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ISSN 2074-1871 Уфимский математический журнал. Том 9. № 3 (2017). С. 158-164.
ON INTEGRABILITY OF A DISCRETE ANALOGUE OF KAUP-KUPERSHMIDT EQUATION
R.N. GARIFULLIN, R.I. YAMILOV
Abstract. We study a new example of the equation obtained as a result of a recent generalized symmetry classification of differential-difference equations defined on five points of an one-dimensional lattice. We establish that in the continuous limit this new equation turns into the well-known Kaup-Kupershmidt equation. We also prove its integrabilitv by constructing an L — A pair and conservation laws. Moreover, we present a possibly new scheme for constructing conservation laws from L — A pairs.
We show that this new differential-difference equation is similar by its properties to the discrete Sawada-Kotera equation studied earlier. Their continuous limits, namely the Kaup-Kupershmidt and Sawada-Kotera equations, play the main role in the classification of fifth order evolutionary equations made by V.G. Drinfel'd, S.I. Svinolupov and V.V. Sokolov.
Keywords: differential-difference equation, integrabilitv, Lax pair, conservation law.
Mathematics Subject Classification: 37K10, 35G50, 39A10
1. Introduction We consider the differential-difference equation
un,t = (ul — 1) ^un+2^ul+1 — 1 — Un-2^U2n-X — , (1)
where n G Z and un(t) is the unknown function of one diserete variable n and one continuous variable t, and the subscript t denotes the time derivative. Equation (1) is obtained as a result of generalized symmetry classification of five-point differential-difference equations
Un,t = F (Un+2,Un+i ,Un,Un-i,Un-2) (2)
made in [8]. Equation (1) coincides with the equation [8, (E17)] up to a scaling of un and t .
Equations (2) play an important role in the study of four-point discrete equations on the square lattice, which are very relevant for today, see e.g. [1,5,6,15]. No relation between (1) and any other known equation of the form (2) is known. More precisely, here we mean the relations in the form of the transformations
un = <p(un+k,un+k-i,... ,un+m), k>m, (3)
and their compositions, see a detailed discussion of such transformations in [7]. The only information we have at the moment on (1) is that it possesses a nine-point generalized symmetry
P.H. Гарифуллин, P.II. Ямилов, Об интегрируемости дискретного аналога уравнения Каупа-Купершмидта.
© Гарифуллин Р.Н., Ямилов Р.И. 2017.
The research is supported by the Russian Science Foundation (project no. 15-11-20007). Поступила 12 декабря 2016 г.
of the form:
— G{un+4:,un+з, . . . , Un-4). In this article we study equation (1) in details. In Section 2 we find its continuous limit, which is the well-known Kaup-Kupershmidt equation [4,10]:
25 2
Uj- UXXXXX + 5UUXXX +
-Uxuxx + 5U Ux, (4)
where the subscripts t and x denote t and x partial derivatives. In order to justify the integrabilitv of (1), we construct an L — A pair in Section 3 and in Section 4, we show that it provides an infinity hierarchy of conservation laws. In Section 5 we discuss possible generalizations of a scheme for constructing the conservation laws, which is formulated in Section 4 for equation (1),
2. Continuous limit
In the continuous limit, most of the equations of form (2) presented in [8] turns into the Korteweg-de Vries equation. The exceptions are (1) and the following two equations:
Un,t — u2n(un+2Un+i — Un-iUn-2) — Un(un+1 — Un-i), (5)
/ , -n f un+2un{un+i + l)2 Un-2Un(Un-i + l)2 \
Un,t — (Un + 1)----+ (1 + 2un)(un+i — Un-i) ) , (6)
V Un+1 Un-1 J
which correspond to equations (E15) and (E16) in [8], Equation (5) is known for a long time [17],
Equation (6) was found recently in [2] and it is related to (5) by a composition of transformations
of the form (3), In the continuous limit, these three equations correspond to the fifth order
equations of the form:
U-y Uxxxxx + F(UXXXX, Uxxx^ U^ U^
U). (7)
There is a complete list of integrable equations (7), see [3,11,14], Two equations play the main role there, namely, (4) and the Sawada-Kotera equation [16]:
Ur — ^^xxxxx I 5 ^^ Uxxx I 5 Uxx I 5 ^^ Ux. ^
All the other are transformed into these two by transformations of the form:
U — $(U,UX,UXX,...,UX...X).
It is known [1] that in the continuous limit equation (5) becomes the Sawada-Kotera equation (8), The other results below are new. Using the substitution
2^2 yJ2 ( 9 2 \
un(t) —--!--£2U t--e5t, x + -et , x — en,
92
[t--e5t,x + - £t), x — en, (9)
V 80 ' 3 ) ' ' 1 ;
3 16 V 80 3
in equation (1), as e ^ 0 we get the Kaup-Kupershmidt equation (4),
It is interesting that equation (6) has two different continuous limits. The substitution
4 /18 4 \
un(t) = - - — e2U l t ——e51, x + -£t\, x = en, (10)
3 V 5 3 J
in (6) leads us to equation (4), while the substitution
2 / 18 4 \
un{t) = — 3 + £2U It — ye5t,x +3£t\ , x = en, (11)
gives rise to equation (8), As well as (1), equation (6) deserves further study.
In conclusion, let us present a picture that shows the link between discrete and continuous equations:
(1) (6) (5)
(9)
(4)
(10) (11)
(8)
3. L — A pair
As the continuous limit shows, the integrabilitv properties of equation (1) should be close to those of equation (5), Following the L — A pair [1, (15,17)], we look for an L — A pair of the form:
Ln^n = 0, ^n,t = An^n (12)
with the operator Ln of the form:
T = j(2)T 2 + j(1)T + j(0) + j(-1)T-1
where ln\ k = —1, 0,1, 2, depend on finitely many functions un+j. Here T is the shift operator: Thn = hn+1. In this case the operator An can be chosen as
Ar,
Xr.-. I I Oir.] I Oir.] I .
The compatibility condition for the system (12) is
d(Ln^n)
dt
(Ln,t + LnAn)^n = 0
(13)
and it must be satisfied on virtue of equations (1) and Ln^n = 0,
If we suppose that the coefficients depend on un only, as in [1], we can see that a^ depend on un-1 ,un only. However, in this case the problem has no solution. This is why we proceed to the case when the functions depend on un, un+1. Then the coeffieients an) must depend on un-1 ,un,un+1. In this case we succeeded to find the operators Ln and An with one irremovable arbitrary constant A playing the role of a spectral parameter:
n
UnA I
Ar.
1T2 + un+1T + \(^un — un+1^u2n — 1T ^ , (14)
(Vui — 1(un+1T + un— 1T-1) — X~1 un- 1T + \un+1T-1] . (15) un v /
The L — A pair (12,14,15) can be rewritten in the standard matrix form with 3 x 3 matrices
L n, An.
^n +1 Ln^n, ^n,t
Here a new spectral function is given by
A V
■ri n * n-
Vr.
V ^n-1
and the matrices Ln, An read:
L r,
'U:
un
x ^^Fï-1 \
«n+l U„
00
1
0
(16)
/
—n
2
.
0
Ar
( a-1 - v^r-T-r \
-A-1un-1 0 Xun+1^^+un-l{ul-i) . (17)
\un + A-lUn-2^U2n-1 - 1 UnUn-1 A + u-ry/un-1 - 1 /
In this case, unlike (13), the compatibility condition can be represented in matrix form:
L n,t An+iL n L n-An
without using the spectral function
L - A
pairs [5,9,12], However, we do not see how to apply those methods in case of matrices (16) and (17), In the next section, we shall use a different scheme for constructing conservation laws L - A
4. Conservation laws
L - A
pair. The operator Ln depends linearly on A:
Ln = Pn — AQn, (18)
where
Pn = Un\JU2n+1 - 1T2 + Un+1 T, Qn = Un+1 \Ju2n - 1T-1 - Un.
Introducing Ln = Q-1Pn, we get an equation of the form:
L npn = Alpn. (19)
The functions Apn and A-1pn in the second equation of (12) can be expressed in terms of Ln and pn by using (19) and its consequence A-1pn = L-1pn. As a result we have:
pn,t = Anpn, (20)
where
-4n =^Un 1 (^/nT - 1(Un+1T + Un-1T-1) - Un-1TP-1Qn + Un+1T-1Q-1Pn) .
Un V /
It is important that new operators Ln and An in the L - A pair (19,20) are independent of
A
operator form without using p-function:
Ln,t = -A^nLn - LnAn = [An,Ln], (21)
L - A
well-known Lax pairs for the Toda and Volterra equations is that now the operators Ln and An are nonlocal. Nevertheless, using the definition of inverse operators being linear:
PnP-1 = P-1Pn = 1, QnQ-1 = Q--1Qn = 1, (22)
by straightforward calculations we can check that (21) holds true.
The conservation laws of equation (1), which are expressions of the form
pT) = (T - 1)ank), 0,
can be derived from the Lax equation (21), notwithstanding nonlocal structure of the operators Ln, Am see [18], For this we must, first of all, represent the operators Ln, An as formal series in powers of T-1:
Hn = ^ hnk)Tk. (23)
Ln
k<N
Formal series of this kind can be multiplied according the rule:
(dnTk )(bnT) = anbn+k Tk+. The inverse series can be obtained by definition (22), for instance:
Q-1 = -(1 + QnT-1 + (QnT-1)2 + ... + (QnT-1)k + ...) -, qn = ^ VUn-1.
Un Un
The series Ln has the second order:
Ln = ^ ln]Tk = (\J Un+1 - 1T2 +Un+1UnT + Un+1Un-1^U2n - 1 + ...).
The conserved densities pT^ of equation (1) can be found as:
piV — log in2), p™ =resL*, 1, (24)
where the residue of formal series (23) is defined by the rule:
r6S Hn — h2); S66 [18]. The corresponding functions a^ can easily be found by direct calculations.
In this way below we find the conserved densities p^ and then we simplify in accordance with the rule:
P(nk) = CkPnk) + (T - 1) 9nk),
k
№ = log(Un - 1),
pT) = Un+1 Un-1\jU2n - 1,
pn) = (uT - 1)(2uT+2uT-2\JuT+1 - ^UT-1 - 1 + UT+1UT-1)
+ Un+1Un-1Un^uT - 1(Un+2\JU2n+1 - 1 + Un-2^U2n-1 - 1).
5. Discussion of the construction scheme
In the previous section we have outlined the scheme for constructing the conservation laws by example of equation (1), It can easily be generalized for the equations of an arbitrarily high order:
Un,t = F (ut+M ,Un+M-1, . . . ,Un-M).
L - A A Ln
and let the operators Pn, Qn of (18) have the form:
k2
Rn = ^ r(nk)Tk, h ^ k2 e Z, (25)
k=ki
( k )
with the coefficients rn ) depending on finitely many functions un+j. We suppose that
Ln Q- Pn "y ] lT )T
k<N
has a positive order N ^ 1. If N ^ —1, then we ehange A ^ X-1 and introduce Ln = P-1Qr of a positive order. In the ease N = 0, the scheme does not work.
As Xk^n = Ln^n for any integer fc, we can consider operators An of the form:
m-2
An ank) №, m1 ^ m2 G Z,
k=mi
where a^ [T] are operators of the form (25), Then we can rewrite An as
m-2
An = £ ank)iT]Lkn = £ a{nk)Tk.
k=mi ^N
We are led to Lax equation (21) with Ln,An of form (23) and, therefore, we can construct the conserved densities as written above, namelv, according (24) with the onlv difference p^0 =
log e i
It should be remarked that the scheme can easily be applied to equation (5) with the L — A pair [1, (15,17)].
In a quite similar way this scheme can also be applied in the continuous case, namely, to PDEs of the form
^t F(U, Uxi ^xxy . . . ■) ^x...x).
We consider the operators (25) with Dx instead of T, which become the differential operators, where Dx is the operator of total ^-derivative. Besides, k2 ^ k1 ^ 0 and the coefficients r^
depend on finitely many functions u,ux,uxx,____Instead of (23) we consider the formal series
in powers of D-1. A theory of such formal series and, in particular, the definition of the residue were discussed in [13],
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Garifullin Eustem Nailevieh,
Institute of Mathematics, Ufa Scientific Center, EAS, Chenrvshevskv str. 112, 450008, Ufa, Eussia E-mail: [email protected]
Yamilov Eavil Islamovich,
Institute of Mathematics, Ufa Scientific Center, EAS, Chenrvshevskv str. 112, 450008, Ufa, Eussia E-mail: [email protected]
