Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation
We consider five-point differential-difference equations. We aim to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated with transformations of the first, second, and third orders....
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Інститут математики НАН України
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| Цитувати: | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation / R.N. Garifullin, R.I. Yamilov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 26 назв. — англ. |
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| author_facet | Garifullin, R.N. Yamilov, R.I. |
| citation_txt | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation / R.N. Garifullin, R.I. Yamilov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 26 назв. — англ. |
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| description | We consider five-point differential-difference equations. We aim to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated with transformations of the first, second, and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable five-point equations and new transformations have been found. Moreover, we have derived a new, completely discrete equation. There are a few non-standard transformations which are of the Miura type or are linearizable in a non-standard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.
|
| first_indexed | 2025-12-07T21:24:50Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 062, 15 pages
Integrable Modifications
of the Ito–Narita–Bogoyavlensky Equation
Rustem N. GARIFULLIN and Ravil I. YAMILOV
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences,
112 Chernyshevsky Street, Ufa 450008, Russia
E-mail: rustem@matem.anrb.ru, RvlYamilov@matem.anrb.ru
Received April 01, 2019, in final form August 14, 2019; Published online August 23, 2019
https://doi.org/10.3842/SIGMA.2019.062
Abstract. We consider five-point differential-difference equations. Our aim is to find inte-
grable modifications of the Ito–Narita–Bogoyavlensky equation related to it by non-invertible
discrete transformations. We enumerate all modifications associated to transformations of
the first, second and third orders. As far as we know, such a classification problem is solved
for the first time in the discrete case. We analyze transformations obtained to specify their
nature. A number of new integrable five-point equations and new transformations have
been found. Moreover, we have derived one new completely discrete equation. There are
a few non-standard transformations which are of the Miura type or are linearizable in a non-
standard way. We have also proved that the orders of possible transformations are restricted
by the number five in this problem.
Key words: Miura transformation; integrable differential-difference equation; Ito–Narita–
Bogoyavlensky equation
2010 Mathematics Subject Classification: 37K05; 37K10; 35G20
1 Introduction
We consider differential-difference equations of the form
v̇n = f(vn+2, vn+1, vn, vn−1, vn−2), (1)
where vn = vn(t) is an unknown function of the continuous time t and discrete integer vari-
able n, v̇n denotes the time derivative of vn, and f is a function of five variables. The oldest
and the most famous integrable example of this class is the Ito–Narita–Bogoyavlensky (INB)
equation [3, 9, 13]
u̇n = un(un+2 + un+1 − un−1 − un−2). (2)
Lately, equations of this class have been intensively studied by the generalized symmetry
method, see, e.g., [1, 2, 7, 8]. However, the problem of description of all integrable equations
of this class is far from complete. To find new integrable equations, we apply in this article an
alternative approach using non-invertible discrete transformations.
More precisely, we use non-invertible transformations of the following special form
un = g(vn+k1 , vn+k1−1, . . . , vn+k2+1, vn+k2), k1 > k2, (3)
relating (1) and (2). They transform any solution vn of (1) into a solution un of (2). Such
transformation is explicit in one direction and can be called the discrete substitution by analogy
with differential substitutions in the continuous case, see, e.g., [16, 17], where such transforma-
tions were studied. Numerous examples of transformations of the form (3) can be found in [25]
mailto:rustem@matem.anrb.ru
mailto:RvlYamilov@matem.anrb.ru
https://doi.org/10.3842/SIGMA.2019.062
2 R.N. Garifullin and R.I. Yamilov
for the Volterra and Toda type equations and in [7, 8] for the five-point equations (1). Different
methods for the construction of discrete transformations (3) are presented in [6, 24].
Equation (2) is one of the key equations of lists of integrable equations found in [7, 8] as
a result of the generalized symmetry classification of an important subclass of (1). Here we
are going to enumerate all modifications of the INB equation (2), which are associated with
transformations (3) of the orders k = 1, 2, 3, where k = k1 − k2. We will also prove that the
order of a possible transformation in this problem is restricted by the number five: k ≤ 5. This
estimate is accurate in the sense that there exist transformations for all orders 1 ≤ k ≤ 5.
Examples of transformations of any such an order will be given below. As far as we know,
a classification problem of this kind is solved for the first time in the discrete case, as for
continuous case, see, e.g., [10].
A well-known example of a transformation of the form (3) is the discrete Miura transforma-
tion [20]
un = (vn+1 + 1)(vn − 1) (4)
relating the Volterra equation and its modification. There is a class of more simple linearizable
transformations in the terminology of [6]. Such transformations are more simple than Miura
type ones in the sense that the problem of finding vn by (3), starting from a given function un,
is more easy. We analyze transformations obtained in this paper to show that the most of them
are linearizable.
As a result of the classification we obtain a number of new integrable equations and trans-
formations. Most of new equations belong to the class (1) and one of them is the discrete
quad-equation. Among new transformations found here, one is of Miura type and two are
linearizable in a non-standard way.
In Section 2 we discuss some theoretical aspects and prove a boundedness theorem for the
orders of possible transformations. In Section 3 transformations of the orders 1 and 2 are
classified together with corresponding modifications. Section 4 is devoted to transformations of
the order 3. In Section 5 we discuss in detail the most interesting examples of equations and
transformations obtained in the previous section. In conclusion we briefly summarize results
obtained in the paper.
2 Theoretical comments and results
The differential-difference equations (1) and discrete transformations (3) we consider in this
paper are autonomous, i.e., do not explicitly depend on the discrete variable n. That is why, for
brevity, we may write down equations and transformations at the point n = 0:
v̇ = f(v2, v1, v, v−1, v−2), (5)
where v = v0. Moreover, up to the shift of the discrete variable n, transformations (3) can be
rewritten in the form
u = g(vk, vk−1, . . . , v1, v), k ≥ 1, (6)
where u = u0. We also use a natural restriction
∂g
∂vk
6= 0,
∂g
∂v
6= 0. (7)
We are going to enumerate all modifications of the form (5) of the INB equation (2), which
correspond to transformations (6) of the orders k = 1, 2, 3. Such modifications are integrable
equations in the sense that they possess infinitely many conservation laws.
Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation 3
In fact, the INB equation (2) has conservation laws of an arbitrarily high order, which can
easily be constructed by using the well-known Lax representation [3] or recursive operator [26].
Modifications (5) are related to the INB equation by transformations of the form (6) and, for
this reason, also have conservation laws of an arbitrarily high order due to [25, Theorem 18].
That theorem is formulated for the Volterra type equations, but it can easily be reformulated
for the case of five-point equations (5). Conservation laws for a modification (5) are constructed
in an explicit way by using the corresponding transformation (6) and known conservation laws
of (2) [25, Section 2.7].
Let us introduce the notation hj for any function h = h(vm1 , vm1−1, . . . , vm2): hj = T jh,
where T is the shift operator defined by
T jh(vm1 , vm1−1, . . . , vm2) = h(vm1+j , vm1−1+j , . . . , vm2+j)
for any integer j, in particular, h0 = h.
If equation (5) is transformed into (2) by transformation (6), then the functions f , g must
satisfy the determining equation
Dtg = g(g2 + g1 − g−1 − g−2), Dtg =
k∑
j=0
∂g
∂vj
fj , (8)
i.e., here Dt is the operator of differentiation in virtue of (5).
Our main problem can formulated as follows: for any fixed k ≥ 1 we look for pairs of func-
tions f , g satisfying (8). It is important that the functions vj , j ∈ Z, are considered in this
problem as the independent variables. The functions f , g must identically satisfy (8) for all
values of these variables.
Below we prove that the orders of possible transformations in this problem are restricted by
the number five: k ≤ 5. This estimate is accurate in the sense that there exist transformations
for all the orders 1 ≤ k ≤ 5. Examples of transformations of the orders k = 1, 2, 3 will be given in
Sections 3 and 4, while examples of the orders k = 4, 5 are presented in both Sections 5.1 and 5.2.
Theorem 1. An equation of the form (5) cannot be transformed into the INB equation (2) by
a transformation of the form (6) for any order k > 5.
Proof. Let us rewrite condition (8) as
Dt log g = g2 + g1 − g−1 − g−2. (9)
Differentiating (9) with respect to vk+2 and dividing the result by ∂fk
∂vk+2
, one has
∂ log g
∂vk
=
∂g2
∂vk+2
/ ∂fk
∂vk+2
. (10)
We see that, if k ≥ 4, then the right hand side of this relation and therefore the function ∂ log g
∂vk
do not depend on v1, v. Hence, the function g can be represented in the form
g = g(1)(vk, vk−1, . . . , v2)g
(2)(vk−1, . . . , v1, v). (11)
Let us divide (10) by g
(2)
2
1
g
(2)
2
∂ log g(1)
∂vk
=
∂g
(1)
2
∂vk+2
/ ∂fk
∂vk+2
. (12)
4 R.N. Garifullin and R.I. Yamilov
If k ≥ 6, then both sides of this relation depend on vk+1, vk, . . . , v4 only, therefore
∂ log g(1)
∂vk
= g
(2)
2 f (1)(vk+1, vk, . . . , v4),
where f (1) is a new function. As the left hand side of (12) does not depend on vk+1, we apply
the operator T−2 ∂
∂vk+1
log and derive the following consequence
∂ log g(2)
∂vk−1
= −
∂ log f
(1)
−2
∂vk−1
= f (2)(vk−1, . . . , v2) (13)
with a new function f (2).
Differentiating the main equation (9) for g and f with respect to vk+1, we get
∂ log g
∂vk
∂fk
∂vk+1
+
∂ log g
∂vk−1
∂fk−1
∂vk+1
=
∂g2
∂vk+1
+
∂g1
∂vk+1
.
Then we differentiate the result with respect to v1
∂2g1
∂vk+1∂v1
=
∂fk
∂vk+1
∂2 log g
∂vk∂v1
+
∂fk−1
∂vk+1
(
∂2 log g(1)
∂vk−1∂v1
+
∂2 log g(2)
∂vk−1∂v1
)
.
Here ∂2 log g
∂vk∂v1
= 0 due to (11), ∂ log g(1)
∂v1
= 0 due to the definition of g(1) in (11), and ∂2 log g(2)
∂vk−1∂v1
= 0
due to (13). Therefore ∂2g
∂vk∂v
= 0, i.e.,
∂2g
∂vk∂v
=
∂g(1)
∂vk
∂g(2)
∂v
=
1
g
∂g
∂vk
∂g
∂v
= 0,
but the last equality contradicts the conditions (7). �
3 Modifications of the INB equation of the levels 1 and 2
We classify here modifications of the INB equation of the levels 1 and 2, which correspond to
transformations of the orders 1 and 2. In the classification, we use differential consequences of
the determining equation (8) for the functions f and g, as in the proof of Theorem 1, and we
are trying all possible cases.
If necessary, we can change v in an equation (5) and corresponding transformation (6) by
using the point transformation
ṽ = ϑ(v). (14)
The transformation (6) and the time t in (5) can be changed by the transformation
ũ = ηu, t̃ = t/η (15)
leaving the INB equation (2) invariant. The classification will be carried out up to these au-
tonomous point transformations (14) and (15).
An exact statement of the result will be given later. First of all, let us enumerate all possible
modifications of the first level.
List 1. Modifications of the first level.
v̇ = v(v2v1 − v−1v−2), (E1)
Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation 5
v̇ = v
(
v2
v1
+ 2
v1
v
+ 2
v
v−1
+
v−1
v−2
)
+ cv, (E2)
v̇ = v
(
v2v
κ
1 − κ2v1vκ − vvκ−1 + κ2v−1v
κ
−2
)
, (E3)
v̇ = (v2 − v1 + a)(v − v−1 + a) + (v1 − v + a)(v−1 − v−2 + a)
+ (v1 − v + a)(v − v−1 + a) + c, (E4)
v̇ = φ(v2 − v1)φ(v − v−1) + φ(v1 − v)φ(v−1 − v−2)
+ φ(v1 − v)φ(v − v−1) + c. (E5)
In equation (E3) and in all the lists below
κ =
(
1± i
√
3
)
/2, (16)
i.e., κ3 = −1. That is why one has in (E3) two cases corresponding to the signs + and −. In
equation (E5) and in all the lists below, the function φ satisfies the differential equation
φ′ =
φ− 1
φ
. (17)
This function φ can be defined as
φ(x) = 1 + ψ−1(x+ a), ψ(u) = u+ log u.
Here and below a and c are arbitrary constants. The constant c indicates the existence
of a point symmetry. For example, equation (E2) has the point symmetry vτ = v, while equa-
tions (E4) and (E5) have the point symmetry vτ = 1. Let us remind that these point symmetries
correspond to the one-parameter groups of auto-transformations v → eτv and v → v + τ , re-
spectively.
The constants a and c can be removed by using one of the following non-autonomous n- and
t-dependent point transformations
v = ṽect, v = ṽ + ct, v = ṽ − an,
where ṽ is a new unknown function. The same is true for equations of Lists 2 and 3 below,
except for equation (E27). In equations (E23)–(E25) of List 3, we can make a = 1 by the
transformation z = z̃a−n.
Transformations corresponding to the equations of List 1 are presented in List 1′.
List 1′. Transformations of the first order.
u = v1v, (T1)
u =
v1
v
, (T2)
u = v1v
κ, (T3)
u = v1 − v + a, (T4)
u = φ(v1 − v)− 1. (T5)
Here and in all the lists below, a transformation with the number (TX) corresponds to the
equation with the number (EX) for any X.
Theorem 2. If an equation (5) is transformed into the INB equation (2) by a transformation (6)
with k = 1, then up to the autonomous point transformations (14) and (15) it coincides to one
of equations (E1)–(E5) of List 1. Equations (E1)–(E5) are transformed into the INB equation
by transformations (T1)–(T5), respectively.
6 R.N. Garifullin and R.I. Yamilov
Equations (E1) and (E4) together with transformations (T1) and (T2) are known, see [7,
List 4], [8, List 3] and references therein. Transformation (T3) was discussed in [6, Section 2.2].
All these transformations (T1)–(T5) are linearizable in the terminology of [6]. Such transforma-
tions can be represented as compositions of point transformations and of linear transformations
with constant coefficients. For instance, in the case of transformations (T1)–(T3), one has
û = v̂1 + αv̂, u = eû, v = ev̂.
In the next list we enumerate all possible modifications of the INB equation of the second
level.
List 2. Modifications of the second level.
ẇ = w
(
w2
w
+
w1
w−1
+
w
w−2
)
+ cw, (E6)
ẇ = w2(w2w1 − w−1w−2), (E7)
ẇ = w
(
w2w
1+κ
1 wκ − κw1w
1+κwκ−1 + κ2ww1+κ
−1 wκ−2
)
, (E8)
ẇ = w
(
w2w
κ−1
1 w−κ +
(
1− κ2
)
w1w
κ−1w−κ−1 − κ
2wwκ−1−1 w−κ−2
)
+ cw, (E9)
ẇ = (w1 + w)(w + w−1) (w2 + w1 − w−1 − w−2) , (E10)
ẇ = (w1 − w + a)(w − w−1 + a) (w2 − w1 + w−1 − w−2 + 2a) + c, (E11)
ẇ = φ(w1 − w)φ(w − w−1) (φ(w2 − w1) + φ(w−1 − w−2)− 1)
− φ(w2 − w1)φ(w − w−1)− φ(w1 − w)φ(w−1 − w−2) + c, (E12)
ẇ = (w1 − w)(w − w−1)
(
w2
w1
− w−2
w−1
)
, (E13)
ẇ = w(w + 1)(w2w1 − w−1w−2). (E14)
In equations (E8) and (E9), κ is defined by (16), i.e., in each of these equations there are two
cases. In equation (E12) the function φ is defined by (17), while a and c are arbitrary constants.
In the next list, corresponding transformations are presented.
List 2′. Transformations of the second order.
u =
w2
w
, (T6)
u = w2w1w, (T7)
u = w2w
1+κ
1 wκ, (T8)
u = w2w
κ−1
1 w−κ, (T9)
u = (w2 + w1)(w1 + w), (T10)
u = (w2 − w1 + a)(w1 − w + a), (T11)
u = (φ(w2 − w1)− 1)(φ(w1 − w)− 1), (T12)
u =
(w2 − w1)(w1 − w)
w1
, (T13)
u = w2w1(w + 1), (T14.a)
u = (w2 + 1)w1w. (T14.b)
Theorem 3. If an equation (5) is transformed into the INB equation (2) by a transformation (6)
with k = 2, then up to the autonomous point transformations (14) and (15) it coincides to one of
equations (E6)–(E14) of List 2. Equations (E6)–(E13) are transformed into the INB equation by
transformations (T6)–(T13), respectively. Equation (E14) is transformed into the INB equation
by any of transformations (T14.a)–(T14.b).
Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation 7
Equations (E7), (E10), (E11), (E13) together with corresponding transformations were dis-
cussed in [7, 8], see also references therein. The important case (E14) with transformations
(T14.a)–(T14.b) has been found [19, equation (17.6.24)].
All the transformations of List 2′ except for (T14.a) and (T14.b) are linearizable. For trans-
formations (T6)–(T9) one has
û = v̂2 + αv̂1 + βv̂, u = eû, v = ev̂,
where α, β are some constants. Equations (E10)–(E12) are transformed into (E1) by transfor-
mations
v = w1 + w, v = w1 − w + a, v = φ(w1 − w)− 1,
respectively, and therefore corresponding transformations (T10)–(T12) are also linearizable as
the compositions of linearizable transformations. Transformation (T13) is linearizable in a more
complicated way, see [6, Section 3].
Both transformations (T14.a) and (T14.b) are of Miura type, see a comment in [6, Section 2.1].
4 Modifications of the INB equation of the level 3
Here we classify all integrable modifications of the INB equation of the third level. First we give
a list of equations.
List 3. Modifications of the third level.
ż = z
(
z2
z−1
+
z1
z−2
)
+ cz, (E15)
ż = z1z
3z−1(z2z1 − z−1z−2), (E16)
ż = z
(
z2z
κ
1 z
−1z−κ−1 − κ
2z1z
κz−1−1z
−κ
−2
)
+ cz, (E17)
ż = (z2 − z + a)(z1 − z−1 + a)(z − z−2 + a) + c, (E18)
ż = −
(
T + 1 + T−1
) 1
(z1 − z + a)(z − z−1 + a)
+ c, (E19)
ż = −
(
T − κ+ κ2T−1
) 1
(z1 + κz)(z + κz−1)
, (E20)
ż = w1w + ww−1 + w−1w−2 − w1ww−1 − ww−1w−2 + c, (E21)
w =
1
1− φ(z1 − z)
,
ż = z(z1z − 1)(zz−1 − 1)(z2z1 − z−1z−2), (E22)
ż =
(az1 − z)(az − z−1)(az2z−2 + az1z−1 − 2z1z−2)
z1z−1z−2
+ cz, a 6= 0, (E23)
ż
a2
=
z2(az + z−1)
z−1
+
z1z
z−1
+
(az1 + z)z
z−2
+ cz, (E24)
ż = −z
(
T + 1 + T−1
) zz−1
(az1 − z)(az − z−1)
+ cz, a 6= 0, (E25)
ż = −z
(
T − κ+ κ2T−1
) 1
(z1zκ − 1)(zzκ−1 − 1)
, (E26)
ż = θ(z1 − z)θ(z − z−1)[θ(z2 − z1) + θ(z−1 − z−2) + a]
+ a[θ(z2 − z1)θ(z1 − z) + θ(z − z−1)θ(z−1 − z−2)] + c, (E27)
θ′ =
θ(θ + 1)
θ + a
, a 6= 0, a 6= 1,
8 R.N. Garifullin and R.I. Yamilov
ż = 4
(
z2 − 1
)
(2 + (z−1 − 1)T−1 − (z1 + 1)T )Ω, (E28)
Ω =
1
[(z1 + 1)(z − 1) + 4][(z + 1)(z−1 − 1) + 4]
,
ż = −z
(
T + 1 + T−1
) z
(z1 − z)(z − z−1)
. (E29)
Here κ is defined by (16), i.e., for each such an equation with a dependence on κ, there are
two cases. Besides, a and c are arbitrary constants, while in equation (E21) the function φ is
defined by (17). Let us list now the corresponding transformations.
List 3′. Transformations of the third order.
u =
z3
z
, (T15)
u = z3z
2
2z
2
1z, (T16)
u = z3z
κ
2 z
−1
1 z−κ, (T17)
u = (z3 − z1 + a)(z2 − z + a), (T18)
u =
1
(z3 − z2 + a)(z2 − z1 + a)(z1 − z + a)
, (T19)
u =
1
(z3 + κz2)(z2 + κz1)(z1 + κz)
, (T20)
u =
1
(1− φ(z3 − z2))(1− φ(z2 − z1))(1− φ(z1 − z))
, (T21)
u = (z3z2 − 1)(z2z1 − 1)z1z, (T22.a)
u = z3z2(z2z1 − 1)(z1z − 1), (T22.b)
u = a
(az3 − z2)(az2 − z1)
z2z
, (T23.a)
u = a
z3(az2 − z1)(az1 − z)
z2z1z
, (T23.b)
u = a2
z3(az1 + z)
z1z
, (T24.a)
u = a2
az3 + z2
z
, (T24.b)
u =
az2z
2
1
(az3 − z2)(az2 − z1)(az1 − z)
, (T25.a)
u =
az3z1z
(az3 − z2)(az2 − z1)(az1 − z)
, (T25.b)
u =
z1z
κ
(z3zκ2 − 1)(z2zκ1 − 1)(z1zκ − 1)
, (T26.a)
u =
z3z
κ
2
(z3zκ2 − 1)(z2zκ1 − 1)(z1zκ − 1)
, (T26.b)
u = θ(z3 − z2)θ(z2 − z1)(θ(z1 − z) + 1), (T27.a)
u = (θ(z3 − z2) + 1)θ(z2 − z1)θ(z1 − z), (T27.b)
u =
4(z2 + 1)
(
z21 − 1
)
(z − 1)
[(z3 + 1)(z2 − 1) + 4][(z2 + 1)(z1 − 1) + 4][(z1 + 1)(z − 1) + 4]
, (T28)
u =
z2z1
(z3 − z2)(z2 − z1)(z1 − z)
. (T29)
Theorem 4. If an equation (5) is transformed into the INB equation (2) by a transformation of
the form (6) with k = 3, then up to autonomous point transformations (14) and (15) it coincides
Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation 9
with one of equations (E15)–(E29) of the List 3. Equations (E15)–(E29) are transformed into the
INB equation by transformations (T15)–(T29), respectively. For each of equations (E22)–(E27),
there are two transformations.
Equations (E16), (E22) together with corresponding transformations were discussed in [7, 8],
see also references therein. Equation (E18) with a = c = 0 has been presented [6]. Transforma-
tions (T15)–(T17) are linearized as follows
û = v̂3 + αv̂2 + βv̂1 + γv̂, u = eû, v = ev̂, (18)
where α, β, γ are some constants.
For further discussion, we will need some auxiliary transformations.
List 3′′. Auxiliary transformations.
(E18)→ (E1) : v = z2 − z + a, (T∗18)
(E19)→ (E7) : w =
1
z1 − z + a
, (T∗19)
(E20)→ (E7) : w =
1
z1 + κz
, (T∗20)
(E21)→ (E7) : w =
1
1− φ(z1 − z)
, (T∗21)
(E22)→ (E14) : w = z1z − 1, (T∗22)
(E23)→ (E14) : w = a
z1
z
− 1, (T∗23)
(E24)→ (E14) : w = a
z1
z
, (T∗24)
(E25)→ (E14) : w =
z
az1 − z
, (T∗25)
(E26)→ (E14) : w =
1
z1zκ − 1
, (T∗26)
(E27)→ (E14) : w = θ(z1 − z), (T∗27)
(E28)→ (E14) : w = − (z1 + 1)(z − 1)
(z1 + 1)(z − 1) + 4
. (T∗28)
Transformations (T18)–(T21) are compositions of linearizable transformations (T∗18)–(T∗21)
and of one of linearizable transformations (T1) or (T7) and therefore are linearizable too.
All transformations (T∗22)–(T∗27) are linearizable. In the case of equations (E22)–
(E27), transformations (T22.a)–(T27.a) are compositions of transformations (T∗22)–(T∗27)
and of (T14.a), while (T22.b)–(T27.b) are compositions of transformations (T∗22)–(T∗27) and
of (T14.b). As we have mentioned above, transformations (T14.a) and (T14.b) are of Miura
type, and for this reason all the transformations associated with (E22)–(E27) are compositions
of linearizable and Miura type transformations.
Transformation (T28) is a composition of two Miura type transformations, see details in
Section 5.2 below. Transformation (T29) is linearized in a non-standard way, see Section 5.1.
5 The most interesting examples
Here we discuss in detail equations (E29) and (E28) and corresponding transformations (T29)
and (T28).
10 R.N. Garifullin and R.I. Yamilov
5.1 Equation (E29)
Transformation (T29) corresponding to (E29) is quite similar to (T13) which is discussed in
detail in [6, Section 3]. As it will be shown below, unlike the most of linearizable transformations
presented in this paper, it is a composition of linearizable transformations in different directions.
In this case we need one auxiliary equation
ẏ =
1
(y2 − y−1)(y1 − y−2)
(19)
and two obviously linearizable transformations
(19)→ (E29) : z = −y2y1y, (20)
(19)→ (E7) : w =
1
y − y3
. (21)
Transformation (T29) can be decomposed as it is shown in the following diagram
(E29)
(T29)
> (2)
(19)
(20)
∧
(21)
> (E7)
(T7)
∧
(22)
As we have mentioned in Section 2, transformation (T29) allows one to construct conserva-
tion laws for the modified equation (E29) in an explicit way, as this transformation is of the
form (6). The construction of generalized symmetries is a more difficult problem. However,
it is simplified in the case of linearizable transformations, as it is reduced to the use of linear
transformations like (18). The construction of modified equations by using linearizable trans-
formations is discussed in [7, Appendix A]. Using that theory, we construct here a generalized
symmetry for (E29).
The decomposition shown in diagram (22) allows one to construct the generalized symmetry
for (E29) by using a known symmetry for the INB equation (2). It is even easier to use a sym-
metry, presented in [26], for its well-known modification (E7). At first we construct a generalized
symmetry for (19), which is of the form
yτ =
1
(y2 − y−1)(y1 − y−2)
(
T 3 + T 2 + T + 1
) 1
(y1 − y−2)(y − y−3)(y−1 − y−4)
.
Now we can find a generalized symmetry for (E29), which reads
zτ = z
(
T + 1 + T−1
) zΞ
(z1 − z)(z − z−1)
,
Ξ =
(
T 3 + T 2 + T + 1
) z−1z−2
(z − z−1)(z−1 − z−2)(z−2 − z−3)
.
Equation (19) exemplifies a modification of the INB equation (2) of the highest possible fifth
level. Corresponding transformation of (19) into (2) is the composition of transformations shown
in diagram (22) and it reads
u = − 1
y5 − y2
1
y4 − y1
1
y3 − y
.
Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation 11
5.2 Equation (E28)
Transformation (T∗28), unlike all other transformations of List 3′′, is of Miura type. Rewriting
it in the form
w̃ = − 4w
w + 1
= (z1 + 1)(z − 1), (23)
we see that it is the standard Miura transformation of z to w̃, cf. (4).
Transformation (T28) is a composition of transformations (T∗28) and (T14.b). The second
natural composition of transformations (T∗28) and (T14.a) is not written down in List 3′, as it
comes from (T28) by using the point transformation
z̃ = ς(z) =
z + 3
1− z
. (24)
It is the auto-transformation of equation (E28). It is interesting that
ς2(z) = ς(ς(z)) =
z − 3
z + 1
, ς3(z) = z, (25)
i.e., there are two different auto-transformations for equation (E28).
Using the Miura type transformation (T∗28) and corresponding modification (E28), let us
construct in this section one more non-standard transformation and an integrable completely
discrete quad-equation.
Non-standard transformation. A Miura-like transformation can be constructed with the
help of a theory developed in [23, 24].
Using transformation (T∗28), let us construct a relation
w = − (z1 + 1)(z − 1)
(z1 + 1)(z − 1) + 4
= − (ẑ1 + 1)(ẑ − 1)
(ẑ1 + 1)(ẑ − 1) + 4
(26)
for two solutions z, ẑ of equation (E28). This relation is compatible with equation (E28). It is
simplified in terms of the function w̃ given by (23)
w̃ = (z1 + 1)(z − 1) = (ẑ1 + 1)(ẑ − 1).
The latter can be rewritten as
z − 1
ẑ − 1
=
ẑ1 + 1
z1 + 1
= −y1,
where yn is a new unknown function. We get a transformation
y = − ẑ + 1
z + 1
, y1 = −z − 1
ẑ − 1
, (27)
which is invertible on the solutions of (26)
z =
y1y + 2y1 + 1
1− y1y
, (28a)
ẑ =
y1y + 2y + 1
y1y − 1
. (28b)
This transformation allows us to rewrite equation (E28) in terms of yn.
12 R.N. Garifullin and R.I. Yamilov
Differentiating the first of relations (27) with respect to the time in virtue of (E28) and
substituting the functions (28), we get an equation
ẏ =
y(y + 1)(y1y − 1)(yy−1 − 1)(y2y1 − y−1y−2)
(y2y1y + 1)(y1yy−1 + 1)(yy−1y−2 + 1)
. (29)
This equation coincides with [21, equation (5.11)], where χ = −1, and it is equivalent to [12,
equation (59b)] up to a point transformation. Equation (29) is transformed into (E28) by any of
transformations (28). It should be remark that these two transformations are equivalent up to
the point auto-transformation yn → 1/yn of equation (29). So, we have derived modification (29)
of equation (E28) and two equivalent transformations that look like Miura type transformations.
Let us analyze in more detail transformation (28a). It can be rewritten as a discrete Riccati
equation for the function yn
(1 + z)y1y + 2y1 + 1− z = 0. (30)
In accordance with [6] it should be the Miura type transformation. However, a particular solution
of this equation can easily be found: y∗n ≡ −1. Therefore, unlike the case of general position, it
can be linearized in an explicit way and can be solved by quadrature.
In fact, using this particular solution, let us change yn = y∗n + 1/ỹn in order to get a non-
autonomous linear equation
ỹ1 −
1− z
1 + z
ỹ − 1 = 0. (31)
Substituting the relation
1− z
1 + z
=
x− x−1
x1 − x
(32)
in terms of a new function xn, we rewrite (31) as the equation
(T − 1)[(x− x−1)ỹ − x] = 0,
which has the obvious solution
ỹ =
x
x− x−1
.
As a result we have y = −x−1
x . Starting from a given function zn, we can find xn from (32)
by using two discrete integrations, as
x− x−1 = r,
r
r1
=
1− z
1 + z
.
So, the Riccati equation (30) is degenerate in the sense that it is solved by quadrature.
These results can be reformulated in terms of linearizable transformations and modifications
of the INB equation. We are led to the following picture
(29)
(28a)
> (E28)
(34)
y = −x−1
x
∧
r = x− x−1
> (35)
1− z
1 + z
=
r
r1
∧
(33)
Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation 13
where
ẋ =
(x2 − x)(x1 − x−1)(x− x−2)
(x2 − x−1)(x1 − x−2)
, (34)
ṙ =
r(r2 + r1 − r−1 − r−2)(r1 + r)(r + r−1)
(r2 + r1 + r)(r1 + r + r−1)(r + r−1 + r−2)
. (35)
We have that (28a) is a complete analogue of transformations (T29) and (T13), i.e., it is a com-
position of linearizable transformations in different directions. Therefore it is not of Miura type,
but it is the linearizable transformation in accordance with the terminology of [6].
Equation (34) has been found in [14, equation (2.23)], (35) coincides with [15, equation (3.1b)],
while (29) has been discussed above. New objects in this diagram are equation (E28) and trans-
formation (28a). Equations (29), (34), (35) exemplify modifications of the INB equation (2) of
the fourth and fifth levels. Corresponding transformations into (2) are obtained as compositions
of transformation (T28) and transformations shown in diagram (33). Those transformations are
too cumbersome to be written here.
Integrable quad-equation. It is known that differential-difference equations like (5) and
discrete equations on the square lattice (quad-equations) are closely related. Differential-diffe-
rence equations define the generalized symmetries for quad-equations, while quad-equations can
be interpreted as the Bäcklund transformations for differential-difference equations [11]. The
problem of construction of a compatible quad-equation for a given differential-difference equation
was solved in [4, 5]. In [4] the five-point differential-difference equations (5) were considered.
Let us find an integrable quad-equation compatible with equation (E28) by using a different
method. We use here a theory developed in [18, 22]. From [12] we know that equation (E14),
expressed in the form
ẇn,m = wn,m(wn,m + 1)(wn+2,mwn+1,m − wn−1,mwn−2,m), (36)
is the generalized symmetry of quad-equation
wn+1,m+1(wn,m+1 + wn,m + 1) + wn,m(wn+1,m + 1) = 0. (37)
Let us rewrite it in the form
2
wn+1,m + 1
wn+1,m+1
+ 1 = −2
wn,m+1 + 1
wn,m
− 1.
This allows us to introduce a new function zn,m, so that
zn+1,m = −2
wn,m+1 + 1
wn,m
− 1, zn,m = 2
wn,m + 1
wn,m+1
+ 1. (38)
It is obvious that
wn,m = − 2(zn,m + 1)
(zn+1,m + 1)(zn,m − 1) + 4
, wn,m+1 =
2(zn+1,m − 1)
(zn+1,m + 1)(zn,m − 1) + 4
. (39)
Rewriting these relations at the same point wn,m+1, we get an equation for zn,m
(zn+1,m+1 − 1)(zn+1,m − 1)(zn,m+1 − 1) + (zn,m + 1)(zn+1,m + 1)(zn,m+1 + 1) = 0. (40)
So, this new discrete equation is obtained by transformation (38) which is invertible on the
solutions of quad-equation (37).
Transformation (38), (39) allows one to rewrite the generalized symmetries of quad-equa-
tion (37). Differentiating the second of relations (38) with respect to the time in virtue of (36)
14 R.N. Garifullin and R.I. Yamilov
and substituting the functions (39), we get a generalized symmetry for (40). Let us denote the
right hand side of equation (E28) by
Θ(zn+2, zn+1, zn, zn−1, zn−2).
Then the resulting generalized symmetry is of the form
żn,m = Θ(zn+2,m, zn+1,m, zn,m, zn−1,m, zn−2,m), (41)
i.e., it is defined by equation (E28). Due to the invariance of quad-equation (40) under the
change of discrete variables n↔ m, a generalized symmetry in the other direction m reads
z′n,m = Θ(zn,m+2, zn,m+1, zn,m, zn,m−1, zn,m−2). (42)
A generalized symmetry of (37) in the m-direction is presented in [12] and it is equivalent to
symmetry (42) up to the invertible transformation (38), (39). Formulae (39) define two Miura
type transformations which transform (41) into (36). Both of them are equivalent to the discrete
Miura transformation (T∗28) up to auto-transformations given in (24), (25).
6 Conclusion
In this paper we have enumerated all integrable modifications of the form (1) of the INB equa-
tion (2) of the levels k = 1, 2, 3, which are related to it by non-invertible transformations of
the form (3). Resulting Lists 1–3 contain 29 equations, see more precise formulations in Theo-
rems 2–4. As far as we know, a classification problem of this kind is solved for the first time in
the discrete case.
Corresponding discrete non-invertible transformations are presented in Lists 1′–3′. We have
analyzed those transformations to understand their nature. As a result we have shown that the
most of them are linearizable.
We have also proved that the orders of possible transformations in this problem are restricted
by the number five, see Theorem 1. This estimate is accurate in the sense that there exist
transformations for each order k: 1 ≤ k ≤ 5. Transformations of the orders k = 1, 2, 3 have
been completely enumerated in Sections 3 and 4. Examples of the orders k = 4, 5 are presented
in Section 5. The complete classification of transformations of the fourth and fifth orders is left
for a future work.
As a result of the classification we obtain a number of new integrable equations and non-
invertible discrete transformations. Among new transformations we would like to highlight
transformation (T∗28) of Miura type and two transformations (T29) and (28a) that are li-
nearizable in a non-standard way. Among equations we note two five-point differential-difference
equations (E28), (E29) and one quad-equation (40) compatible with (E28).
Acknowledgments
RIY gratefully acknowledges the financial support from Russian Science Foundation grant (pro-
ject 15-11-20007).
References
[1] Adler V.E., Necessary integrability conditions for evolutionary lattice equations, Theoret. and Math. Phys.
181 (2014), 1367–1382, arXiv:1406.1522.
[2] Adler V.E., Integrable Möbius-invariant evolutionary lattices of second order, Funct. Anal. Appl. 50 (2016),
257–267, arXiv:1605.00018.
https://doi.org/10.1007/s11232-014-0218-2
https://arxiv.org/abs/1406.1522
https://doi.org/10.1007/s10688-016-0157-9
https://arxiv.org/abs/1605.00018
Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation 15
[3] Bogoyavlensky O.I., Integrable discretizations of the KdV equation, Phys. Lett. A 134 (1988), 34–38.
[4] Garifullin R.N., Gubbiotti G., Yamilov R.I., Integrable discrete autonomous quad-equations admitting,
as generalized symmetries, known five-point differential-difference equations, J. Nonlinear Math. Phys. 26
(2019), 333–357, arXiv:1810.11184.
[5] Garifullin R.N., Yamilov R.I., Integrable discrete nonautonomous quad-equations as Bäcklund auto-
transformations for known Volterra and Toda type semidiscrete equations, J. Phys. Conf. Ser. 621 (2015),
012005, 18 pages, arXiv:1405.1835.
[6] Garifullin R.N., Yamilov R.I., Levi D., Non-invertible transformations of differential-difference equations,
J. Phys. A: Math. Theor. 49 (2016), 37LT01, 12 pages, arXiv:1604.05634.
[7] Garifullin R.N., Yamilov R.I., Levi D., Classification of five-point differential-difference equations,
J. Phys. A: Math. Theor. 50 (2017), 125201, 27 pages, arXiv:1610.07342.
[8] Garifullin R.N., Yamilov R.I., Levi D., Classification of five-point differential-difference equations II,
J. Phys. A: Math. Theor. 51 (2018), 065204, 16 pages, arXiv:1708.02456.
[9] Itoh Y., An H-theorem for a system of competing species, Proc. Japan Acad. 51 (1975), 374–379.
[10] Kuznetsova M.N., Pekcan A., Zhiber A.V., The Klein–Gordon equation and differential substitutions of the
form v = φ(u, ux, uy), SIGMA 8 (2012), 090, 37 pages, arXiv:1111.7255.
[11] Levi D., Petrera M., Scimiterna C., Yamilov R., On Miura transformations and Volterra-type equations
associated with the Adler–Bobenko–Suris equations, SIGMA 4 (2008), 077, 14 pages, arXiv:0802.1850.
[12] Mikhailov A.V., Xenitidis P., Second order integrability conditions for difference equations: an integrable
equation, Lett. Math. Phys. 104 (2014), 431–450, arXiv:1305.4347.
[13] Narita K., Soliton solution to extended Volterra equation, J. Phys. Soc. Japan 51 (1982), 1682–1685.
[14] Papageorgiou V.G., Nijhoff F.W., On some integrable discrete-time systems associated with the Bogoyavlen-
sky lattices, Phys. A 228 (1996), 172–188.
[15] Scimiterna C., Hay M., Levi D., On the integrability of a new lattice equation found by multiple scale
analysis, J. Phys. A: Math. Theor. 47 (2014), 265204, 16 pages, arXiv:1401.5691.
[16] Sokolov V.V., On the symmetries of evolution equations, Russian Math. Surveys 43 (1988), no. 5, 165–204.
[17] Startsev S.Ya., On hyperbolic equations that admit differential substitutions, Theoret. and Math. Phys. 127
(2001), 460–470.
[18] Startsev S.Ya., On non-point invertible transformations of difference and differential-difference equations,
SIGMA 6 (2010), 092, 14 pages, arXiv:1010.0361.
[19] Suris Yu.B., The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics,
Vol. 219, Birkhäuser Verlag, Basel, 2003.
[20] Wadati M., Transformation theories for nonlinear discrete systems, Progr. Theoret. Phys. Suppl. 59 (1976),
36–63.
[21] Xenitidis P., Determining the symmetries of difference equations, Proc. A. 474 (2018), 20180340, 20 pages.
[22] Yamilov R.I., Invertible changes of variables generated by Bäcklund transformations, Theoret. and Math.
Phys. 85 (1990), 1269–1275.
[23] Yamilov R.I., On the construction of Miura type transformations by others of this kind, Phys. Lett. A 173
(1993), 53–57.
[24] Yamilov R.I., Construction scheme for discrete Miura transformations, J. Phys. A: Math. Gen. 27 (1994),
6839–6851.
[25] Yamilov R.I., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math.
Gen. 39 (2006), R541–R623.
[26] Zhang H., Tu G.Z., Oevel W., Fuchssteiner B., Symmetries, conserved quantities, and hierarchies for some
lattice systems with soliton structure, J. Math. Phys. 32 (1991), 1908–1918.
https://doi.org/10.1016/0375-9601(88)90542-7
https://doi.org/10.1080/14029251.2019.1613050
https://arxiv.org/abs/1810.11184
https://doi.org/10.1088/1742-6596/621/1/012005
https://arxiv.org/abs/1405.1835
https://doi.org/10.1088/1751-8113/49/37/37LT01
https://arxiv.org/abs/1604.05634
https://doi.org/10.1088/1751-8121/aa5cc3
https://arxiv.org/abs/1610.07342
https://doi.org/10.1088/1751-8121/aaa14e
https://arxiv.org/abs/1708.02456
https://doi.org/10.3792/pja/1195518557
https://doi.org/10.3842/SIGMA.2012.090
https://arxiv.org/abs/1111.7255
https://doi.org/10.3842/SIGMA.2008.077
https://arxiv.org/abs/0802.1850
https://doi.org/10.1007/s11005-013-0668-8
https://arxiv.org/abs/1305.4347
https://doi.org/10.1143/JPSJ.51.1682
https://doi.org/10.1016/0378-4371(95)00440-8
https://doi.org/10.1088/1751-8113/47/26/265204
https://arxiv.org/abs/1401.5691
http://dx.doi.org/10.1070/RM1988v043n05ABEH001927
https://doi.org/10.1023/A:1010359808044
https://doi.org/10.3842/SIGMA.2010.092
https://arxiv.org/abs/1010.0361
https://doi.org/10.1007/978-3-0348-8016-9
https://doi.org/10.1143/PTPS.59.36
https://doi.org/10.1098/rspa.2018.0340
https://doi.org/10.1007/BF01018403
https://doi.org/10.1007/BF01018403
https://doi.org/10.1016/0375-9601(93)90086-F
https://doi.org/10.1088/0305-4470/27/20/020
https://doi.org/10.1088/0305-4470/39/45/R01
https://doi.org/10.1088/0305-4470/39/45/R01
https://doi.org/10.1063/1.529205
1 Introduction
2 Theoretical comments and results
3 Modifications of the INB equation of the levels 1 and 2
4 Modifications of the INB equation of the level 3
5 The most interesting examples
5.1 Equation (E29)
5.2 Equation (E28)
6 Conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-210233 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:50Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
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| spelling | Garifullin, R.N. Yamilov, R.I. 2025-12-04T13:05:21Z 2019 Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation / R.N. Garifullin, R.I. Yamilov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K05; 37K10; 35G20 arXiv: 1903.11893 https://nasplib.isofts.kiev.ua/handle/123456789/210233 https://doi.org/10.3842/SIGMA.2019.062 We consider five-point differential-difference equations. We aim to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated with transformations of the first, second, and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable five-point equations and new transformations have been found. Moreover, we have derived a new, completely discrete equation. There are a few non-standard transformations which are of the Miura type or are linearizable in a non-standard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem. RIY gratefully acknowledges the financial support from the Russian Science Foundation grant (project 15-11-20007). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation Article published earlier |
| spellingShingle | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation Garifullin, R.N. Yamilov, R.I. |
| title | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation |
| title_full | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation |
| title_fullStr | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation |
| title_full_unstemmed | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation |
| title_short | Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation |
| title_sort | integrable modifications of the ito-narita-bogoyavlensky equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210233 |
| work_keys_str_mv | AT garifullinrn integrablemodificationsoftheitonaritabogoyavlenskyequation AT yamilovri integrablemodificationsoftheitonaritabogoyavlenskyequation |