 Олександра Локазюк, Софія Гурака http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Huraka, Sofiia Lokaziuk, Oleksandra Гурака, Софія Локазюк, Олександра On group classification of differential equations: algebraic approach |
| title | On group classification of differential equations: algebraic approach |
| title_alt | Групова класифікація диференціальних рівнянь: алгебраїчний підхід |
| title_full | On group classification of differential equations: algebraic approach |
| title_fullStr | On group classification of differential equations: algebraic approach |
| title_full_unstemmed | On group classification of differential equations: algebraic approach |
| title_short | On group classification of differential equations: algebraic approach |
| title_sort | on group classification of differential equations: algebraic approach |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/539 |
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