A novel 8-parameter integrable map in R 4
aa r X i v : . [ n li n . S I] J u l A novel 8-parameter integrable map in R G R W Quispel, D I McLaren and P H van der Kamp
Department of Mathematics, La Trobe University, Bundoora, VIC 3083, AustraliaE-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
Abstract.
We present a novel 8-parameter integrable map in R . The map is measure-preserving and possesses two functionally independent 2-integrals, as well as a measure-preserving 2-symmetry.
1. Introduction
Discrete integrable systems have attracted a lot of attention in recent years [9]. One of the reasonsfor this comes from physics: many physical models include discreteness at a fundamental level.Another reason for the increased interest in discrete integrable systems comes from mathematics:in several instances it turns out that discrete integrable systems are arguably richer, or morefundamental than continuous (i.e. non-discrete) ones. Prime examples are (i) Integrable partialdifference equations (P∆Es), where a single P∆E yields (through the use of vertex operators)an entire infinite hierarchy of integrable partial differential equations [19]; (ii) Discrete Painlev´eequations, where the Sakai classification is much richer in the discrete case than in the continuousone [18]; (iii) Darboux polynomials, where in the discrete case unique factorization of the so-calledco-factors can be used (which does not exist in the continuous (additive) case) ‡ .In this Letter we will be interested in autonomous integrable ordinary difference equations (ormaps). Much interest was generated by the discovery of the 18-parameter integrable QRT mapin R ( [6, 16, 17]). For some other examples in higher dimensions, cf. e.g. Chapter 6 of [9].A special aspect of the maps we consider in this Letter is that they are an example of integrablemaps arising as discretisations of ordinary differential equations (ODEs). Earlier examples ofthis arose using the Kahan discretisation of first-order quadratic ODEs (cf. [12], [10], [15], [5]and references therein), or by the discretisation of ODEs of order 1 and arbitrary degree usingpolarisation methods [4], and by the methods in [11] for the discretisation of ODEs of order o and degree o + 1, cf. also [13, 14].In section 3 we present a novel integrable 8-parameter map in R . This map generalizes a 5-parameter map in R found earlier in [4] to the inhomogeneous case, and because the derivationof the novel map may be somewhat mysterious if the reader is unfamiliar with the previous mapand its derivation, we summarise the latter in section 2. ‡ cf [2, 3] for the discrete case, and [7] for a very nice introduction to the continuous case. novel 8-parameter integrable map in R
2. What went before
In [4] Celledoni, McLachlan, McLaren, Owren and Quispel introduced a novel integrable map in R . It was constructed as follows.The authors considered the homogeneous quartic Hamiltonian H = aq + 4 bq p + 6 cq p + 4 dqp + ep , (1)where a, b, c, d and e are 5 arbitrary parameters.This gave rise to an ordinary differential equation (ODE) ddt (cid:18) qp (cid:19) = (cid:18) − (cid:19) ∇ H = f (cid:18) qp (cid:19) , (2)where the cubic vector field f is defined by f (cid:18) qp (cid:19) = (cid:18) bq + 12 cq p + 12 dqp + 4 ep − aq − bq p − cqp − dp (cid:19) . (3)Defining x := (cid:18) qp (cid:19) , and introducing the timestep h , the vector field (2) was then discretized: x n +2 − x n h = F ( x n , x n +1 , x n +2 ) , (4)where F was defined using polarization, i.e. F ( x n , x n +1 , x n +2 ) := 16 ∂∂α ∂∂α ∂∂α f ( α x n + α x n +1 + α x n +2 ) | α =0 (5)It is not difficult to check that the multilinear function F defined by (5) is equivalent to F ( x n , x n +1 , x n +2 ) := 92 f (cid:18) x n + x n +1 + x n +2 (cid:19) − f (cid:18) x n + x n +1 (cid:19) − f (cid:18) x n + x n +2 (cid:19) − f (cid:18) x n +1 + x n +2 (cid:19) + 16 f ( x n ) + 16 f ( x n +1 ) + 16 f ( x n +2 ) , (6)cf [4] and page 110 of reference [8].By construction, the rhs of (5) is linear in x n +2 and x n for cubic vector fields, i.e. (4) represents abirational map (see [4]), and it was shown that this map possesses two functionally independent2-integrals (recall that a 2-integral of a map φ is defined to be an integral of φ ◦ φ ): I ( q n , p n , q n +1 , p n +1 ) = q n p n +1 − p n q n +1 (7) I ( q n +1 , p n +1 , q n +2 , p n +2 ) = q n +1 p n +2 − p n +1 q n +2 , (8)where q n +2 and p n +2 should be eliminated from (7) using (4).Note that (7) above does not depend on the parameters a, b, c, d, e (in contrast to (8), which willdepend on the parameters once expressed in q n , q n +1 , p n , p n +1 ).The map (4) also preserves the measure dq n ∧ dp n ∧ dq n +1 ∧ dp n +1 h ∆ , (9) novel 8-parameter integrable map in R § ∆ = (cid:12)(cid:12)(cid:12)(cid:12) c dd e (cid:12)(cid:12)(cid:12)(cid:12) p n p n +1 + (cid:12)(cid:12)(cid:12)(cid:12) b cd e (cid:12)(cid:12)(cid:12)(cid:12) ( p n p n +1 q n +1 + p n q n q n +1 ) (10)+ (cid:12)(cid:12)(cid:12)(cid:12) b cc d (cid:12)(cid:12)(cid:12)(cid:12) ( p n q n +1 + q n p n +1 ) + (cid:12)(cid:12)(cid:12)(cid:12) a bc d (cid:12)(cid:12)(cid:12)(cid:12) ( q n p n +1 q n +1 + p n q n q n +1 )+ (cid:12)(cid:12)(cid:12)(cid:12) a cc e (cid:12)(cid:12)(cid:12)(cid:12) p n q n p n +1 q n +1 + (cid:12)(cid:12)(cid:12)(cid:12) a bb c (cid:12)(cid:12)(cid:12)(cid:12) q n q n +1 . Finally, the map (4) is invariant under the scaling symmetry group x n → λ ( − n x n . (11)
3. A novel 8-parameter integrable map in R We now generalise the treatment of section 2 to the non-homogeneous Hamiltonian H = aq + 4 bq p + 6 cq p + 4 dqp + ep + 12 ρq + σqp + 12 τ p , (12)where a, b, c, d, e, ρ, σ and τ are 8 arbitrary parameters.This gives rise to an ODE ddt (cid:18) qp (cid:19) = (cid:18) − (cid:19) ∇ H = f (cid:18) qp (cid:19) + f (cid:18) qp (cid:19) , (13)where the cubic part of the vector field, f , is again given by (3), whereas the linear part f isgiven by f (cid:18) qp (cid:19) = (cid:18) σq + τ p − ρq − σp (cid:19) . (14)We now discretise the cubic part resp. the linear part of the vector field in different ways: x n +2 − x n h = F ( x n , x n +1 , x n +2 ) + F ( x n , x n +2 ) , (15)where F is again defined by (5), but F is defined by a kind of midpoint rule: F ( x n , x n +2 ) = f (cid:18) x n + x n +2 (cid:19) . (16)It follows that equation (15) again defines a birational map, and, importantly, it again preservesthe scaling symmetry (11). (Indeed the latter is the primary reason we use the discretization(16)).Two questions thus remain:(i) Does eq (15) preserve two 2-integrals?(ii) Is eq (15) measure-preserving?The answer to both these questions will turn out to be positive.We actually had numerical evidence several years ago that the map (15) (or at least a special caseof it) was integrable. However it has taken us until now to actually find closed-form expressionsfor the preserved measure and for the 2-integrals. § Erratum: In eqs (4.1) of [4], 1 − h ∆ should read 1 + 4 h ∆. Their ∆ is our ∆ . novel 8-parameter integrable map in R
4A first clue to the identity of a possible 2-integral of (15) came when we were carryingout experimental mathematical computations (in the sense of [1]) to find “discrete Darbouxpolynomials” for the map (15) (cf. [2] and [3]). This gave a hint that a possible quadratic2-integral I ( q n , p n , q n +1 , p n +1 ) generalising (7), might exist for the map (15).However, the mathematical complexity of the general 8-parameter map (15) was too great tocarry out these computations for a completely general quadratic 2-integral in four variables withall 8 parameters symbolic.Our process of discovery thus proceeded in two steps:Step 1: Taking all parameters a, b, c, d, e, ρ, σ, τ and h to be random integers, and assuming the 2-integral was an arbitrary quadratic function in four variables (with coefficients to be determined),we computed the 2-integral for a large number of random choices of the integer parameters. Ineach case, it turned out that the same six coefficients in the quadratic function were zero, i.e.the 2-integral always had the form I ( q n , p n , q n +1 , p n +1 ) = Aq n q n +1 + Bp n p n +1 + Cq n p n +1 + Dp n q n +1 , (17)where A, B, C , and D depended on the parameters in a way as yet to be determined.Step 2: Now taking all parameters a, b, c, d, e, ρ, σ, τ and h symbolic, and assuming the 2-integral I had the special quadratic form (17), we found I ( q n , p n , q n +1 , p n +1 ) = ( hσ + 1) p n q n +1 + ( hσ − q n p n +1 (18)+ hρq n q n +1 + hτ p n p n +1 . Notes:(i) The 2-integral (18) is invariant under the scaling symmetry group (11) k .(ii) In the continuum limit h →
0, and using eq (13), the integral I ( q n , p n , q n +1 , p n +1 ) /h → H ( q, p ).(iii) Like equation (7), equation (18) does not explicitly depend on the parameters a, b, c, d, e .(iv) Note that it is a common feature of many dynamical systems that one has a choice toeither study a given phenomenon for a single system containing as many free parameters aspossible, or alternatively for multiple systems in so-called normal form (obtained by suitabletransformations of the variables), containing fewer parameters. Both in our earlier workson the QRT map [16, 17], and on the 5-parameter map in R [4], as well as in the currentLetter, we have chosen the former option.Once we had the putative equation (18), it was not difficult to verify using symbolic computationthat I ( q n , p n , q n +1 , p n +1 ) and I ( q n +1 , p n +1 , q n +2 , p n +2 ) are indeed functionally independent 2-integrals of (15).The map (15) preserves the measure dq n ∧ dp n ∧ dq n +1 ∧ dp n +1 h (∆ + ∆ ) , (19)where the quartic function ∆ is given by (10) and the quadratic function ∆ is given by∆ = 12 (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) a bσ τ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) c bσ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) q n q n +1 + 12 (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) c dσ τ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) e dσ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) p n p n +1 (20)+ 12 (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) b cσ τ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d cσ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ( p n q n +1 + q n p n +1 ) + 14 (cid:12)(cid:12)(cid:12)(cid:12) ρ σσ τ (cid:12)(cid:12)(cid:12)(cid:12) . k The scaling symmetry (11) is an essential ingredient in our proof of the Theorem in the current Letter that themap (15) is integrable (as well as in our proof in [4] that the map (4) is integrable). novel 8-parameter integrable map in R Theorem
The birational map defined by (15) is integrable.
Proof
The proof of integrability is identical to the proof in [4]. The second iterate of the mapdefined by (15) has a one-dimensional measure-preserving symmetry group. The map thusdescends to a measure-preserving map on the three-dimensional quotient. The two integralsof the second iterate of the map are invariant under the symmetry and therefore also pass to thequotient. This yields a three-dimensional measure-preserving map with two integrals, which isthus integrable.
Acknowledgements
We are grateful to R. McLachlan for early discussions on scaling symmetry, and to E. Celledoni,B. Owren and B. Tapley for many discussions on discrete Darboux polynomials.
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