Algebraic entropy of a class of five-point differential-difference equations
aa r X i v : . [ n li n . S I] M a r ALGEBRAIC ENTROPY OF A CLASS OF FIVE-POINTDIFFERENTIAL-DIFFERENCE EQUATIONS
GIORGIO GUBBIOTTI
Abstract.
We compute the algebraic entropy of a class of integrable Volterra-like five-point differential-difference equations recently classified using the gen-eralised symmetry method. We show that, when applicable, the results of thealgebraic entropy agrees with the result of the generalised symmetry method,as all the equations in this class have vanishing entropy. Introduction
One of the most important topic in modern Mathematical Physics is the study ofthe so-called integrable systems . Roughly speaking integrable systems are importantboth from theoretical and practical point of view since they can be regarded as universal models for physics going beyond the linear regime [10]. The birth moderntheory of integrable systems is usually recognized in the seminal works of Zabuskyand Kruskal [62], Gardner, Greene, Kruskal and Miura [16] and Lax [38] on theKorteweg-deVries (KdV) equation [37].The concept of integrability come from Classical Mechanics and means the ex-istence of a sufficiently high number of first integrals . To be more specific anHamiltonian with Hamiltonian H = H ( p, q ) system with N degrees of freedom issaid to be integrable if there exist N − well defined functionally independentand Poisson-commuting first integrals [42,58]. In the case of systems with infinitelymany degrees of freedom, e.g. partial differential equations like the KdV equation,the existence of infinitely many conservation laws is then required. One of the mostefficient way to find these infinitely many is the existence a so-called Lax pair [38].A Lax pair is an associated overdetermined linear problem whose compatibilitycondition is guaranteed if and only if the desired non-linear equation is satisfied.Nowadays a purely algorithmic method to prove or disprove the existence of aLax pair is not available, so during the years many integrability detectors have beendeveloped. Integrability detectors are algorithmic procedure which are sufficientconditions for integrability, or alternative definitions of integrability. This meansthat integrability detectors can be used to prove the integrability of a given equa-tion without the need of a Lax pair. One of the fundamental integrability detectors,which works both at continuous and discrete level, is the generalised symmetry ap-proach . The generalised symmetry approach was mainly developed by the scientificschool of A. B. Shabat in Ufa during the 80s and has obtained many important resultin the classification of partial differential equations [4,32,43–45,50,51], differential-difference equations [23,24,39,59,60] and partial difference equations [17,20,40,41].
Date : March 12, 2019.2010
Mathematics Subject Classification. We say that a function is well-defined on the phase space if it is analytic and single-valued . Another integrability detector is the algebraic entropy test. The algebraic entropytest is specific to systems with discrete degrees of freedom which can be put into abi-rational form. The basic idea, given a bi-rational map, which can be an ordinarydifference equation, a differential-difference equation or even a partial differenceequation, is to examine the growth of the degree of its iterates, and extract acanonical quantity, which is an index of complexity of the map. This canonicalquantity is what is called the algebraic entropy. The idea of algebraic entropy asmeasure of the complexity of the growth of bi-rational maps comes from the notionof complexity introduced by Arnol’d in [5] and was discussed for the first time inrelation of discrete systems by Veselov [55].In this paper we will compute the algebraic entropy of two classes of first order,five-point differential-difference equation which were classified recently in [23, 24]through the generalised symmetry method. We mention that these equations wereused to produce and classify new examples of quad-equations in [17]. These equa-tions are autonomous fourth order differential-difference equations of the followingform:(1) d u n d t = A ( u n +1 , u n , u n − ) u n +2 + B ( u n +1 , u n , u n − ) u n − + C ( u n +1 , u n , u n − ) , where u n = u n ( t ) , is the dependent variable depending on n ∈ Z and on t ∈ R . Dueto the similarity of equation (1) with the well-known three-point Volterra equation(2) d u n d t = u n ( u n +1 − u n − ) , we will call equations of this form Volterra-like equations . Remark . Throughout this paper we are going to consider only autonomous equa-tions of the form (1). Therefore, we will make use of the short-hand notation u n + k = u k to simplify the formulæ.Explicitly, we are going to consider are the following equations divided in sixlists: List 1:
Equations related to the double Volterra equation: d u d t = u ( u − u − ) , (E.1) d u d t = u ( u − u − ) , (E.2) d u d t = ( u + u )( u − u − ) , (E.3) d u d t = ( u + u )( u + u − ) − ( u + u )( u − + u − ) , (E.4) d u d t = ( u − u + a )( u − u − + a )+ ( u − u + a )( u − − u − + a ) + b, (E.5) d u d t = u u u ( u u − + 1) − ( u u + 1) u u − u − + u ( u − − u ) , (E.6) d u d t = u [ u ( u − u ) + u − ( u − u − )] , (E.1 ′ ) LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 3 d u d t = u u u − ( u − u − ) . (E.2 ′ ) Transformations ˜ u k = u k or ˜ u k = u k +1 turn equations (E.1)-(E.3) intothe well-known Volterra equation and its modifications in their standardform. The other equations are related to the double Volterra equation (E.1)through some autonomous non-invertible non-point transformations. Wenote that equation (E.2 ′ ) was presented in [3]. List 2:
Linearizable equations: d u d t = ( T − a ) (cid:20) ( u + au + b )( u − + au − + b ) u + au − + b + u + au − + b (cid:21) + cu + d, (E.7) d u d t = u u u + u − a (cid:18) u − + u u − u − (cid:19) + cu . (E.8) In both equations a = 0 , in (E.7) ( a + 1) d = bc , and T is the translationoperator T f n = f n +1 .Both equations of List 2 are related to the linear equation:(3) d u d t = u − a u − + c u through an autonomous non-invertible non-point transformations. We notethat (E.7) is linked to (3) with a transformation which is implicit in bothdirections, see [23] for more details. List 3:
Equations related to a generalised symmetry of the Volterra equation: d u d t = u [ u ( u + u + u ) − u − ( u + u − + u − )]+ cu ( u − u − ) , (E.3 ′ ) d u d t = ( u − a ) (cid:2) ( u − a )( u + u ) − ( u − − a )( u + u − ) (cid:3) + c ( u − a ) ( u − u − ) , (E.4 ′ ) d u d t = ( u − u + a )( u − u − + a )( u − u − + 4 a + c ) + b, (E.5 ′ ) d u d t = u [ u ( u − u + u ) − u − ( u − u − + u − )] , (E.6 ′ ) d u d t = ( u − a ) (cid:2) ( u − a )( u − u ) + ( u − − a )( u − u − ) (cid:3) , (E.7 ′ ) d u d t = ( u + u )( u + u − )( u − u − ) . (E.8 ′ ) These equations are related between themselves by some transformations,for more details see [24]. Moreover equations (E.3 ′ ,E.4 ′ ,E.5 ′ ) are the gen-eralised symmetries of some known three-point autonomous differential-difference equations [61]. List 4:
Equations of the relativistic Toda type: d u d t = ( u − (cid:18) u ( u − u u − u ( u − − u − u − − u + u − (cid:19) , (E.9) GIORGIO GUBBIOTTI d u d t = u u u ( u u − + 1) u u + 1 − ( u u + 1) u u − u − u u − + 1 − ( u − u − )(2 u u u − + u + u − ) u ( u u + 1)( u u − + 1) , (E.10) d u d t = ( u u − u u − − u − u − ) . (E.13 ′ ) Equation (E.13 ′ ) was known [18, 19] to be is a relativistic Toda type equa-tion. Since in [23] it was shown that the equations of List 4 are relatedthrough autonomous non-invertible non-point transformations, it was sug-gested that (E.9) and (E.10) should be of the same type. Finally, we notethat equation (E.9) appeared in [1] earlier than in [23]. List 5:
Equations related to the Itoh-Narita-Bogoyavlensky (INB) equation: d u d t = u ( u + u − u − − u − ) , (E.11) d u d t = ( u − u + a )( u − u − + a )+ ( u − u + a )( u − − u − + a )+ ( u − u + a )( u − u − + a ) + b, (E.12) d u d t = ( u + au )( u u − u − u − ) , (E.13) d u d t = ( u − u )( u − u − ) (cid:18) u u − u − u − (cid:19) , (E.14) d u d t = u ( u u − u − u − ) , (E.9 ′ ) d u d t = ( u − u + a )( u − u − + a )( u − u + u − − u − + 2 a ) + b, (E.10 ′ ) d u d t = u ( u u − a )( u u − − a )( u u − u − u − ) , (E.11 ′ ) d u d t = ( u + u )( u + u − )( u + u − u − − u − ) . (E.12 ′ ) Equation (E.11) is the well-known INB equation [8,34,47]. Equations (E.12)with a = 0 and (E.13) with a = 0 are simple modifications of the INB andwere presented in [46] and [9], respectively. Equation (E.13) with a = 1 hasbeen found in [52]. Up to an obvious linear transformation, it is equation(17.6.24) with m = 2 in [52],. Equation (E.9 ′ ) is a well-known modifica-tion of INB equation (E.11), found by Bogoyalavlesky himself [8]. Finally,equation (E.11 ′ ) with a = 0 was considered in [3]. All the equations in thislist can be reduced to the INB equation using autonomous non-invertiblenon-point transformations. Moreover, equations (E.12),(E.14) and (E.9 ′ )are related through non-invertible transformations to the equation: d u d t = ( u − u ) ( u − u − ) ( u − u − ) . (4) For this reason, as it was done in [17], we will consider equation (4), asindependent. We note that equation (4) and its relationship with equation(E.12),(E.14) and (E.9 ′ ) were first discussed in [22]. LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 5
List 6:
Other equations: d u d t = u ( u u − u − u − ) − u ( u − u − ) , (E.15) d u d t = ( u + 1) × (cid:20) u u ( u + 1) u − u − u ( u − + 1) u − + (1 + 2 u )( u − u − ) (cid:21) , (E.16) d u d t = ( u + 1) (cid:18) u q u + 1 − u − q u − + 1 (cid:19) , (E.17) d u d t = u u u − ( u u − u − u − ) − u ( u − u − ) . (E.14 ′ ) Equation (E.15) has been found in [54] and it is called the discrete Sawada-Kotera equation [2, 54]. Equation (E.14 ′ ) is a simple modification of thediscrete Sawada-Kotera equation (E.15). Equation (E.16) has been foundin [1] and is related to (E.15). On the other hand equation (E.17) hasbeen found as a result of the classification in [23] and seems to be a newequation. It was shown in [21] that equation (E.17) is a discrete analogue ofthe Kaup-Kupershmidt equation. Then we will refer to equation (E.17) asthe discrete Kaup-Kupershmidt equation. No transformation into knownequations of equation (E.17) is known.In this paper we will show that all the bi-rational equations of Lists 1–6 possessquadratic (linear) growth, and hence are integrable (linearizable) according to thealgebraic entropy test. We note that the only non-bi-rational equations equation inClasses I and II is the discrete Kaup-Kupershmidt equation (E.17) which containssquare root terms. In section 2 we will give some details on how algebraic entropyis computed, then in section 3 we will show the results for the equations of Lists1–6 except the discrete Kaup-Kupershmidt equation (E.17). In section 4 we willdiscuss the results obtained in sections 3 in the framework of the existing literatureand we will give an outlook on future research in the field.Before going on we would like to present a new rational form of the discreteKaup-Kupershmidt equation (E.17). That is we have the following proposition: Proposition 1.
There exists a point transformation which brings the discreteKaup-Kupershmidt equation (E.17) into the following rational form: (5) d v d t = (cid:0) v (cid:1) (cid:20) v − v v − v − v − − v − v − − v − (cid:21) . Proof.
We start with the substitution:(6) u n = sinh ( ϕ n ) , which brings the discrete Kaup-Kupershmidt equation in hyperbolic form :(7) ˙ ϕ = cosh ( ϕ n ) [cosh ( ϕ n +1 ) sinh ( ϕ n +2 ) − cosh ( ϕ n − ) sinh ( ϕ n − )] . Using the hyperbolic identities:(8) sinh α = 2 tanh ( α/ − tanh ( α/ , cosh α = 1 + tanh ( α/ − tanh ( α/ , D. Levi, private communication.
GIORGIO GUBBIOTTI and putting(9) tanh (cid:16) ϕ n (cid:17) = v n equation (5) follows. (cid:3) Remark . We note that under the scaling:(10) u n ( t ) = i " √ √ ε U (cid:18) τ − ε t, x + 49 εt (cid:19) , x = nε, equation (5) admits the Kaup-Kaupershmit equation as continuum limit:(11) U τ = U xxxxx + 5 U U xxx + 252 U x U xx + 5 U U x , just as the original (E.17) equation.2. Algebraic entropy
Heuristically integrability deals with the regularity of the solutions of a givensystem. In this sense a simple characterisation of chaotic behaviour is when two ar-bitrarily near initial values give rise to solutions diverging at infinity. For recurrencerelations, i.e. equations where the solution is given by iteration of a formula, wecould just try to compute the iteration to extract information about integrability,even if we cannot solve the equation explicitly. However it is usually impossible tocalculate explicitly these iterates by hand or even with any state-of-the-art formalcalculus software, simply because the expressions one should manipulate are ratio-nal fractions of increasing degree of the various initial conditions. The complexityand size of the calculation make it impossible to calculate the iterates.It was nevertheless observed that “integrable” maps are not as complex as genericones. This was done primarily experimentally, by an accumulation of examples, andlater by the elaboration of the concept of algebraic entropy for difference equations[7,13,15,49,55]. In [53,56] the method was developed in the case of quad equationsand then used as a classifying tool [33]. Finally in [12] the same concept wasintroduced for differential-difference equation and later [57] to the very similar caseof differential-delay equations. For a more complete discussion of the method inthe context of the so-called integrability indicators we refer to [25, 26].As we stated in the introduction algebraic entropy is a measure of the growthof bi-rational maps. The most natural space for considering bi-rational maps isthe projective space over a closed field rather than in the affine space one. Wethen transforms a recurrence relation into a polynomial map in the homogeneouscoordinates of the proper projective space over some closed field:(12) ϕ : x i ϕ i ( x k ) , with x i , x k ∈ IN where IN is the space of the initial conditions. The recurrence isthen obtained by iterating the polynomial map ϕ . The map ϕ has to be bi-rational in the sense that it has to possesses an inverse map which is again a rational map.The space of the initial condition depends on which type of recurrence relationwe are considering. In the case of differential-difference equation of the discrete k − k ′ -th order and of the p -th continuous order:(13) u n + k = f n (cid:18)(cid:26) d i u n + k − d t i , . . . , d i u n + k ′ +1 d t i (cid:27) pi =0 ; u n + k ′ (cid:19) , k ′ , k, n ∈ Z , k ′ < k LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 7 the space of initial conditions is infinite dimensional. Indeed, in the case the orderof the equation is k − k ′ , we need the initial value of k − k ′ -tuple as a function ofthe parameter t , but also the value of all its derivatives :(14) IN = (cid:26) d i u k − d t i , d i u k − d t i , . . . , d i u k ′ d t i , (cid:27) i ∈ N . We need all the derivatives of u i ( t ) and not just the first p because at every iterationthe order of the equation is raised by p . Therefore, to describe infinitely manyiterations we need infinitely many derivatives. To obtain the map one just need topass to homogeneous coordinates in the equation and in (14). Remark . We remark that if we restrict to compute a finite number of iterates of adifferential-difference equation of discrete k − k ′ -th order and of the p -th continuousorder (13) then the space of initial conditions is finite dimensional. Indeed, let usassume that we wish to compute the N th iterate of a differential-difference equation(13), then at most we will need the derivatives of order N ( p + 1) . That is, we needto consider the following restricted space of initial conditions:(15) IN ( N ) = (cid:26) d i u k − d t i , d i u k − d t i , . . . , d i u k ′ d t i , (cid:27) N ( p +1) i =0 . If we factor out any common polynomial factors we can say that the degree withrespect to the initial conditions is well defined. We can therefore form the sequenceof degrees of the iterates of the map ϕ and call it d N = deg ϕ N :(16) , d , d , d , d , d , . . . , d N , . . . . The degree of the bi-rational projective map ϕ has to be understood as the max-imum of the total polynomial degree in the initial conditions IN of the entries of ϕ . The same definition in the affine case just translates to the maximum of thedegree of the numerator and of the denominator of the N th iterate in terms ofthe affine initial conditions. Degrees in the projective and in the affine setting canbe different, but the global behaviour will be the same due to the properties ofhomogenization and de-homogenization.The sequence of degree (16) is fixed in a given system of coordinates, but it is notinvariant with respect to changes of coordinates. Therefore we need to introduce acanonical measure of the growth. It turn out that a good definition is the followingone: consider the following number(17) η ϕ = lim N →∞ N log d N , called the algebraic entropy of the map ϕ When no confusion is possible about themap ϕ we will usually omit the subscript ϕ in (17).Algebraic entropy for bi-rational maps has the following properties [7, 25, 26]:(1) The algebraic entropy as given by (17) always exists.(2) The algebraic entropy has the following upper bound:(18) η ϕ ≤ deg ϕ. (3) If η ϕ = 0 , i.e. the algebraic entropy is zero, then(19) d N ∼ N ν , with ν ∈ N , as N → ∞ . GIORGIO GUBBIOTTI (4) The algebraic entropy is a bi-rational invariant of bi-rational maps . Thatis, if two bi-rational maps ϕ and ψ are conjugated by a bi-rational map χ ,(20) ϕ = χ ◦ ψ ◦ χ − then:(21) η ϕ = η ψ . Property 4 tell us that the algebraic entropy is a canonical measure of growth forbi-rational maps.We will then have the following classification of equations according to theirAlgebraic Entropy [33]:
Linear growth:
The equation is linearizable.
Polynomial growth:
The equation is integrable.
Exponential growth:
The equation is chaotic.In our the following sections we will be dealing with differential-difference equa-tions of first continuous order and fourth discrete order of the particular form:(22) u n +2 = f (cid:18) u n +1 , u n , u n − , u n − , d u n d t (cid:19) . n ∈ Z , To practically compute the algebraic entropy we introduce some technical meth-ods to reduce the computational complexity [26,27]. First, we fix the desired numberof iterations to be some fixed N ∈ N . Following remark 3 this means that we needonly finitely many initial conditions given by (15). Then we assume that the spaceof initial conditions is linearly parametrised in the appropriate projective space, i.e.in inhomogenous coordinates it has the following form:(23) u i = α i t + β i α t + β , u i ∈ IN ( N ) . We will assume that the parameter t is the same which describes the “time” evolutionof the problem. To simplify the problem we choose all the parameters involved inthe equations to be integers. Moreover, to avoid accidental factorisations which mayalter the results we choose these integers to be prime numbers . A final simplificationto speed up the computations is given by considering the factorisation of the iteratesin some finite field K r , with r prime number. Remark . Several equations in Lists 1–6, e.g. (E.5) or (E.5 ′ ), depend on some pa-rameters. Depending on the value of the parameters their integrability propertiescan be, in principle, different. As it was done in [17], in order to avoid ambiguities,we use some simple autonomous transformations to fix the values of some parame-ters. The remaining free parameters are then treated as free coefficients and thenfixed to integers following the above discussion. We will describe these subcaseswhen needed in the next section.Using the rules above we are able to avoid accidental cancellations and producea finite sequence of degrees:(24) , d , d , d , d , d , . . . , d N . To extract the asymptotic behaviour from the finite sequence (24): we compute itsgenerating function, i.e. a function g = g ( s ) such that the coefficients of its Taylor LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 9 series(25) g ( z ) = ∞ X l =0 d l z l up to order N coincides with the finite sequence (24). Assuming that such gener-ating function is rational it can be computed using the method of Padé approxi-mants [6, 48].This generating function is predictive tool. Indeed one can readily compute thesuccessive terms in the Taylor expansion for (25) and confront them with the degreescalculated with the iterations. This means that the assumption that the value ofthe algebraic entropy given by the approximate method is in fact very strong andvery unlikely the real value will differ from it.Having a rational generating function will also yield the value of the AlgebraicEntropy from the modulus of the smallest pole of the generating function:(26) η ϕ = log min (cid:26) | z | ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) g ( z ) = 0 (cid:27) . From the generating function one can also find an asymptotic fit for the degrees(24). This can be done by using the Z -transform [14,36]. Indeed, from the definitionof Z -transform it can be readily proved that:(27) d l = Z (cid:20) g (cid:18) ζ (cid:19)(cid:21) l , where Z [ f ( ζ )] l is the Z -transform of the function f ( ζ ) . Remark . We note that the general asymptotic behaviour of the sequence { d l } l ∈ N can be obtained even without computing the Z -transform. Indeed, let us assumethat the given generating g function has radius of convergence ρ > . Then, let usassume that the generating function can be written in the following way:(28) g = A ( z ) + B ( z ) (cid:18) − zρ (cid:19) − β , where A and B are analytic functions for | z | < r such that B ( ρ ) = 0 . Then wehave the following estimate:(29) d N ∼ B ( ρ )Γ ( β ) N β − ρ − N , N → ∞ where Γ( z ) is the Euler Gamma function. When the radius of convergence is one,i.e. when the given equation is integrable, we have the simpler estimate(30) d N ∼ N β − , N → ∞ . Results
In this section we describe the results of the procedure outlined in section 2 for thedifferential-difference equations of Lists 1–6. Specifically, as described in Remark4, we will underline the particular cases in which the parametric equations can bedivided. We notice that certain equations are symmetric under the involution(31) u n → ˜ u n = u − n . This implies that the recurrence defined by solving the equation with respect to u and u − is the same. For equations satisfying this property the growth of thedegree of the iterates can be computed just in one direction, as the growth in theother direction will be the same. Computations are performed using the python program for differential-difference equations presented in [27]. We remark thatthis program was already employed to discuss the integrability of some three-pointdifferential-difference equations in [31].3.1. List 1.
Equation (E.1) : Equation (E.1) is symmetric and has the following growthof degrees:(32) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (32) is:(33) g ( z ) = − z − z + z + 1( z − ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.1)has quadratic growth.3.1.2. Equation (E.2) : Equation (E.2) is symmetric and has the following growthof degrees:(34) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (34) is:(35) g ( z ) = − z − z + 2 z + 1( z − ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.2)has quadratic growth.3.1.3. Equation (E.3) : Equation (E.3) is symmetric and has the same growth ofdegrees as equation (E.2). Therefore we have that equation (E.3) has zero entropyand quadratic growth.3.1.4.
Equation (E.4) : Equation (E.4) is symmetric and has the following growthof degrees:(36) , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (36) is:(37) g ( z ) = − z + z − z − z + z + z + 1( z − ( z + 1) ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.4)has quadratic growth. LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 11
Equation (E.5) : Equation (E.5) depends on the parameter a . Using a simplescaling if a = 0 it is possible to set a = 1 . For this reason we can consider the twocases a = 1 and a = 0 . If a = 1 equation (E.5) is asymmetric, but it has thefollowing growth of degrees in both directions:(38) , , , , , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (38) is:(39) g ( z ) = − z + z − z − z + z + z + 1( z − ( z + 1) ( z + 1) . If a = 0 equation (E.5) is symmetric, but its growth of degrees is still given by thesequence (38) and fitted by the generating function (39). Therefore in both casesthe entropy is zero since all the poles of g lie on the unit circle. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.5)has quadratic growth for all values of a .3.1.6. Equation (E.6) : Equation (E.6) is symmetric and has the following growthof degrees:(40) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (40) is:(41) g ( z ) = − z − z + z + 4 z + 1( z − ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.6)has quadratic growth.3.1.7. Equation (E.1 ′ ) : Equation (E.1 ′ ) is symmetric and has the following growthof degrees:(42) , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (42) is:(43) g ( z ) = − z − z + 4 z − z + 4 z − z + 2 z − z + 1( z − ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.1 ′ )has quadratic growth.3.1.8. Equation (E.2 ′ ) : Equation (E.2 ′ ) is symmetric and has the following growthof degrees:(44) , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (44) is:(45) g ( z ) = − z − z + z − z + 2 z + 2 z + z + 4 z + 1( z − ( z + 1) ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.2 ′ )has quadratic growth.3.2. List 2.
Equation (E.7) : Equation (E.7) depends on four parameters a , b , c and d linked among themselves by the condition ( a + 1) d = bc . Using a linear transfor-mation u n,m → αu n,m + β we need to consider only three different cases:(1) a = 0 , a = − , b = 0 , d = 0 ,(2) a = − , b = 1 , c = 0 ,(3) a = − , b = 0 .Recall that a = 0 in all cases. See [17] for more details. In all the three casesequation (E.7) is asymmetric. However, it has the same growth of degrees in bothdirections and in all the three cases:(46) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (46) is:(47) g ( z ) = − z + z − z − z + z + z + 1( z − ( z + 1) ( z + 1) . In all cases the entropy is zero since all the poles of g lie on the unit circle. Moreover,due to the presence of the factor ( z − following remark 5 we have that equation(E.7) has linear growth for all values of the parameters.3.2.2. Equation (E.8) : Equation (E.8) is not symmetric, but in both directions hasthe following growth of degrees:(48) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (48) is:(49) g ( z ) = z + 3 z + 1( z − ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.8)has linear growth.3.3. List 3.
Equation (E.3 ′ ) . Equation (E.3 ′ ) is symmetric and has the following growthof degrees:(50) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (50) is:(51) g ( z ) = − z − z + z + 1( z − . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.3 ′ )has quadratic growth. LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 13
Equation (E.4 ′ ) . Equation (E.4 ′ ) depends on the parameter a . Using a sim-ple scaling if a = 0 it is possible to set a = 1 . For this reason we can consider thetwo cases a = 1 and a = 0 . Equation (E.4 ′ ) is symmetric for both a = 1 and a = 0 .Moreover, in both cases it has the following growth of degrees:(52) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (52) is:(53) g ( z ) = − z − z + 2 z − z + 2 z + 1 z − . Therefore in both cases the entropy is zero since all the poles of g lie on the unitcircle. Moreover, due to the presence of the factor ( z − following remark 5 wehave that equation (E.4 ′ ) has quadratic growth for all values of a .3.3.3. Equation (E.5 ′ ) : Equation (E.5 ′ ) is not symmetric, but it has the samegrowth of degrees in both directions:(54) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (54) is:(55) g ( z ) = − z − z + 2 z − z + z + 1( z − ( z + 1)( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.5 ′ )has quadratic growth.3.3.4. Equation (E.6 ′ ) : Equation (E.6 ′ ) is symmetric and has the same growth ofdegrees as equation (E.3 ′ ). Therefore we have that equation (E.6 ′ ) has zero entropyand quadratic growth.3.3.5. Equation (E.7 ′ ) : Equation (E.7 ′ ) depends on the parameter a . Using a sim-ple scaling if a = 0 it is possible to set a = 1 . For this reason we can considerthe two cases a = 1 and a = 0 . Equation (E.7 ′ ) is symmetric for both a = 1 and a = 0 . However, in both cases equation (E.7 ′ ) has the same growth of degreesas equation (E.4 ′ ). Therefore we have that equation (E.7 ′ ) has zero entropy andquadratic growth for all values of a .3.3.6. Equation (E.8 ′ ) : Equation (E.8 ′ ) is symmetric and has the following growthof degrees:(56) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (56) is:(57) g ( z ) = − z − z + 2 z − z + z + 1( z − ( z + 1)( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.8 ′ )has quadratic growth.3.4. List 4.
Equation (E.9) . Equation (E.9) is symmetric and has the following growthof degrees:(58) , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (58) is:(59) g ( z ) = − z − z + 2 z − z + z + 2 z − z + 2 z + 1( z − ( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.9)has quadratic growth.3.4.2. Equation (E.10) . Equation (E.10) is symmetric and has the following growthof degrees:(60) , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (60) is:(61) g ( z ) = − z − z − z + z + 8 z + 2 z + 7 z + 1( z − ( z + z + z + z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.10)has quadratic growth.3.4.3. Equation (E.13 ′ ) . Equation (E.13 ′ ) is symmetric and has the following growthof degrees:(62) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (62) is:(63) g ( z ) = − z − z + 4 z − z + 3 z + 1( z − ( z + 1)( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.13 ′ )has quadratic growth.3.5. List 5.
Equation (E.11) . Equation (E.11) is symmetric and has the following growthof degrees:(64) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (64) is:(65) g ( z ) = − z − z + 1( z − ( z + z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.11)has quadratic growth. LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 15
Equation (E.12) . Equation (E.12) depends on the parameter a . Using asimple scaling if a = 0 it is possible to set a = 1 . For this reason we can considerthe two cases a = 1 and a = 0 . If a = 1 equation (E.12) is asymmetric, but it hasthe following growth of degrees in both directions:(66) , , , , , , , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (66) is:(67) g ( z ) = − z + z + z − z + 2 z + z + z + z + 2 z + 1( z − ( z + 1) ( z − z + 1)( z + z + 1) . If a = 0 equation (E.12) is symmetric, but its growth of degrees is still given by thesequence (66) and fitted by the generating function (67). Therefore in both casesthe entropy is zero since all the poles of g lie on the unit circle. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.12)has quadratic growth for all values of a .3.5.3. Equation (E.13) . Equation (E.13) depends on the parameter a . Using asimple scaling if a = 0 it is possible to set a = 1 . For this reason we can considerthe two cases a = 1 and a = 0 . Equation (E.13) is symmetric for both a = 1 and a = 0 . Moreover, in both cases it has the following growth of degrees:(68) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (68) is:(69) g ( z ) = − ( z − z + 1)( z − z − z + 2 z + 1)( z − . If a = 0 equation (E.13) is symmetric, but its growth of degrees is still given by thesequence (68) and fitted by the generating function (69). Therefore in both casesthe entropy is zero since all the poles of g lie on the unit circle. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.13)has quadratic growth for all values of a .3.5.4. Equation (E.14) . Equation (E.14) is symmetric and has the following growthof degrees:(70) , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (70) is:(71) g ( z ) = − ( z + z + 1)( z − z + z + 1)( z − ( z + 1)( z − z + 1)( z + z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.14)has quadratic growth. Equation (4) . Equation (4) is symmetric and has the following growth ofdegrees:(72) , , , , , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (72) is:(73) g ( z ) = − z + z + z − z + z + z + 3 z + 1( z − ( z + 1) ( z − z + 1)( z + z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Using the Z -transform we obtain the following expression for the degrees:(74) d n = n n − n
12 + ( − n n √
336 sin (cid:16) nπ (cid:17) − √ (cid:18) n
54 + 736 (cid:19) sin (cid:18) nπ (cid:19) + 112 cos (cid:16) nπ (cid:17) + (cid:18) n
18 + 79108 (cid:19) cos (cid:18) nπ (cid:19) . Therefore the growth (74) is quadratic as n → ∞ , but we notice also the unusualpresence of oscillating term proportional to ( − n n which explains the high oscil-lations of the sequence (72). A similar was found in [30] on the degree pattern ofsome linearisable quad-equations.3.5.6. Equation (E.9 ′ ) . Equation (E.9 ′ ) is symmetric and has the following growthof degrees:(75) , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (75) is:(76) g ( z ) = − z − z − z + z + z − z + z + 1( z − ( z + z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.9 ′ )has quadratic growth.3.5.7. Equation (E.10 ′ ) . Equation (E.10 ′ ) depends on the parameter a . Using asimple scaling if a = 0 it is possible to set a = 1 . For this reason we can considerthe two cases a = 1 and a = 0 . Equation (E.10 ′ ) is not symmetric for both a = 1 and a = 0 . However, in both cases it has the following growth of degrees:(77) , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (77) is:(78) g ( z ) = − z + z + z + 3 z + 2 z + 2 z + 3 z + 2 z + 1( z − ( z + 1)( z − z + 1)( z + z + 1) ) . Therefore in both cases the entropy is zero since all the poles of g lie on the unitcircle. Moreover, due to the presence of the factor ( z − following remark 5 wehave that equation (E.10 ′ ) has quadratic growth for all values of a . LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 17
Equation (E.11 ′ ) . Equation (E.11 ′ ) depends on the parameter a . Using asimple scaling if a = 0 it is possible to set a = 1 . For this reason we can considerthe two cases a = 1 and a = 0 . Equation (E.11 ′ ) is symmetric for both a = 1 and a = 0 . When a = 1 equation (E.11 ′ ) has the following growth of degrees:(79) , , , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (79) is:(80) g a =1 ( z ) = − z + 2 z + 5 z + 8 z + 5 z + 8 z + 8 z + 6 z + 1( z − ( z + 1)( z − z + 1)( z + z + 1) . When a = 0 equation (E.11 ′ ) has the following growth of degrees:(81) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (79) is:(82) g a =0 ( z ) = − z − z + 2 z − z + 2 z − z − z + 6 z + z + 6 z + 3 z + 10 z + 5 z + 5 z + 3 z + 5 z + 1 ! ( z − ( z + 1)( z − z + 1) ( z + z + 1) . All the poles of g a =1 and g a =0 lie on the unit circle, so that the entropy is zero inboth cases. Moreover, due to the presence of the factor ( z − following remark5 we have that equation (E.11 ′ ) has quadratic growth in both cases.3.5.9. Equation (E.12 ′ ) . Equation (E.12 ′ ) is symmetric and has the same growthof degrees as equation (E.10 ′ ). Therefore we have that equation (E.12 ′ ) has zeroentropy and quadratic growth.3.6. List 6.
Equation (E.15) . Equation (E.15) is symmetric and has the following growthof degrees:(83) , , , , , , , , , , , , . . .. The generating function corresponding to the growth (83) is:(84) g ( z ) = − z − z + 4 z − z + z + 1( z − . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.15)has quadratic growth. Equation (E.16) . Equation (E.16) is symmetric and has the following growthof degrees:(85) , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (85) is:(86) g ( z ) = − z − z − z + z + 4 z + 2 z + 4 z + 1( z − ( z + z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.16)has quadratic growth.3.6.3. Equation (E.17) . We proved that equation (E.17) can be brought in in ra-tional form (6), but this form is not bi-rational. So we cannot apply the algebraicentropy method to this equation.3.6.4.
Equation (E.14 ′ ) . Equation (E.14 ′ ) is symmetric and has the following growthof degrees:(87) , , , , , , , , , , , , , , , , , . . .. The generating function corresponding to the growth (87) is:(88) g ( z ) = − z − z + z + 2 z − z + z + z + 2 z + 5 z + 1( z − ( z + 1)( z + 1) . All the poles of g lie on the unit circle, so that the entropy is zero. Moreover, due tothe presence of the factor ( z − following remark 5 we have that equation (E.14 ′ )has quadratic growth. 4. Discussion
In the previous section we computed the algebraic entropy of all the integrableVolterra-like five-point differential-difference equations recently classified in [23,24].When possible, we showed that the method of algebraic entropy and the method ofgeneralised symmetries agree. That is, we showed that all the equations integrableaccording to the generalised symmetry test are also integrable according to the al-gebraic entropy method, i.e. the algebraic entropy is zero. The algebraic entropymethod is unfortunately unable to treat the semi-discrete Kaup-Kaupershmidtequation (E.17). This is because the generalised symmetry approach, differentlyfrom the algebraic entropy, makes no assumption on the nature of the recurrenceand algebraic or even transcendental terms are allowed. That is, we proved that thefor integrable Volterra-like five-point differential-difference equations the followingversion of the algebraic entropy conjecture holds true:
Conjecture 2.
The condition that algebraic entropy is zero is equivalent to thedefinition of integrability for bi-rational maps.
Except for the two known non-trivially linearisable equations (E.7) and (E.8)the growth is always quadratic. Equation (4) possesses an interesting non-standardhighly oscillating growth, observed for the first time in differential-difference equa-tions. Nevertheless the asymptotic growth is still quadratic.In the case of two-dimensional difference equation it is known that the onlypossible polynomial, i.e. integrable, growth is quadratic [13]. Integrable higher
LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 19 order maps can exhibit higher rate of growth, see e.g. [28, 29, 35]. Despite beinginfinite-dimensional all the integrable Volterra-like five-point differential-differenceequations possess this “minimal” integrable growth.In the case of difference equations it has been observed that degree growth greaterthan quadratic is related to a procedure called deflation [35]. That is, a five-pointequation is reduced to a four-point one using a non-point potential-like transfor-mations of the form:(89) v n = a u n u n +1 + a u n +1 + a u n + a b u n u n +1 + b u n +1 + b u n + b . Let us notice, that the inflated version of a Volterra-like differential-difference equa-tion is not always a Volterra-like differential-difference equation. This fact makesmore difficult to make predictions on the integrability properties of the inflatedforms of differential-difference equations.Finally, we notice that another interesting problem is to study the integrabil-ity properties of the stationary reductions of the integrable Volterra-like five-pointdifferential-difference equations. The stationary reduction of a bi-rational five-point differential-difference equation is a fourth-order difference equation, i.e. afour-dimensional map of the projective space into itself. It will be important tounderstand how integrability arises inside these families of equations and if it fitswith known cases of integrable families of fourth-order differential difference equa-tions [11, 28, 29].
Acknowledgment
GG thanks Prof. R. N. Garifullin, Prof. D. Levi and Prof. R. I. Yamilov forinteresting and helpful discussions during the preparation of this paper.GG is supported by the Australian Research Council through Nalini Joshi’sAustralian Laureate Fellowship grant FL120100094.
References [1] V. E. Adler. Integrable Möbius invariant evolutionary lattices of second order. Preprint on arXiv:1605.00018 .[2] V. E. Adler. On a discrete analog of the Tzitzeica equation. preprint on arXiv:1103.5139 .[3] V. E. Adler and V. V. Postnikov. On vector analogs of the modified volterra lattice.
J. Phys.A: Math. Theor. , 41:455203 (16 pp), 2008.[4] V. E. Adler, A. B. Shabat, and R. I. Yamilov. Symmetry approach to the integrability prob-lem.
Theor. Math. Phys. , 125(3):1603–1661, 2000.[5] V. I. Arnol’d. Dynamics of complexity of intersections.
Bol. Soc. Bras. Mat. , 21:1–10, 1990.[6] G. A. Baker and P. R. Graves-Morris.
Padé approximants . Cambridge University Press, 1996.[7] M. Bellon and C-M. Viallet. Algebraic entropy.
Comm. Math. Phys. , 204:425–437, 1999.[8] O. I. Bogoyavlensky. Integrable discretizations of the kdv equation.
Phys. Lett. A , 134:34–38,1988.[9] O. I. Bogoyavlensky. Algebraic constructions of integrable dynamical systems-extensions ofthe Volterra system.
Russ. Math Surveys , 46:1–64, 1991.[10] F. Calogero. Why are certain nonlinear PDEs both widely applicable and integrable? In V. E.Zakharov, editor,
What is integrability?
Springer, Berlin-Heidelberg, 1991.[11] H. W. Capel and R. Sahadevan. A new family of four-dimensional symplectic and integrablemappings.
Physica A , 289:80–106, 2001.[12] D. K. Demskoy and C-M. Viallet. Algebraic entropy for semi-discrete equations.
J. Phys. A:Math. Theor. , 45:352001 (10 pp), 2012.[13] J. Diller. Dynamics of birational maps of P . Indiana Univ. Math. J. , pages 721–772, 1996.[14] S. Elaydi.
An introduction to Difference Equations . Springer, 3rd edition, 2005. [15] G. Falqui and C-M. Viallet. Singularity, complexity, and quasi-integrability of rational map-pings.
Comm. Math. Phys. , 154:111–125, 1993.[16] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura. Method for solving theKorteweg-deVries equation.
Phys. Rev. Lett. , 19(19):1095–1097, 1967.[17] R. N. Garifullin, G. Gubbiotti, and R. I. Yamilov. Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-differenceequations, 2019. To appear on
J. Nonlinear Math. Phys. , preprint on arXiv:1810.11184[nlin.SI] .[18] R. N. Garifullin, A. V. Mikhailov, and R. I. Yamilov. Discrete equation on a square latticewith a nonstandard structure of generalized symmetries.
Theor. Math. Phys. , 180(1):765–780,2014.[19] R. N. Garifullin and R. I. Yamilov. Generalized symmetry classification of discrete equationsof a class depending on twelve parameters.
J. Phys. A: Math. Theor. , 45:345205 (23pp), 2012.[20] R. N. Garifullin and R. I. Yamilov. Integrable discrete nonautonomous quad-equations asBäcklund auto-transformations for known Volterra and Toda type semidiscrete equations.
J.Phys.: Conf. Ser. , 621:012005 (18pp), 2015.[21] R. N. Garifullin and R. I. Yamilov. On integrability of a discrete analogue of Kaup-Kupershmidt equation.
Ufa Math. J. , 9:158–164, 2017.[22] R. N. Garifullin, R. I. Yamilov, and D. Levi. Non-invertible transformations of differential-difference equations.
J. Phys. A: Math. Theor. , 49:37LT01 (12pp), 2016.[23] R. N. Garifullin, R. I. Yamilov, and D. Levi. Classification of five-point differential-differenceequations.
J. Phys. A: Math. Theor. , 50:125201 (27pp), 2017.[24] R. N. Garifullin, R. I. Yamilov, and D. Levi. Classification of five-point differential-differenceequations II.
J. Phys. A: Math. Theor. , 51:065204 (16pp), 2018.[25] B. Grammaticos, R. G. Halburd, A. Ramani, and C-M. Viallet. How to detect the integrabilityof discrete systems.
J. Phys A: Math. Theor. , 42:454002 (41 pp), 2009. Newton InstitutePreprint NI09060-DIS.[26] G. Gubbiotti. Integrability of difference equations through algebraic entropy and generalizedsymmetries. In D. Levi, R. Verge-Rebelo, and P. Winternitz, editors,
Symmetries and Integra-bility of Difference Equations: Lecture Notes of the Abecederian School of SIDE 12, Montreal2016 , CRM Series in Mathematical Physics, chapter 3, pages 75–152. Springer InternationalPublishing, Berlin, 2017.[27] G. Gubbiotti.
Integrability properties of quad equations consistent on the cube . PhD thesis,Università degli Studi Roma Tre, 2017.[28] G. Gubbiotti, N. Joshi, D. T. Tran, and C-M. Viallet. Complexity and integrability in 4Dbi-rational maps with two invariants, 2019. Preprint on arXiv:1808.04942 [nlin.SI] .[29] G. Gubbiotti, N. Joshi, D. T. Tran, and C-M. Viallet. Integrability properties of a class of4D bi-rational maps with two invariants, 2019. In preparation.[30] G. Gubbiotti, C. Scimiterna, and D. Levi. Algebraic entropy, symmetries and linearization ofquad equations consistent on the cube.
J. Nonlinear Math. Phys. , 23(4):507–543, 2016.[31] G. Gubbiotti, C. Scimiterna, and D. Levi. The non autonomous YdKN equation and gener-alized symmetries of Boll equations.
J. Math. Phys. , 58(5):053507, 2017.[32] I. T. Habibullin, Sokolov V. V., and R. I. Yamilov. Multi-component integrable systems andnonassociative structures. In E. Alfinito, M Boiti, L. Martina, and F. Pempinelli, editors,
Proceedings of 1st Int. Workshop on Nonlinear Physics: Theory and Experiment, Gallipoli,Italy, 29 June - 7 July 1995 , pages 139–168. World Scientific Publishing, 1996.[33] J. Hietarinta and C-M. Viallet. Searching for integrable lattice maps using factorization.
J.Phys. A: Math. Theor. , 40:12629–12643, 2007.[34] Y. Itoh. An H -theorem for a system of competing species. Proc. Japan Acad. , 51:374–379,1975.[35] N. Joshi and C-M. Viallet. Rational Maps with Invariant Surfaces.
J. Int. Sys. , 3:xyy017(14pp), 2018.[36] E. Jury.
Theory and applications of the Z -transform method . Robert E. Krieger, 1964.[37] D. J. Korteweg and G. de Vries. XLI. on the change of form of long waves advancing in arectangular canal, and on a new type of long stationary waves. Phil. Mag. Ser. 5 , 39(240):422–443, 1895.[38] P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves.
Comm. Pure Appl.Math. , 21(5):467–490, 1968.
LGEBRAIC ENTROPY OF A CLASS OF DIFFERENTIAL-DIFFERENCE EQUATIONS 21 [39] D. Levi and R. I. Yamilov. Conditions for the existence of higer symmetries of the evolutionaryequations on the lattice.
J. Math. Phys. , 38(12):6648–6674, 1997.[40] D. Levi and R. I. Yamilov. The generalized symmetry method for discrete equations.
J. Phys.A: Math. Theor. , 42:454012 (18pp), 2009.[41] D. Levi and R. I. Yamilov. Generalized symmetry integrability test for discrete equations onthe square lattice.
J. Phys. A: Math. Theor. , 44:145207 (22pp), 2011.[42] J. Liouville. Note sur l’intégration des équations différentielles de la Dynamique, présentéeau Bureau des Longitudes le 29 juin 1853.
J. Math. Pures Appl. , 20:137–138, 1855.[43] A. V. Mikhailov and A. B. Shabat. Symmetries – test of integrability. In A. S. Fokas and V. E.Zakharov, editors,
Important developments in soliton theory , pages 355–372. Springer-Verlag,1993.[44] A. V. Mikhailov, A. B. Shabat, and V. V. Sokolov. The symmetry approach to classificationof integrable equations. In V. E. Zakharov, editor,
What is Integrability? , pages 115–184.Springer-Verlag, 1991.[45] A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov. The symmetry approach to the classificationof nonlinear equations. complete lists of integrable systems.
Russian Math. Survey , 42(4):1–63, 1987.[46] A. V. Mikhailov and P. Xenitidis. Second order integrability conditions for difference equa-tions: an integrable equation.
Lett. Math. Phys. , pages doi:10.1007/s11005–013–0668–8(20pp), 2013.[47] K. Narita. Soliton solution to extended Volterra equation.
J. Phys. Soc. Japan , 51:1682–1685,1982.[48] H. Padé. Sur la répresentation approchée d’une fonction par des fractions rationelles.
Ann.École Nor. , 3(9):1–93, 1892. Thesis, suppement.[49] A. Russakovskii and B. Shiffman. Value distribution of sequences of rational mappings andcomplex dynamics.
Indiana U. Math. J. , 46:897–932, 1997.[50] V. V. Sokolov. On the symmetries of evolution equations.
Russian Math. Surveys , 43(5):165–204, 1988.[51] V. V. Sokolov and A. B. Shabat. Classification of integrable evolution equations. In
SovietScientific Reviews, Section C, Mathematical Physics Reviews , volume 4, pages 221–280. Har-wood Academic Publishers, New York, 1984.[52] Yu. B. Suris.
The problem of integrable discretization: Hamiltonian approach . Birkhäuser,Basel, 2003.[53] S. Tremblay, B. Grammaticos, and A. Ramani. Integrable lattice equations and their growthproperties.
Phys. Lett. A , 278(6):319–324, 2001.[54] S. Tsujimoto and R. Hirota. Pfaffian representation of solutions to the discrete BKP hierarchyin bilinear form.
J. Phys. Soc. Jpn. , 65:2797–2806, 1996.[55] A. P. Veselov. Growth and integrability in the dynamics of mappings.
Comm. Math. Phys. ,145:181–193, 1992.[56] C-M. Viallet. Algebraic Entropy for lattice equations, 2006. arXiv:0609.043 .[57] C-M. Viallet. Algebraic entropy for differential–delay equations, 2014. arXiv:1408.6161 .[58] E. T. Whittaker.
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies .Cambridge University Press, Cambridge, 1999.[59] R. I. Yamilov. Classification of discrete evolution equations.
Usp. Mat. Nauk. , 38:155–156,1983. in Russian.[60] R. I. Yamilov.
Classification of discrete integrable equations . PhD thesis, Ufa, 1984. in Rus-sian.[61] R. I. Yamilov. Symmetries as integrability criteria for differential difference equations.
J.Phys. A. , 39:R541–R623, 2006.[62] N. J. Zabusky and M. D. Kruskal. Interaction of "Solitons" in a Collisionless Plasma and theRecurrence of Initial States.
Phys. Rev. Lett. , 15(6):240–243, 1965.
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