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ISSN 2074-1863
???????? ?????????????? ??????. ??? 6. ? 1 (2014). ?. 115-120.
? ????????? ?????????? ??????
?????????? ?????????
?.?. ?????????
?????????. ? ?????? ??????????? ????????? ?????????? ?????? ?????????? ?????????. ??????????????? ?????? ?????????? ??????? ????, ??????????????? ??????? ??
????? A2 .
???????? ?????: ??????? ?????????? ?????????, ?????? ????? ??????????.
Mathematics Subject Classification: 35Q53, 37K10
1.
????????
????????? ?????? ????????? ???????? ????????? ?????????? ??????? ????????? ??
?????????? ??????? ?????? ????:
ui1,1 = f i (n, m, u, u1,0 , u0,1 ),
i = 1, 2, . . . , N,
(1)
??? u = u(n, m) ??????-??????? ???? ?????????? ??????????, ???????????? ?? CN :
u = (u1 , u2 , . . . , uN )T . ?????? ??????? ?????????? ???????? ?????? ?????????? ???????:
q
u(n, m) = u(n + p, m + q), ??? Dm , Dn ????????? ??????. ????? ????????????
up,q = Dnp Dm
?????????? ???????? ? ???? ?????????? u ? ?? ?????? up,0 , u0,q , ??? p, q ? Z. ????????
?????, ??? ??????? ????????? (1) ????????? ???????????? ?????????? u?1,?1 , u?1,1 , u1,?1 .
PN ?I 2
6= 0 (???????
??????????? 1. i) ??????? I(n, m, u, u1,0 , u2,0 , . . . , uk,0 ), i=1 ?ui
k,0
2
P
?F
F (n, m, u, u0,1 , u0,2 , . . . , u0,l ), N
6= 0) ?????????? m-?????????? (???????i=1 ?ui
0,l
??????? n-??????????) ??????? ????????? (1), ???? ??????????? ????????? Dm I = I
(??? Dn F = F ).
ii) ????????? ???? I = I(n) (F = F (m)) ?????????? ????????????.
iii) ??????? ?????????, ??????????? N ????????????? ??????????? ?????????? ? ?????? ???????????, ?????????? ????????????? ?? ?????.
iv) ????????? ?????????? ????????????, ???? ?? ???? ?? ??? ?? ?????????? ?????
????????? ? ?? ??????.
??????? ??????????????? ?? ????? ???????????????? ????????? ????????????????
???? ? ?????? ????? ????????? ??????? ????????? ? ??????? ?????? ??????????? [1],
[2]. ? ??????? [3], [4] ????????, ??? ??????????????? ??????? ???????????? ?? ????? ????? ? ?????? ?????, ????? ?? ?????????????????? ?????? ?? ????? ???????? ???????????
(??. ????? [5]). ? ??????? [6], [7] ??????? ??????? ??????????????????? ?????? ??????????? ????????? ? ? ??????? ????? ??????? ????????? ????????????? ?????????????
?? ????? ???????????????-?????????? ????????? ???? u1,x = f (u, u1 , ux ). ? ?????? [8],
[9] ????????? ???????? ?????????? ??????? ?????? ?????????? ??????????????? ???????.
M.V. Yangubaeva, On structure of integrals for systems of discrete equations.
c ????????? ?.?. 2014.
?????? ?????????? ???? (?????? 13-01-00070-?, 14-01-97008 ?-????????-?).
????????? 25 ??????? 2013 ?.
115
116
?.?. ?????????
????????? ?????????? ???????????????? ????????? ???? ??????? ? [5]. ? ??????? [10],
[11] ??????????? ????? ??????????????? ?? ????? ? ?????????? ???? ??????????? ??????? ???????????????? ?????????. ?????? [12], [13] ????????? ???????? ?????????????
?? ????? ?????????? ????????? ? ?????? ????????????? ???????.
1.1. ??????? ??????? ?????? ?????????? ?????? ?????????? ?????????.
???????? ????????? I = I(n, m, u, u1,0 , u2,0 , . . . , uk,0 ) ??????? ?????????? ????????? (1)
?I
?I
?????????? ????? k. ??????????????, ??? ?????????? ????? i, j, ??? ?u
6= 0.
i 6= 0,
?uj
k,0
????? ??????? (1) ????? N ??????????? m-?????????? I 1 , I 2 , . . . , I N ??????????? ???????? k1 6 k2 6 . . . 6 kN , (ordI i = ki ). ??? ??????, ??? ????????? ????????? ???????
1. Dm I(n, m, u, u1,0 , . . . , uk,0 ) = I(n, m, u, u1,0 , . . . , uk,0 ), k 6 k1 ????? ? ?????? ?????, ?????
I = I(n);
2. Dm I(n, m, u, u1,0 , . . . , uk,0 ) = I(n, m, u, u1,0 , . . . , uk,0 ), k ? [kj?1 , kj ), j ? [2; N ] ????? ?
?????? ?????, ????? I ???????? ????????, ????????? ?????? ?? n, I 1 , I 2 , . . . , I j?1 ? ??
??????? ?? n.
??????????? ???????????:
????? 1. ????? ??????? ????????? (1) ????? N ??????????? m-?????????? ???????????? ??????? k. ????? ????? ?????? m-???????? ???????? ???????? ?????????? n,
?????????? I 1 , I 2 , . . . , I N ? ?? ??????? ?? n.
??????????????. ????? ????????? ???????
?I 1
?u1k,0
6= 0. ???? ??? ?? ???, ?? ????????
?????????? ????? ??????? ???????????????? ??????????. ??????? ?????????? u1k,0 ?????
????????? ?????????? ? ???????? I 1 :
1
u1k,0 = g 1 (n, m, u, u1,0 , . . . , uk?1,0 , u2k,0 , . . . , uN
k,0 , I ).
(2)
????? ???????? ?? ?????? ?????????? n, m, u, u1,0 , . . . , uk,0 ? ?????? ?????? ???????1
s
??? n, m, u, u1,0 , . . . , uk?1,0 , u2k,0 , . . . , uN
k,0 , I . ? ?????????? ??? ????????? I , s = 2, 3, . . . , N
1
??????????? ? ???? I s = I s (n, m, u, u1,0 , . . . , uk?1,0 , u2k,0 , . . . , uN
k,0 , I ). ????? ?????
2
PN
?I 2
6= 0. ?????????????, ? ????????? ?????? ?????? ?? ?????????? ?????????
i=2 ?ui
k,0
?????? ????? ???? ?? ????????? ? ??????????? ????? I 1 = i0 , ??? i0 ? ????????? ??????????? ?????, ? c???????? ???
?
X
2
I =
Ij (n, m, u, u1,0 , . . . , uk?1,0 )(I 1 ? i0 )j .
(3)
j=0
??? ??? I 1 , I 2 ???????? m-???????????, ?? ??? ???????????? ???? (3) ????? ????? m??????????? ??????? ?? ?????????????? k ? 1. ?????????????, ?????
!
?
X
Dm I 2 = Dm
Ij (n, m, u, u1,0 , . . . , uk?1,0 )(I 1 ? i0 )j ,
j=0
Dm I 2 =
I2 =
?
X
j=0
?
X
Dm Ij (n, m, u, u1,0 , . . . , uk?1,0 )Dm (I 1 ? i0 )j ,
Dm Ij (n, m, u, u1,0 , . . . , uk?1,0 )(I 1 ? i0 )j .
(4)
j=0
????????? ???? (3) ? (4) ? ????????? ??????????????? ?????????? ? ??? ???????, ???????? ?????????
Dm Ij (n, m, u, u1,0 , . . . , uk?1,0 ) = Ij (n, m, u, u1,0 , . . . , uk?1,0 ),
? ????????? ?????????? ?????? ?????????? ?????????
117
? ???????? I 2 ???????? ???????? ?? ?????????? n, I 1 , ??? ???????????? ?????????? ????????????? ???????????????? ?????? ??????????.
?I 2
?????????? ?????????? ???????????? ????? ???????, ??? ?u
6= 0. ????? ???????2
k,0
??? u2k,0 ??????? ????? ?????????? ?????????? ? ????????? I 1 , I 2
1 2
u2k,0 = g 2 (n, m, u, u1,0 , . . . , uk?1,0 , u3k,0 , . . . , uN
k,0 , I , I ).
(5)
1
?? ?????????? n, m, u, u1,0 , . . . , uk?1,0 , u2k,0 , . . . , uN
k,0 , I ???????? ? ?????? ??????????
1 2
s
n, m, u, u1,0 , . . . , uk?1,0 , u3k,0 , . . . , uN
k,0 , I , I ? ?????????? I , s = 3, 4, . . . , N .
????????? ???? ??????????? ???? ?? ?????? ?????????? u3k,0 ? ????????? I 3 , ?????
P ?I 3 2
?I 3
6= 0, ???????
6= 0. ??????????? ?u
????????, ??? N
3
i=3 ?ui
k,0
k,0
u3k,0
= g (n, m, u, u1,0 , . . . , uk?1,0 , u4k,0 , . . . , upk,0 , I 1 , I 2 , I 3 ).
3
(6)
?? i-?? ???? ??????? ???????????
N
1
i
uik,0 = g i (n, m, u, u1,0 , . . . , uk?1,0 , ui+1
k,0 , . . . , uk,0 , I , . . . , I ), i 6 N.
(7)
? ?????????? ??????????? ?????????????? ???????? ? ????????
uik,0 = g? i (n, m, u, u1,0 , . . . , uk?1,0 , I 1 , . . . , I N ), i = 1, 2, . . . , N.
(8)
???????? ???????? ?????? Dnj ? ????????? ????????????, ?????, ???
uik+j,0 = ?i (n, m, u, u1,0 , . . . , uk?1,0 , I 1 , . . . , I N ,
Dn I 1 , . . . , Dn I N , . . . , Dnj I 1 , . . . , Dnj I N ),
i = 1, 2, . . . , N,
j = 1, 2, . . . .
(9)
????? G ? ???????????? m-???????? ??????? q ? k:
G = G(n, m, u, u1,0 , . . . , uq,0 ).
?? ?????????? n, m, u, u1,0 , . . . , uq,0 ???????? ? ?????????? n, m, u, u1,0 , . . . , uk?1,0 , I 1 , . . . ,
I N , Dn I 1 , . . . , Dn I N , . . . , Dnq?k I 1 , . . . , Dnq?k I N ? ? ??????????? ?????
(1)
(2)
(N )
(1) (1)
(2) (2)
(N ) (N )
? , ?1,0 , . . . , ?q?k,0 , ? , ?1,0 , . . . , ?q?k,0 , . . . , ? , ?1,0 , . . . , ?q?k,0 ,
??????? G ???????? ? ????????? ???:
?
X
G=
?
X
?[1,0]+?[1,1]+иии+?[1,q?k]=0 ?[2,0]+?[2,1]+иии+?[2,q?k]=0
...
?
X
?[N,0]+?[N,1]+иии+?[N,q?k]=0
?[1,1]
?[1,0] (1)
G?1 ?2 ...?N (n, m, u, u1,0 , . . . , uk?1,0 ) I 1 ? ? (1)
. Dn I 1 ? ?1
...
?[1,q?k]
?[2,1]
?[2,0] (1)
(2)
Dnq?k I 1 ? ?q?k
. I 2 ? ? (2)
. Dn I 2 ? ?1
...
?[2,q?k]
?[N,q?k]
?[N,0]
(2)
(N )
. I N ? ? (N )
. . . Dnq?k I N ? ?q?k
,
Dnq?k I 2 ? ?q?k
??? ?j = (?[k, 0], ?[j, 1], ?[j, 2], . . . , ?[j, q ? k]) , j = 1, 2, . . . , N. ??? ??? ??????? G, I 1 , . . . ,
I N , Dn I 1 , . . . , Dn I N , . . . , Dnq?k I 1 , . . . , Dnq?k I N ???????? m-???????????, ?? ??? ???????????? ????? ???? ?????? ???? ????? m-??????????? ???????, ?? ?????????????? k ? 1. ???
??? ????? k ???????? ??????????? ???????? ?????????, ?? G?1 ?2 ...?N ???????? ????????
?? ?????????? n.
? ?????????? ????????, ??? ???????????? m-???????? G ?????????? ????? ?????????
????????? ?????? ? ?? ?????? ?? n
G = G(n, I 1 , . . . , I N , Dn I 1 , . . . , Dn I N , . . . , Dnq?k I 1 , . . . , Dnq?k I N ).
????? ????????.
118
?.?. ?????????
????? ??????? ??????? ? ????????? ?????????? ??????? (1) ? ????? ??????.
??????? 1. ????? ??????? ????????? (1) ????? p ??????????? m-?????????? I j ,
j = 1, 2, . . . , N ??????????? ???????? k1 6 k2 6 . . . 6 kN . ????? ????? ?????? m-????????
???????? ???????? ?? ?????????? n, ?????????? I 1 , . . . , I N ? ?? ??????? ?? n.
??????????????. ???????? K = kN . ???????? ????????? I j , j = 1, 2, . . . , N ?????????
K?k
?????? ? ?????? ??????? K, ???????????? ?? ??? ?????????? ?????? Dn j ? ????????
I j , ???????
DnK?kj I j + I j = I?j (n, m, u, u1 , . . . , uK?k ,0 , uK?k +1,0 , . . . , uK,0 ), j = 1, 2, . . . , N ? 1. (10)
j
j
????? ???????, ?? ????? N ?????????? I?1 , I?2 , . . . , I?N ?1 , I N ?????? ??????? K. ???????,
??? ??????????? ????????? ???????? ????????????. ?????, ??????? ??????????? ???????????, ??? ? ? ?????????????? ????? 1, ????????, ??? ?????????? uiK,0 , i = 1, 2, . . . , N
????? ???????? ????? ???????????? ?????????? u, u1,0 , . . . , uK?1,0 ? ????? ??????????
I?1 , I?2 , . . . , I?N ?1 , I N
uiK,0 = g i (n, m, u, u1,0 , . . . , uK?1 , I?1 , I?2 , . . . , I?N ?1 , I N ), i = 1, 2, . . . , N
???, ???????? (10),
uiK,0 = g i (n, m, u, u1,0 , . . . , uK?1 , I 1 , . . . , I N ?1 , I N , DnK?k1 I 1 , . . . , DnK?kN ?1 I N ?1 ),
i = 1, 2, . . . , N.
???????? ???????? ?????? Dnr ? ????????? ????????????, ??????? ???????
uiK+r,0 = ? i (n, m, u, u1,0 , . . . , uK?1,0 , DnK?k1 I 1 , DnK?k1 +1 I 1 , . . . , DnK?k1 +r I 1 ,
DnK?k2 I 2 , DnK?k2 +1 I 2 , . . . , DnK?k2 +r I 2 , . . . , DnK?kN ?1 I N ?1 , DnK?kN ?1 +1 I N ?1 , . . . ,
r N
DnK?kN ?1 +r I N ?1 , I N , Dn I N , . . . , Dm
I ),
i = 1, 2, . . . , N,
k = 1, 2, . . . .
????? G ? ???????????? m-???????? ??????? s ? K:
G = G(n, m, u, u1,0 , u2,0 , . . . , us,0 ).
?????, ??? ? ????, ?? ?????????? n, m, u, u1,0 , u2,0 , . . . , us,0 ???????? ? ??????????
n, m, u, u1,0 , u2,0 , . . . , uK?1,0 , DnK?k1 I 1 , DnK?k1 +1 I 1 , . . . , DnK?k1 +r I 1 , DnK?k2 I 2 , DnK?k2 +1 I 2 , . . . ,
K?k
K?k
+1
K?k
+r
r N
DnK?k2 +r I 2 , . . . , Dn N ?1 I N ?1 , Dn N ?1 I N ?1 , . . . , Dn N ?1 I N ?1 , I N , Dn I N , . . . , Dm
I .?
(1)
(1)
(1)
(2)
(2)
(2)
(N ?1)
??????????? ????? (?K?k1 ,0 , ?K?k1 +1,0 , . . . , ?r?k1 ,0 , ?K?k2 ,0 , ?K?k2 +1,0 , . . . , ?r?k2 ,0 , . . . , ?K?kN ?1 ,0 ,
(N ?1)
(N ?1)
(N )
(N )
?K?kN ?1 +1,0 , . . . , ?s?kN ?1 ,0 , ? (N ) , ?1,0 , . . . , ?s?K,0 ) ????????? ??????? G ?????????? ? ????
?????????? ????
?
?
?
X
X
X
G=
...
?[1,K?k1 ]+иии+?[1,s?k1 ]=0
?[N ?1,K?kN ?1 ]+иии+?[N ?1,s?kN ?1 ]=0 ?[N,0]+?[N,1]+иии+?[N,s?K]=0
DnK?k1 I 1
(1)
?K?k1
?[1,K?k1 ]
G?1 ?2 ...?N (n, m, u, u1,0 , . . . , uK?1,0 )
?
.
?[1,K?k1 +1]
?[1,s?k1 ]
(1)
(1)
DnK?k1 +1 I 1 ? ?K?k1 +1
. . . Dns?k1 I 1 ? ?s?k1
...
?[N ?1,K?kN ?1 ] ?[N ?1,K?kN ?1 +1]
(N ?1)
(N ?1)
DnK?kN ?1 I N ?1 ? ?K?kN ?1
DnK?kN ?1 +1 I N ?1 ? ?K?kN ?1 +1
...
?[N ?1,s?kN ?1 ]
?[N,1]
?[N,0] (N ?1)
(N )
Dns?kN ?1 I N ?1 ? ?s?kN ?1
. I N ? ? (N )
. Dn I N ? ?1
...
?[N,s?K?1] ?[N,s?K]
(N )
(N )
Dns?K?1 I N ? ?s?K?1
. Dns?K I N ? ?s?K
.
????? ?j = (?[k, K ? kj ], ?[j, K ? kj + 1], . . . , ?[j, s ? kj ]) , j = 1, 2, . . . , N ? 1,
?N = (?[N, 0], ?[N, 1], . . . , ?[N, s ? K]). ??? ??? G ? ??? ??????? DnK?k1 I 1 , DnK?k1 +1 I 1 , . . . ,
? ????????? ?????????? ?????? ?????????? ?????????
K?k
K?k
119
+1
Dns?k1 I 1 , DnK?k2 I 2 , DnK?k2 +1 I 2 , . . . , Dns?k2 I 2 , . . . , Dn N ?1 I N ?1 , Dn N ?1 I N ?1 , . . . ,
s?k
Dn N ?1 I N ?1 , I N , Dn I N , . . . , Dns?K I N ???????? m-???????????, ?? ??? ???????????? ?????
???? ?????? ???? ????? m-??????????? ??????? ?? ?????????????? K ? 1. ?? ??????????? ?????????? ??????????? ???????? ????????, ??? G?1 ?2 ...?N ???????? ???????? ??
?????????? n, I 1 , I 2 , . . . , I N ?1 , I N ? ?? ??????? ?? n.
????? ???????, ???????????? m-???????? G ????? ???
G = G(n, I 1 , . . . , I N , Dn I 1 , . . . , Dn I N , . . . , Dns?k1 I 1 , . . . , Dns?kN I N ),
?, ?????????????, ????????? I 1 , I 2 , . . . , I N ???????? ?????? ?????.
??????? ????????.
1.2. ?????? ??????? ?????????? ?????????, ??????? ?????? ??????
??????????.
?????????? ?????????? ??????? ????, ??????????????? ??????? ??????? ?? ????? A2
a0,0 a1,1 ? a1,0 a0,1 = b1,0 ,
b0,0 b1,1 ? b1,0 b0,1 = a0,1 ,
(11)
????? a = a(n, m), b = b(n, m) ? ??????????? ??????? ???? ?????????? ?????????? n, m.
? ?????? [14] ???? ????????? ?????????????????? ?????? ?? Lm , ??????? ?????
X1 =
?
,
?a0,?1
X2 =
?
,
?b0,?1
!
?
a
b
b
1
?
?
1,0
1,0 0,?1
?1
Dm =
+
?
+
+
Y1 = Dm
?a0,1
?a0,0
a0,0 a0,0 b0,0 a0,?1 a0,?1 b 0,0 ?a1,0
a?1,0
b0,?1
1
a?1,0
1
?
?
?
+
+
+ иии +
?
+
+ ...,
a0,0
a0,0 a0,?1 ?a?1,0
b0,?1 ?b1,0
a0,0 b0,?1 a0,0 a0,?1 ?b?1,0
?
b1,0
a0,0
?
b?1,0
a?1,0
?
?1 ?
Dm =
+
?
+
+
+ ...,
Y2 = Dm
?b0,1
?b0,0
b0,0 b0,0 b0,?1 ?b1,0
b0,0
b0,0 b0,?1 ?b?1,0
P1 = [X1 , Y1 ], P2 = [X2 , Y1 ], P3 = [X2 , Y2 ]. ? ?????? ?? ?????????? ???????? ?????????,
?????????? ?????????? ? ?????????????? ????????, ? ????? ??????? ?????????
X1 (F ) = 0,
X2 (F ) = 0,
P1 (F ) = 0,
Y1 (F ) = 0,
P2 (F ) = 0,
Y2 (F ) = 0,
P3 (F ) = 0.
(12)
??? ??????? ??????? (12) ?????????? ???????, ??? F ??????? ?? b0,0 , a0,0 , b1,0 , a1,0 , b2,0 , a2,0
??? ?? a0,0 , b1,0 , a1,0 , b2,0 , a2,0 , b3,0 . ???? ????????????, ??? F ??????? ?? ???????? ?????
??????????, ?? ??????? ?????? ??????????? ?????????. ? ?????????? ?????????? ??? ??????? ???????? ???????????????? ????????? ? ??????? ??????????? ??????? ???????.
????? ??? ???????, ?? ??????? ??? ??????????? m-????????? ?????? ???????
I1 =
b0,0 a0,0 b2,0 a2,0
+
+
,
b1,0 a1,0 b1,0 a1,0
I2 =
a0,0 a2,0 b1,0 b3,0
+
+
.
a1,0 a1,0 b2,0 b2,0
????? ???????, ?????? ?????? ????? m-?????????? ??????????? ????????. ??????? ??????? 1 ?????????, ??????? ????? ?????? m-???????? ???????? ????????, ????????? ??
n, I 1 , I 2 ? ?? ???????.
????? ?????????? ?.?. ??????????? ?? ?????????? ?????? ? ?????????? ???????? ?
??????.
120
?.?. ?????????
?????? ??????????
1. G. Darboux Lecons sur la the?orie ge?ne?rale des surfaces et les applications geometriques du calcul
infinitesimal. Gauthier-Villars. Paris. 1896. V. 1?4.
2. E. Goursat Recherches sur quelques e?quations aux de?rive?es partielles du second ordre // Annales
de la faculte? des Sciences de l?Universite? de Toulouse 2e se?rie. V. 1. ? 1. 1899. P. 31?78.
3. ?????? ?.?., ??????? ?.?., ????? ?.?. ?????? ?????????? ????????? ? ??????? ??????????????? ????????? ???????????? ?????? // ???. ?. 51, ? 1. 1982. ?. 10-?21.
4. ????? ?.?., ?????? ?.?. ???????????????? ??????? ???? I ? ??????? ??????? // ????????. ?????????? ?????? ?? ????. ???. 1981. 22 ?.
5. ????? ?.?., ????????? ?.?., ?????????? ?.?., ????? ?.?. ?????????????????? ?????? ??
? ????????? ????????????? ?????????. ?. ??????: ???????? ???????????? ????????????.
2012. 376 ?.
6. I. Habibullin, N. Zheltukhina, A. Pekcan On the classification of Darboux integrable equations //
Journal of Mathematical Physics. V. 49, ? 102702. 2008.
7. I. Habibullin, N. Zheltukhina, A. Pekcan Complete list of Darboux integrable chains of the form
t1,x = tx + d(t, t1 ) // Journal of Mathematical Physics. 2009. ?. 50, ? 1.
8. ????? ?.?., ??????? ? ?????????????????? ????????? ????????????? ??????????????? ??????? ????????? // ??????? ?????. ???. ?.6, ? 2 (13). 2005. ?. 26?34.
9. ????? ?.?., ?????????? ?.?. ?????????????????? ??????? ?????????? ??????????????? ?????? ????????? // ????. ???. ???. ?????. ? ???. 2010. ?. 3, ? 2. ?. 173-?184.
10. ????? ?.?., ??????? ?.?. ? ?????????? ???????? ????????? ???????? // ???. 1999. ?. 121,
? 2. ?. 271-?284.
11. ????? ?.?., ??????? ?.?. ????? ????????????? ??????????????? ????????? ??????????????
???? // ???. 2001. ?. 56, ? 1 (337). ?. 63-?106.
12. R.N. Garifullin, R.I. Yamilov Examples of Darboux Integrable Discrete Equations Possessing First
Integrals of an Arbitrarily High Minimal Order // ???????? ?????????????? ??????. 2012. ?. 4,
? 3. ?. 177?183.
13. R.N. Garifullin, R.I. Yamilov Generalized symmetry classification of discrete equations of a class
depending on twelve parameters // J. Phys. A: Math. Theor. 2012. V. 45, ? 345205.
14. R.N. Garifullin, I.T. Habibullin, M.V. Yangubaeva Affine and Finite Lie Algebras and Integrable
Toda Field Equations on Discrete Space-Time // SIGMA. V. 8. 2012. 33 p.
?????? ?????????? ?????????,
???????? ??????????????? ??????????? ????????? ? ???????,
??. ?????????????, 145,
450078, ?. ???, ??????
E-mail: marina.yangubaeva@mail.ru
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