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ISSN 2074-1871 Уфимский математический журнал. Том 4. № 3 (2012). С. 177-183.
УДК 517.929
EXAMPLES OF DARBOUX INTEGRABLE DISCRETE EQUATIONS POSSESSING FIRST INTEGRALS OF AN ARBITRARILY HIGH MINIMAL ORDER
R.N. GARIFULLIN, R.I. YAMILOV
Abstract. We consider a discrete equation, defined on the two-dimensional square lattice, which is linearizable, namely, of the Burgers type and depends on a parameter a. For any natural number N we choose a so that the equation becomes Darboux integrable and the minimal orders of its first integrals in both directions are greater or equal than N.
Keywords: discrete equation, Darboux integrability, first integral.
1. Introduction
In the discrete case Darboux integrable equations with first integrals of low orders are well-known. The existence for such equations of first integrals with arbitrarily high minimal orders was an open problem up to now. In this paper we give a positive answer to this question.
The most general form of the discrete Burgers equation introduced in [1] reads:
(^n+1,m+1 ftn+1 ,m)(^n,m^n,m + ^fn,m)^n,m+1 (^n,m+1 fin,m)
* (&n+1 ,m,Un+1 ,m + ln+1 ,m),M’n,m'i &n,mi Pn ,mi ^in,m _ 0.
Here n,m are arbitrary integers, an,m, Pn,m,ln,m are known complex parameters, while un,m
is an unknown complex-valued function. Eq. (1) can be obtained by the discrete Hopf-Cole
transform (see e.g. [2])
? _ vn+1,m (2)
^n,m — (2)
^n,m
from the following non-autonomous linear equation:
^ra+1,m+1 &n,mVn+1,m + fin,mVn,m+1 + r^n,mVn,m. (3)
In the completely autonomous case, the discrete Burgers equation (1) can be rewritten in the form:
(^ra+1,m+1 £ )(u n,m + 7 )un,m+1
(4)
(^n,m+1 @ )(^n+1,m + ry')'^n,m.
This equation was known [3] earlier than (1). It has been noticed in [4] that there is one more autonomous particular case of eq. (1), namely, the equation
(^ra+1,m+1 £ )(u n,m + 7 )un,m+1
(5)
&(Un,m+1 P)(Un+1,m + 7 )^n,m
R.N. Garifullin and R.I. Yamilov, Examples of Darboux Integrable Discrete Equations Possessing First Integrals of an Arbitrarily High Minimal Order.
© R.N. Garifullin and R.I. Yamilov 2012.
This work has been supported by the Russian Foundation for Basic Research (grant numbers: 10-01-00088-a, 11-01-97005-r-povolzhie-a, 12-01-31208-mol-a) and by Federal Task Program (agreement 8499).
Поступила 12 июля 2012 г.
generalizing (4). Unlike eq. (4), the last equation (5) is related by (2) to a non-autonomous linear equation.
In this paper we consider the particular case /3 = 7 of eq. (5) which can be expressed in the following form after a rescale:
(^n+1,m+1 + 1')(^n,m 1)'U/n,m+1 ^(^n,m+1 + 1')(^n+1,m 1')^n,m,. (6)
Here a = 0 is a complex parameter. A particular case of the linear equation (3) corresponding
to eq. (6) is:
^n+1,m+1 + Vn,m+1 & n+1,m Vn,m). (7)
Our aim is to show that there exist equations of the form (6) possessing first integrals of an arbitrarily high minimal order.
Discrete equations of the form
^n+1,m+1 f ('^n,mj H,n+1,m'l ^n,m+1) , (8)
defined on the two-dimensional square lattice, are analogues of the hyperbolic equations
'Uxy F (X,y,U,Ux ,Uy). (9)
There is an example similar to eq. (6) in the class (9), see [5] and a discussion at the very end of the present paper.
2. Definitions
An equation of the form (8) is Darboux integrable if it has two first integrals W1, W2:
(T1 1)^^2 ° (^n,m+l 2 jUn,m+l2 + 1J . . . ,^n,m+k2^ (10)
(T2 1) 1 0 , 1 Wn(Un+li,mj Un+li + 1,mj . . . , ^n+ki,m). (11)
Here l1,l2,k1,k2 are integers, such that l1 < k1} l2 < k2, and T1,T2 are operators of the shift in the first and second directions, respectively: T1hn,m = hn+1,m, T2hn,m = hn,m+1. We suppose that the relations (10,11) are satisfied identically on the solutions of the corresponding equation (8). The form of W1, W2 given in (10,11) is the most general possible form. The functions un+i,m+j, with i, j = 0, are expressed in terms of un+i,m, un,m+j by using eq. (8). The dependence of W1
on un,m+j, j = 0, and of W2 on un+i,m, i = 0, is impossible. We will call W1 the first integral in
the first (or n) direction, and W2 will be called the first integral in the second (or m) direction.
It is obvious that for any first integrals W1, W2, arbitrary functions Q2 of the form
Q = Q1(W1,T±1W1,.. .,T±jl W1),
1 ^ 1 1 ^ 1 (12) Q2 = Q2(W2,T2±1W2, ..., T±n W2)
are also the first integrals. In particular, using the shifts, we can rewrite the first integrals W1, W2 of (10, 11) as:
1 — ^nm(r^n,mj Un+1,mj . . . j Un+k,m) j k ^ 1 j (13)
^^2 — Wn ,m (Un,m,Un,m+1j . . . jUn,m+l)j I ^ 1.
We assume here that ?Wl = 0, » dWl = 0, ~W2 = 0, J)W'2 = 0 for at least some n, m. The
oun,m ' ’ aun+k,m / ’ dun,m ' ’ aun,m+i ' ’
numbers k,l are called the orders of these first integrals W1,W2, respectively:
ord W1 = k, ord W2 = I.
It is clear due to (12) that the orders of first integrals of a given equation are unbounded above. We are going to construct such examples for which the minimal (or lowest) orders of their first integrals may be arbitrarily high.
3. Transition to the linear equation (7)
In spite of the fact that the transform (2) is not invertible, sometimes such transformations allow one to rewrite conservation laws and generalized symmetries from one equation to another, see e.g. [6] for the discrete-differential case. We are going to transfer first integrals from (6) to (7) and backward and to reduce in this way the problem to the case of the linear equation (7).
Here we present four lemmas on the transfer of first integrals from (6) to (7) and backwards together with the corresponding explicit formulae. More precisely, we discuss some relations between the first integrals (13) of eq. (6) and first integrals W1, W2 of eq. (7):
W1 = W^m(vn,m, vn+1,m, . . . , Vn+k,m), 1, (14)
^2 <m(vn,m, 'Vn,m+1, . . . , Vn,m+l), ^ 1
Lemma 1. If eq. (6) possesses a first integral W1 of an order k, then eq. (7) has a first
integral W1 of the order k = k + 1.
Proof. Using the transform (2), we obtain
W = n(1) ( Vn+1 ,m ^n+2,m ^n+k+1,m ^
W\ Wn,m\ , , ... , )
\ Vn,m Vn+1,m Vn+k,m /
It is clear that ~Wl = 0^ dWl— = 0 for at least some n,m, and therefore W1 is nontrivial
avn,m ' ’ avn+k+l,m I ’ ’ 1
(i.e. cannot depend on n,m only) but also it has the order k + 1. Due to (12) this order may become not minimal.
Lemma 2. If eq. (6) possesses a first integral W2 of an order I, then eq. (7) has a first integral W2 of an order I, such that 1 ^ I ^ I.
Proof. The transform (2) leads us to:
W = i (2) ( Vn+1 ,m ^n+1,m+1 ^n+1,m+l \
Vv 2 rrn I , , ... , I .
n,m
\ V n,m V n,m+1 Vn,m+l
By induction on I we can prove that
^n+1,m+l OL Vn+1,m ^n,m+l + Vn,l ^nmi Vn,m+1, . . . , Vn,m+l-1).
It is clear that
dW2 dWn)m 9 / OLnlVn+1,m — Vn,m+l + "A
\ Vn,m+l /
=0
9Vn,m+l 9^n,m+l n,m+l \ ^n,m+l
for some n,m, and therefore W2 is nontrivial. Its order is not greater than I.
Lemma 3. If eq. (7) has a first integral W1 of an order k, and W1 is linear w.r.t. v n,m, vn+1,m,..., vn+jem, then eq. (6) possesses a first integral W1 of the order k = k.
Proof. Using the transform (2) and the property (12), we obtain the relations
T1W1 a0,n+1 ,m^n+1 ,m + a1,n+1 ,m^n+2,m + . . . + ak n+1 mVn+k+1 m
W1
^^1 a0,n,mVn,m + a1 ,n,mVn+1,m + . . . + ak n m^n+k m
a0,n+1,m'^n+1,m/^n,m + a1,n+1,m'^n+2,m/'^n,m + . . . + ak n+1 m^n+k+1 m/^n,m
,m
a0,n,m + a1,n,m^n+1,m/'^n,m + . . . + ak nmPn+k m/n,i
_ a0,n+1,mUn,m + . . . + ak,n+1,mUn+k,mUn+k-1,m . . . un,m &0,n,m + ... + ak,n,mUn+k-1,mUn+k-2,m . . . Un,m
One can readily verify that J^1 = 0, d®w\ = 0 for at least some values of n, m.
Lemma 4. If eq. (7) has a first integral W2 of an order I, and W2 is linear w.r.t. vn,m, vn,m+1,..., vn m+i, then eq. (6) possesses a first integral W2 of the order I = I + 1.
Proof. By using the property (12) we obtain
T2W2 bo,n,m+1 Vn,m+1 + b1,n,m+1 Vn,m+2 + . . . + b"l,n,m+1Vn,m+1+l
yy 2 = --“--- = -------T-------------T----------------------T-----------------
2 bo,n,m^n,m + &1,n,mVn,m+1 + . . . + &k n m^n m+k
bo,n,m+1^n,m+1/f^n,m + b1,n,m+1^n,m+2/f^n,m + . . . + bjk ,n m+1^n m+1+k/'^n,m
bo,n,m + b\,n,'m^n,m+\/f^n,m + . . . + b^ ,n mPn m+k/'^n,m
= bo,n,m+1^n,m + ... + b^,n,m+1^n,m+i^n,m+i-1 . . . ^n,m b0,n,m + ... + bi,n,m^n,m+i-1^n,m+i-2 . . . ^n,m
It follows from eqs. (2) and (7) that
ry ____ ^n,m+1 ___ n ^n,m 1
^n.m &
16)
4. First integrals of the linear equation (7)
We will use some necessary conditions of the Darboux integrability derived for the discrete case in [7]. Those conditions were formulated there for autonomous equations of the form (8). We can reformulate and prove those conditions for the case of the non-autonomous linear equation
(7).
We can rewrite eq. (7) in the form
{T2 — an)(T1 + 1)Vn,m + 2&aVn,m = °. (17)
By using the discrete Laplace transformation [7]
Vn,m,j — (T1 + Op ')Vn,m,j—1, j — ~1, Vn,m,0 — '^n,m, (18)
we can introduce a sequence of unknown functions vn,mj satisfying the equations
(T2 — an+^ )(T1 + )v,n,m,j + oia+^(1 + a? )v,n,m,j = 0. (19)
The last relations are proved by induction on j. One of the necessary conditions is formulated
in terms of functions Kn,j:
Kn>] = an+i(1 + ), j — 1, Kno = 2an.
The following theorem has been taken from [7].
Theorem 1. If eq. (17) possesses a first integral W1 of an order k, then there exists k,
0 ^ k < k, such that Knjt = 0.
In a similar way, we can rewrite eq. (7) in the form
(T1 + 1)(T2 — an 1)v,n,m + &n 1(1 + a)Vn,m = 0. (20)
Using the second discrete Laplace transformation
'^n,m,j — (T2 OL ^ )Vn,m,j—1, j — ~1, Vn,m,0 — '^n,m, (21)
we can define a sequence of unknown functions vn,mj which satisfy the equations
(T1 + 1)(T2 — an-1-1)vn,m,j + an-3-1(1 + aj+1)i)n,m,j = 0. (22)
V
The last relations are proved by induction on j. The second of the necessary conditions is formulated in terms of functions Hnj:
HnJ = an-j-1(1 + aj+1), j — 0.
The following theorem has been taken from [7].
Theorem 2. If eq. (20) possesses a first integral W2 of an order I, then there exists I, 0 ^ I < I, such that Hnj = 0.
Let aN be a root of —1, such that:
aN = —1, °^N = —1, 1 ^ 3 < N. (23)
For any N — 1, such a root always exists, for example, = exp(in/N). Let us now consider eq. (7) with a = . It follows from Theorems 1 and 2 for this equation that the orders of its
any first integrals in the first and second directions must be such that:
ord W1 — N +1, ord W2 — N. (24)
On the other hand, we can construct first integrals for eq. (7) with a = an of such orders in an explicit form. Eqs. (19) and (22) with j = N and j = N — 1, respectively, take the form:
{T2 + a% )(T1 — \)Vn, m, n = 0 (T1 + 1)(T2 + 0% )Vn,m,N-1 = 0.
So we can find first integrals for these equations (25) in the first and second directions, respectively:
W1 — (—1)majT № - 1)v„,m,K,
W2 — (—l)n(Ti + a% )inm,N-1.
By using the Laplace transformations (18) and (21), we obtain first integrals for eq. (7) in the following explicit form:
W1 = (—1)ma~Nnm(T1 — 1)(T1 + a*-1 )(T1 + a*-2)... (T1 + 1)vn>m,
W2 = (—1)n(T2 + anN)(T2 — anN~N+1)(T2 — anN~N+2)... (T2 — anN~1)vn,m.
We can see that both these first integrals are nontrivial, W1 is expressed in terms of
Vn,m, Vn+1,m, ..., Vn+N+1,m, while IV2 is expressed in terms of Vn,m ,Vn,m+1,. . . , Vn,m+N. Both W1
and W2 are linear w.r.t. vn,m and its shifts. We derive that
ord IV1 = N +1, ord IV2 = N (28)
and that these orders are minimal, taking into account the property (24).
For example, if N = 1, then = — 1, and we have first integrals of the minimal orders:
W1 = ( —1)(1 n'lm(vn+2,m — ^n,m) , W2 = ( — 1)nVn,m+1 + ^n,m.
If N = 2, then aN = ±i, and the first integrals read:
1 ( ^ (^n+3,m ^n+1,m + (^n+2,m '^n,m)'),
W2 = (—1)n (Vn,m+2 + 1(®N — 1)v-n,m+1 — Q?n 1^n,m).
5. First integrals of eq. (6)
We consider here the equation (6) with a = an , where an is defined by (23). Using Lemmas 3 and 4 together with the formulae (15) and (16), we construct first integrals W1 and W2 for eq. (6), starting from the first integrals (27). Their orders are:
ord W1 = ord W2 = N + 1. (29)
From Lemmas 1 and 2 and the relations (28) it follows that the minimal orders of first integrals of eq. (6) in both directions must be greater than or equal to N. We are led to the following theorem:
Theorem 3. Eq. (6) with a = an is Darboux integrable. The minimal orders of its first integrals in both directions must be equal to N or N +1.
It follows from this theorem that there exist Darboux integrable discrete equations among equations of the form (6), such that the minimal orders of their first integrals in both directions are arbitrarily high.
For eq. (6) with a = a1 = —1, we obtain the following first integrals:
\—m Un,m(Un+2,mUn+1,m ^ f 1\n (^n,m+1 + Un,m+2)(Un,m ^
w1 = (—1)-mn ,m^n+2,m'^n+1,m_______tL, w2 = (—1)
Un+1,m Un,m 1 (^n,m + Un,m+1)(^n,m+2 + ^
In the case when a = a2 = ±i, the first integrals read:
_m Un,m (^n+3 ,m^n+2,m^n+1,m + ®2Un+2 ,m^n+1,m ^n+1,m ~ 012)
W1 = (a2)~ W2 = (a2)r
r^n+2,mr^n+1,mr^n,m + ®2Un+1 ,m^n,m ^n,m — &2
(^n,m 1')(<^2(^n,m+3 + '^n,m+2)(^n,m+1 ^ (^n,m+3 + 1%)(^n,m+2 + ^n,m+1)
(1 + U,n,m+3 )(^2(^n,m+2 + '^n,m+1)(^n,m ^ (^n,m+2 + 1')(^n,m+1 + ^n,m))
In the paper [4] a method is presented which uses the so-called annihilation operators [8] and which allows one to construct first integrals of low orders and to show that those orders are minimal. By using this method we have checked that four first integrals given just above have the minimal orders. We believe that all first integrals of eq. (6) with a = which can
be constructed by the scheme presented in this paper have the minimal orders.
As it has been said above, there is a hyperbolic equation [5] of the form (9), namely,
2N _________
^xy ^ ^ \JUxUy (30)
which is analogues to eq. (6) with a = in the sense that the minimal orders of first integrals of such equations (30) may be arbitrarily high. Unlike eq. (30), which is symmetric under the involution x o y, the discrete equation (6) is not symmetric under n o m, and its first integrals in different directions have quite different forms. The second difference is that eq. (6) is of the Burgers type with linearizing transformation (2), while linearizing transformation for eq. (30) has the form
where v(x,y) is a solution of a hyperbolic linear equation.
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Rustem Nailevich Garifullin,
Ufa Institute of Mathematics, Russian Academy of Sciences,
112 Chernyshevsky Street,
Ufa 450008, Russian Federation http://matem.anrb.ru/garifullinrn E-mail: rustem@matem .anrb.ru
Ravil Islamovich Yamilov,
Ufa Institute of Mathematics, Russian Academy of Sciences,
112 Chernyshevsky Street,
Ufa 450008, Russian Federation http://matem.anrb.ru/en/yamilovri E-mail: RvlYamilov@matem.anrb.ru
