- Email: [email protected]
Integrable mappings with transcendental invariants
Communications in Nonlinear Science and Numerical Simulation 12 (2007) 350–356 www.elsevier.com/locate/cnsns
Integrable mappings with transcendental invariants B. Grammaticos a
a,*
, A. Ramani
b
GMPIB, Universite´ Paris VII Tour 24-14, 5e e´tage, case 7021 75251 Paris, France b CPT, Ecole Polytechnique CNRS, UMR 7644 91128 Palaiseau, France
Received 14 January 2005; received in revised form 11 March 2005; accepted 14 March 2005 Available online 17 May 2005
Abstract We examine a family of integrable mappings which possess rational invariants involving polynomials of arbitrarily high degree. Next we extend these mappings to the case where their parameters are functions of the independent variable. The resulting mappings do not preserve any invariant but are solvable by linearisation. Using this result we then proceed to construct the solution of the initial autonomous mappings and use it to explicitly construct the invariant, which turns out to be transcendental in the generic case. 2005 Elsevier B.V. All rights reserved. PACS: 02.30.Ik Keywords: Mappings; Invariants; Integrability; Miura transformations
1. Introduction The construction of concrete examples plays a major role in the study of integrable discrete systems. It was in fact after the discovery of a whole family of integrable mappings by Quispel, Roberts and Thompson [1] (hereafter referred to as QRT) that conjectures on discrete integrability could be formulated [2], leading eventually to the derivation of discrete Painleve´ equations [3].
*
Corresponding author. Fax: +33 1 44277979. E-mail address: [email protected] (B. Grammaticos).
1007-5704/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.03.005
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
351
The QRT mapping is presented traditionally in two forms, dubbed ‘‘symmetric’’ and ‘‘asymmetric’’. In the generic case one starts by introducing two 3 · 3 matrices, A0 and A1 of the form 0 1 ai bi ci B C C Ai ¼ B ð1Þ @ di i fi A ji
ki
li
If both these matrices are symmetric the mapping is called symmetric. Otherwise it is called asym0 1 0 21 f1 x ~ ¼ @ x A and constructs the two vectors ~ metric. Next one introduces the vector X F @ f2 A and 0 1 f3 1 g1 ~ @ A G g2 through g3 ~ ~Þ ðA1 X ~Þ; F ¼ ðA0 X ð2aÞ ~ ¼ ðA ~ 0X ~1X ~ Þ ðA ~Þ; G
ð2bÞ
~ are, where the tilde denotes the transpose of the matrix. The components fi, gi of the vectors ~ F; G in general, quartic polynomial of x. Given the fi, gi the mapping assumes the form: f1 ðy n Þ xn f2 ðy n Þ f2 ðy n Þ xn f3 ðy n Þ g ðxnþ1 Þ y n g2 ðxnþ1 Þ ¼ 1 g2 ðxnþ1 Þ y n g3 ðxnþ1 Þ
xnþ1 ¼
ð3aÞ
y nþ1
ð3bÞ
In the symmetric case we have gi = fi and (2.3) reduces to a single equation xmþ1 ¼
f1 ðxm Þ xm1 f2 ðxm Þ f2 ðxm Þ xm1 f3 ðxm Þ
ð4Þ
with the identification xn ! x2n, yn ! x2n+1. As we have explained in [4] the number of genuine parameters in this system is 8 for the asymmetric mapping and 5 for the symmetric one. The QRT mapping possesses an invariant which is biquadratic in x and y: ða0 þ Ka1 Þx2 y 2 þ ðb0 þ Kb1 Þx2 y þ ðc0 þ Kc1 Þx2 þ ðd0 þ Kd1 Þxy 2 þ ð0 þ K1 Þxy þ ðf0 þ Kf1 Þx þ ðj0 þ Kj1 Þy 2 þ ðk0 þ Kk1 Þy þ ðl0 þ Kl1 Þ ¼ 0
ð5Þ
where K plays the role of the integration constant. In the symmetric case the invariant becomes just: ða0 þ Ka1 Þx2nþ1 x2n þ ðb0 þ Kb1 Þxnþ1 xn ðxnþ1 þ xn Þ þ ðc0 þ Kc1 Þðx2nþ1 þ x2n Þ þ ð0 þ K1 Þxnþ1 xn þ ðf0 þ Kf1 Þðxnþ1 þ xn Þ þ ðl0 þ Kl1 Þ ¼ 0
ð6Þ
The ‘‘symmetric’’ QRT mapping has been studied in [5] from the point of view of the singularity confinement discrete integrability criterion [6]. We have shown that under specific assumptions on the structure of the singularities, the symmetric QRT mapping is the only one that satisfies this integrability criterion. This results could have been interpreted as an indication that the only
352
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
integrable mapping of the form (4) is the QRT one. However this is not the case: the result obtained in [5] relies explicitly on an assumption of a specific singularity pattern of the system. While this pattern does indeed correspond to the QRT mapping, one can imagine confining systems with different patterns (which would still possess solutions exhibiting the same growth at infinity, in the Nevanlinna sense [6], as the QRT mapping). Such mappings do indeed exist. In [7] we have obtained a class of systems which do not belong to the QRT family. While the QRT mappings possess an invariant which is a ratio of two biquadratic polynomials, this new class is comprised of mappings with invariants which are ratios of two biquartic polynomials. The, probably, simplest example of such a system is the mapping xnþ1 xn1 ¼
x2n a2 xn 1
ð7Þ
with invariant 2
K¼
2
x2n x2n1 ðxn xn1 Þ 2xn xn1 ðxn þ xn1 Þððxn xn1 Þ a2 Þ þ ðx2n þ x2n1 a2 Þ x2n x2n1
2
ð8Þ
The integration of the mappings of this class were presented in [8] where we have shown that their solution can be expressed in terms of elliptic functions. The examples of mappings with biquartic invariants presented in [7] were obtained by the autonomisation of q-discrete Painleve´ equations. While more examples than the ones presented there do certainly exist, they do not exhaust all the classes of mappings with invariants with degree higher than two. In this paper we shall present a new class of mappings the invariants of which can be rational expressions involving polynomials of degree arbitrarily high. 2. A mapping with high degree invariants The mapping we will study in what follows is 1 1 k þ ¼ xn þ xnþ1 xn þ xn1 xn
ð9Þ
Mappings of this form do exist within the QRT family. Indeed the form (9) can be obtained from an A1 matrix: 0
1
0
0
1
B A1 ¼ @ 0
2
C 0A
1
0
0
ð10Þ
and looking for the appropriate A0. It turns out that for k = 0, 1, 2 there exists an A0 with all elements zero except one, l, and a, respectively, which makes possible to interpret (9) as a QRT mapping. In these cases (9) possesses a biquadratic invariant. What is more interesting is that a mapping of the form (9) can be obtained also starting from an asymmetric QRT system. Indeed if we consider a matrix
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
0
0 @ A0 ¼ 0 0
0 0 1
1 0 1A 0
353
ð11Þ
we obtain the system 3y 2n þ xn y n xn y n 3x2 þ xn y n ¼ n xn y n
xnþ1 ¼
ð12aÞ
y nþ1
ð12bÞ
which can be rewritten as 1 1 1 þ ¼ y n þ xnþ1 xn þ y n 2y n 1 1 1 þ ¼ xn þ y nþ1 xn þ y n 2xn
ð13aÞ ð13bÞ
It suffices now to introduce a new variable X such that X2n = yn, X2n1 = xn. Then Eq. (13a) becomes simply 1 1 1 þ ¼ X n þ X nþ1 X n þ X n1 2X n while (13b) is its 0 0 B A0 ¼ @ 1 0
upshift. Similarly starting from 1 1 0 C 0 0A 0 0
ð14Þ
ð15Þ
we obtain (9) with k = 3/2. The invariants of the mapping for these values can be easily constructed. For (9) withpkffiffiffiffi= 0 we find readily that (xn + xn1) must be a constant with alternating sign, i.e. ðxn þ xn1 Þ ¼ C ð1Þn . It suffices in order to construct a genuine invariant to square this quantity leading to C = (xn + xn1)2. The invariant for k = 1 is even simpler C = xn/xn1. Similarly for k = 2 we find the invariant C = (xn + xn1)2/(xnxn1)2 which is still biquadratic as expected. In the case of the mappings with k = 1/2 and k = 3/2 the invariants are not biquadratic any more. This is due to the fact that, while for k = 1/2 and k = 3/2 the mappings are asymmetric QRT ones, we write them in a form resembling that of the symmetric QRT family without really belonging to it. For k = 1/2 we find C¼
ðxn þ xn1 Þ4 ðxn xn1 Þ2
ð16Þ
while for k = 3/2 the invariant is C¼
ðxn þ xn1 Þ4 x2n x2n1 ðxn xn1 Þ2
ð17Þ
354
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
Higher degree invariants can be obtained for other values of k, in which case no relation to the QRT family exists any more. In the next section we show how to construct them systematically. 3. The Miura transformations The key ingredient for the construction of these mappings is that they are related by simple Miura transformations which moreover can be used to generate mappings with different values of k not in the QRT class. We start from the mapping (9) and introduce the change of variable y = 1/x. We obtain, after some elementary manipulations a mapping for y of the same form with parameter k 0 = 2 k. Thus under this transformation the mapping k = 0 is transformed into that for k = 2, and vice versa, while the k = 1 mapping is invariant. Next we introduce the transformation yn = (xn + xn+1)1. From (9) we get thus the system k ð18aÞ y n þ y n1 ¼ xn 1 xn þ xnþ1 ¼ ð18bÞ yn and eliminating x we obtain for y a mapping of the same form as (9) with parameter k 0 = 1/k. Thus while the mapping k = 1 is invariant, from the one for k = 2, we can obtain the mapping with k = 1/2. Combining the two transformations we can, starting from (9) with a given value of k, obtain a mapping of the same form with parameter k 0 = 1/(2 k). In particular if we start with k of the form k = N/(N + 1) we find k 0 = (N + 1)/(N + 2). Similarly if k = (N + 1)/N we get k 0 = N/(N 1). Thus with starting point k = 1/2 we obtain the sequence 3/2, 2/3, 4/3, 3/4, 5/4, . . . , i.e. all kÕs of the form N/(N ± 1). Following the construction of these mappings, based on the Miura chain, one can also derive the invariants of the mappings. We have for instance, for k = 2/3, C¼
ð2xn1 xn Þ2 ð2xn xn1 Þ2 ðxn þ xn1 Þ6
ð19Þ
for k = 4/3, C¼
ð2xn1 xn Þ2 ð2xn xn1 Þ2 x2n x2n1 ðxn þ xn1 Þ6
ð20Þ
for k = 3/4, C¼
ð3xn1 xn Þ2 ð3xn xn1 Þ2 ðxn xn1 Þ2 ðxn þ xn1 Þ8
ð21Þ
and rational invariants involving polynomials of still higher degrees for higher values of N. 4. Nonautonomous extensions and linearisation The existence of invariants involving polynomials of arbitrarily high degree hints at the existence of deeper integrability properties for mappings of the form (9). This brings us to the investigation of a generalised form
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
a b c þ ¼ xn þ xnþ1 xn þ xn1 xn
355
ð22Þ
where a, b, c can be functions of n. We examine (22) using the algebraic entropy integrability criterion [9]. This criterion is based on the study of the growth of the degree of the iterates of a mapping. The characteristic quantity which can be easily obtained and computed for a rational mapping is the degree of the numerators or denominators of (the irreducible forms of) the iterates of some initial condition. (In order to obtain the degree one must introduce homogeneous coordinates and compute the homogeneity degree). The algebraic entropy is defined as E ¼ limn!1 logðd n Þ=n where dn is the degree of the nth iterate. A generic, nonintegrable, mapping leads to exponential growth of the degrees of the iterates and thus has a nonzero algebraic entropy, while an integrable mapping has zero algebraic entropy. As we have shown in a previous work [10], the degree growth contains information that can be an indication as to the precise integration method to be used and thus should be studied in detail. (Here, we must stress the fact that the degrees of the iterates are not invariant under transformation of the variables. However the degree growth is invariant and characterises the system at hand). As we have shown in [11], second-order mappings which are integrable by spectral methods have quadratic degree growth while linearisable mappings are characterised by linear growth. It turns out that the degree growth of (22) is linear without any constraint whatsoever on a, b, c. Thus we expect (22) to be linearisable. This is indeed the case. Introducing the variable yn = xn/xn1 we transform (22) into a homographic mapping y nþ1 ¼
ða þ b cÞy n þ a c ðc bÞy n þ c
ð23Þ
ItÕs linearisation is straightforward using a Cole-Hopf transformation y = P/Q. What is more interesting is the explicit construction of the solution of the autonomous mapping (9). We take for convenience a = b = m(=1/k) and c = 1. From (23) we obtain the linear system P nþ1 ¼ ð2m 1ÞP n þ ðm 1ÞQn
ð24aÞ
Qnþ1 ¼ ð1 mÞP n þ Qn
ð24bÞ
and eliminating P we find Qnþ1 2mQn þ m2 Qn1 ¼ 0
ð25Þ
The solution of the latter is Qn = mn(an + b), which we can simplify by taking a = 1 (by division) and b = 0 (by changing the origin of n). Introducing M = m/(1 m) we find yn = (n + M)/n. The solution for x can be obtained from xn = ynxn1 and leads to an expression in terms of gamma functions: xn ¼ ð1Þn K
Cðn þ M þ 1Þ Cðn þ 1Þ
ð26Þ
where K is the integration constant. As a matter of fact, xnC(n + 1)/C(n + M + 1), when squared because of the (1)n factor, is just the invariant C = K2. From the expression of y in terms of n we have n = M/(yn + 1) = Mxn1/(xn + xn1). The invariant is then simply
356
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
12 0 Mxn1 xn C 1 ðxn þxn1 Þ A C¼@ Mxn C 1 þ ðxn þx n1 Þ
ð27Þ
What is particularly interesting is the case where M is an integer. In this case the ratio of the two gamma functions in (26) is a finite polynomial in n. The case of integer M corresponds precisely to a k in Eq. (9) of the form (N ± 1)/N. We can easily verify that in this case the invariants obtained from (27) are precisely the ones we found in the previous sections.
5. Conclusion The family of mappings we have presented here furnish a natural framework for the construction of examples of systems with rational invariants involving high-degree polynomials. This shows that the examples given in [7] are certainly not exceptional (although, admittedly, their derivation method is different). Moreover in the present case we were able to exhibit a mapping with transcendental invariant, a very rare occurence for both continuous [12] and discrete systems. While all the mappings we analysed in this paper are one-component, second-order systems, they do not belong (generically) to the QRT family. An interesting case is that of the mappings with k = 1/2, 3/2 which do have an asymmetric QRT form but invariants of degree higher than biquadratic. We intend to return to the study of such systems in some future work. In the present case the integrability of the mappings we examined is related to their linearisability. This study is an example of the interplay between linearisability and, more conventional, invariant-related, integrability.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
Quispel GRW, Roberts JAG, Thompson CJ. Physica 1989;D34:183. Grammaticos B, Ramani A, Papageorgiou V. Phys. Rev. Lett. 1991;67:1825. Ramani A, Grammaticos B, Hietarinta J. Phys. Rev. Lett. 1991;67:1829. Ramani A, Carstea S, Grammaticos B, Ohta Y. Physica A 2002;305:437. Grammaticos B, Ramani A. Meth. Appl. An. 1997;4:196. Hille E. Ordinary differential equations in the complex domain. New York: John Wiley and Sons; 1976. Kimura K, Yahagi H, Hirota R, Ramani A, Grammaticos B, Ohta Y. J. Phys. A 2002;35:9205. Grammaticos B, Tamizhmani T, Ramani A, Carstea AS, Tamizhmani KM. J. Phys. Soc. Jpn. 2002;71:443. Hietarinta J, Viallet C. Phys. Rev. Lett. 1998;81:325. Ohta Y, Tamizhmani KM, Grammaticos B, Ramani A. Phys. Lett. A 1999;262:152. Ramani A, Grammaticos B, Lafortune S, Ohta Y. J. Phys. A 2000;33:L287. Hietarinta J. Phys. Rep. 1987;147:87.
Integrable mappings with transcendental invariants B. Grammaticos a
a,*
, A. Ramani
b
GMPIB, Universite´ Paris VII Tour 24-14, 5e e´tage, case 7021 75251 Paris, France b CPT, Ecole Polytechnique CNRS, UMR 7644 91128 Palaiseau, France
Received 14 January 2005; received in revised form 11 March 2005; accepted 14 March 2005 Available online 17 May 2005
Abstract We examine a family of integrable mappings which possess rational invariants involving polynomials of arbitrarily high degree. Next we extend these mappings to the case where their parameters are functions of the independent variable. The resulting mappings do not preserve any invariant but are solvable by linearisation. Using this result we then proceed to construct the solution of the initial autonomous mappings and use it to explicitly construct the invariant, which turns out to be transcendental in the generic case. 2005 Elsevier B.V. All rights reserved. PACS: 02.30.Ik Keywords: Mappings; Invariants; Integrability; Miura transformations
1. Introduction The construction of concrete examples plays a major role in the study of integrable discrete systems. It was in fact after the discovery of a whole family of integrable mappings by Quispel, Roberts and Thompson [1] (hereafter referred to as QRT) that conjectures on discrete integrability could be formulated [2], leading eventually to the derivation of discrete Painleve´ equations [3].
*
Corresponding author. Fax: +33 1 44277979. E-mail address: [email protected] (B. Grammaticos).
1007-5704/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.03.005
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
351
The QRT mapping is presented traditionally in two forms, dubbed ‘‘symmetric’’ and ‘‘asymmetric’’. In the generic case one starts by introducing two 3 · 3 matrices, A0 and A1 of the form 0 1 ai bi ci B C C Ai ¼ B ð1Þ @ di i fi A ji
ki
li
If both these matrices are symmetric the mapping is called symmetric. Otherwise it is called asym0 1 0 21 f1 x ~ ¼ @ x A and constructs the two vectors ~ metric. Next one introduces the vector X F @ f2 A and 0 1 f3 1 g1 ~ @ A G g2 through g3 ~ ~Þ ðA1 X ~Þ; F ¼ ðA0 X ð2aÞ ~ ¼ ðA ~ 0X ~1X ~ Þ ðA ~Þ; G
ð2bÞ
~ are, where the tilde denotes the transpose of the matrix. The components fi, gi of the vectors ~ F; G in general, quartic polynomial of x. Given the fi, gi the mapping assumes the form: f1 ðy n Þ xn f2 ðy n Þ f2 ðy n Þ xn f3 ðy n Þ g ðxnþ1 Þ y n g2 ðxnþ1 Þ ¼ 1 g2 ðxnþ1 Þ y n g3 ðxnþ1 Þ
xnþ1 ¼
ð3aÞ
y nþ1
ð3bÞ
In the symmetric case we have gi = fi and (2.3) reduces to a single equation xmþ1 ¼
f1 ðxm Þ xm1 f2 ðxm Þ f2 ðxm Þ xm1 f3 ðxm Þ
ð4Þ
with the identification xn ! x2n, yn ! x2n+1. As we have explained in [4] the number of genuine parameters in this system is 8 for the asymmetric mapping and 5 for the symmetric one. The QRT mapping possesses an invariant which is biquadratic in x and y: ða0 þ Ka1 Þx2 y 2 þ ðb0 þ Kb1 Þx2 y þ ðc0 þ Kc1 Þx2 þ ðd0 þ Kd1 Þxy 2 þ ð0 þ K1 Þxy þ ðf0 þ Kf1 Þx þ ðj0 þ Kj1 Þy 2 þ ðk0 þ Kk1 Þy þ ðl0 þ Kl1 Þ ¼ 0
ð5Þ
where K plays the role of the integration constant. In the symmetric case the invariant becomes just: ða0 þ Ka1 Þx2nþ1 x2n þ ðb0 þ Kb1 Þxnþ1 xn ðxnþ1 þ xn Þ þ ðc0 þ Kc1 Þðx2nþ1 þ x2n Þ þ ð0 þ K1 Þxnþ1 xn þ ðf0 þ Kf1 Þðxnþ1 þ xn Þ þ ðl0 þ Kl1 Þ ¼ 0
ð6Þ
The ‘‘symmetric’’ QRT mapping has been studied in [5] from the point of view of the singularity confinement discrete integrability criterion [6]. We have shown that under specific assumptions on the structure of the singularities, the symmetric QRT mapping is the only one that satisfies this integrability criterion. This results could have been interpreted as an indication that the only
352
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
integrable mapping of the form (4) is the QRT one. However this is not the case: the result obtained in [5] relies explicitly on an assumption of a specific singularity pattern of the system. While this pattern does indeed correspond to the QRT mapping, one can imagine confining systems with different patterns (which would still possess solutions exhibiting the same growth at infinity, in the Nevanlinna sense [6], as the QRT mapping). Such mappings do indeed exist. In [7] we have obtained a class of systems which do not belong to the QRT family. While the QRT mappings possess an invariant which is a ratio of two biquadratic polynomials, this new class is comprised of mappings with invariants which are ratios of two biquartic polynomials. The, probably, simplest example of such a system is the mapping xnþ1 xn1 ¼
x2n a2 xn 1
ð7Þ
with invariant 2
K¼
2
x2n x2n1 ðxn xn1 Þ 2xn xn1 ðxn þ xn1 Þððxn xn1 Þ a2 Þ þ ðx2n þ x2n1 a2 Þ x2n x2n1
2
ð8Þ
The integration of the mappings of this class were presented in [8] where we have shown that their solution can be expressed in terms of elliptic functions. The examples of mappings with biquartic invariants presented in [7] were obtained by the autonomisation of q-discrete Painleve´ equations. While more examples than the ones presented there do certainly exist, they do not exhaust all the classes of mappings with invariants with degree higher than two. In this paper we shall present a new class of mappings the invariants of which can be rational expressions involving polynomials of degree arbitrarily high. 2. A mapping with high degree invariants The mapping we will study in what follows is 1 1 k þ ¼ xn þ xnþ1 xn þ xn1 xn
ð9Þ
Mappings of this form do exist within the QRT family. Indeed the form (9) can be obtained from an A1 matrix: 0
1
0
0
1
B A1 ¼ @ 0
2
C 0A
1
0
0
ð10Þ
and looking for the appropriate A0. It turns out that for k = 0, 1, 2 there exists an A0 with all elements zero except one, l, and a, respectively, which makes possible to interpret (9) as a QRT mapping. In these cases (9) possesses a biquadratic invariant. What is more interesting is that a mapping of the form (9) can be obtained also starting from an asymmetric QRT system. Indeed if we consider a matrix
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
0
0 @ A0 ¼ 0 0
0 0 1
1 0 1A 0
353
ð11Þ
we obtain the system 3y 2n þ xn y n xn y n 3x2 þ xn y n ¼ n xn y n
xnþ1 ¼
ð12aÞ
y nþ1
ð12bÞ
which can be rewritten as 1 1 1 þ ¼ y n þ xnþ1 xn þ y n 2y n 1 1 1 þ ¼ xn þ y nþ1 xn þ y n 2xn
ð13aÞ ð13bÞ
It suffices now to introduce a new variable X such that X2n = yn, X2n1 = xn. Then Eq. (13a) becomes simply 1 1 1 þ ¼ X n þ X nþ1 X n þ X n1 2X n while (13b) is its 0 0 B A0 ¼ @ 1 0
upshift. Similarly starting from 1 1 0 C 0 0A 0 0
ð14Þ
ð15Þ
we obtain (9) with k = 3/2. The invariants of the mapping for these values can be easily constructed. For (9) withpkffiffiffiffi= 0 we find readily that (xn + xn1) must be a constant with alternating sign, i.e. ðxn þ xn1 Þ ¼ C ð1Þn . It suffices in order to construct a genuine invariant to square this quantity leading to C = (xn + xn1)2. The invariant for k = 1 is even simpler C = xn/xn1. Similarly for k = 2 we find the invariant C = (xn + xn1)2/(xnxn1)2 which is still biquadratic as expected. In the case of the mappings with k = 1/2 and k = 3/2 the invariants are not biquadratic any more. This is due to the fact that, while for k = 1/2 and k = 3/2 the mappings are asymmetric QRT ones, we write them in a form resembling that of the symmetric QRT family without really belonging to it. For k = 1/2 we find C¼
ðxn þ xn1 Þ4 ðxn xn1 Þ2
ð16Þ
while for k = 3/2 the invariant is C¼
ðxn þ xn1 Þ4 x2n x2n1 ðxn xn1 Þ2
ð17Þ
354
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
Higher degree invariants can be obtained for other values of k, in which case no relation to the QRT family exists any more. In the next section we show how to construct them systematically. 3. The Miura transformations The key ingredient for the construction of these mappings is that they are related by simple Miura transformations which moreover can be used to generate mappings with different values of k not in the QRT class. We start from the mapping (9) and introduce the change of variable y = 1/x. We obtain, after some elementary manipulations a mapping for y of the same form with parameter k 0 = 2 k. Thus under this transformation the mapping k = 0 is transformed into that for k = 2, and vice versa, while the k = 1 mapping is invariant. Next we introduce the transformation yn = (xn + xn+1)1. From (9) we get thus the system k ð18aÞ y n þ y n1 ¼ xn 1 xn þ xnþ1 ¼ ð18bÞ yn and eliminating x we obtain for y a mapping of the same form as (9) with parameter k 0 = 1/k. Thus while the mapping k = 1 is invariant, from the one for k = 2, we can obtain the mapping with k = 1/2. Combining the two transformations we can, starting from (9) with a given value of k, obtain a mapping of the same form with parameter k 0 = 1/(2 k). In particular if we start with k of the form k = N/(N + 1) we find k 0 = (N + 1)/(N + 2). Similarly if k = (N + 1)/N we get k 0 = N/(N 1). Thus with starting point k = 1/2 we obtain the sequence 3/2, 2/3, 4/3, 3/4, 5/4, . . . , i.e. all kÕs of the form N/(N ± 1). Following the construction of these mappings, based on the Miura chain, one can also derive the invariants of the mappings. We have for instance, for k = 2/3, C¼
ð2xn1 xn Þ2 ð2xn xn1 Þ2 ðxn þ xn1 Þ6
ð19Þ
for k = 4/3, C¼
ð2xn1 xn Þ2 ð2xn xn1 Þ2 x2n x2n1 ðxn þ xn1 Þ6
ð20Þ
for k = 3/4, C¼
ð3xn1 xn Þ2 ð3xn xn1 Þ2 ðxn xn1 Þ2 ðxn þ xn1 Þ8
ð21Þ
and rational invariants involving polynomials of still higher degrees for higher values of N. 4. Nonautonomous extensions and linearisation The existence of invariants involving polynomials of arbitrarily high degree hints at the existence of deeper integrability properties for mappings of the form (9). This brings us to the investigation of a generalised form
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
a b c þ ¼ xn þ xnþ1 xn þ xn1 xn
355
ð22Þ
where a, b, c can be functions of n. We examine (22) using the algebraic entropy integrability criterion [9]. This criterion is based on the study of the growth of the degree of the iterates of a mapping. The characteristic quantity which can be easily obtained and computed for a rational mapping is the degree of the numerators or denominators of (the irreducible forms of) the iterates of some initial condition. (In order to obtain the degree one must introduce homogeneous coordinates and compute the homogeneity degree). The algebraic entropy is defined as E ¼ limn!1 logðd n Þ=n where dn is the degree of the nth iterate. A generic, nonintegrable, mapping leads to exponential growth of the degrees of the iterates and thus has a nonzero algebraic entropy, while an integrable mapping has zero algebraic entropy. As we have shown in a previous work [10], the degree growth contains information that can be an indication as to the precise integration method to be used and thus should be studied in detail. (Here, we must stress the fact that the degrees of the iterates are not invariant under transformation of the variables. However the degree growth is invariant and characterises the system at hand). As we have shown in [11], second-order mappings which are integrable by spectral methods have quadratic degree growth while linearisable mappings are characterised by linear growth. It turns out that the degree growth of (22) is linear without any constraint whatsoever on a, b, c. Thus we expect (22) to be linearisable. This is indeed the case. Introducing the variable yn = xn/xn1 we transform (22) into a homographic mapping y nþ1 ¼
ða þ b cÞy n þ a c ðc bÞy n þ c
ð23Þ
ItÕs linearisation is straightforward using a Cole-Hopf transformation y = P/Q. What is more interesting is the explicit construction of the solution of the autonomous mapping (9). We take for convenience a = b = m(=1/k) and c = 1. From (23) we obtain the linear system P nþ1 ¼ ð2m 1ÞP n þ ðm 1ÞQn
ð24aÞ
Qnþ1 ¼ ð1 mÞP n þ Qn
ð24bÞ
and eliminating P we find Qnþ1 2mQn þ m2 Qn1 ¼ 0
ð25Þ
The solution of the latter is Qn = mn(an + b), which we can simplify by taking a = 1 (by division) and b = 0 (by changing the origin of n). Introducing M = m/(1 m) we find yn = (n + M)/n. The solution for x can be obtained from xn = ynxn1 and leads to an expression in terms of gamma functions: xn ¼ ð1Þn K
Cðn þ M þ 1Þ Cðn þ 1Þ
ð26Þ
where K is the integration constant. As a matter of fact, xnC(n + 1)/C(n + M + 1), when squared because of the (1)n factor, is just the invariant C = K2. From the expression of y in terms of n we have n = M/(yn + 1) = Mxn1/(xn + xn1). The invariant is then simply
356
B. Grammaticos, A. Ramani / Commun. in Nonlinear Science and Numerical Simulation 12 (2007) 350–356
12 0 Mxn1 xn C 1 ðxn þxn1 Þ A C¼@ Mxn C 1 þ ðxn þx n1 Þ
ð27Þ
What is particularly interesting is the case where M is an integer. In this case the ratio of the two gamma functions in (26) is a finite polynomial in n. The case of integer M corresponds precisely to a k in Eq. (9) of the form (N ± 1)/N. We can easily verify that in this case the invariants obtained from (27) are precisely the ones we found in the previous sections.
5. Conclusion The family of mappings we have presented here furnish a natural framework for the construction of examples of systems with rational invariants involving high-degree polynomials. This shows that the examples given in [7] are certainly not exceptional (although, admittedly, their derivation method is different). Moreover in the present case we were able to exhibit a mapping with transcendental invariant, a very rare occurence for both continuous [12] and discrete systems. While all the mappings we analysed in this paper are one-component, second-order systems, they do not belong (generically) to the QRT family. An interesting case is that of the mappings with k = 1/2, 3/2 which do have an asymmetric QRT form but invariants of degree higher than biquadratic. We intend to return to the study of such systems in some future work. In the present case the integrability of the mappings we examined is related to their linearisability. This study is an example of the interplay between linearisability and, more conventional, invariant-related, integrability.
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