MATSUKIDAIRA, Junta

NOBE, Atsushi

TANAKA, Hironori

TSUDA, Teruhisa

Kyoto University Research Information Repository

Title Constructing two-dimensional integrable mappings that possessinvariants of high degree (Expansion of Integrable Systems)Author(s)TANAKA, Hironori; MATSUKIDAIRA, Junta; NOBE,Atsushi; TSUDA, TeruhisaCitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2009), B13: 75-84Issue Date2009-10URL http://hdl.handle.net/2433/176807RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Koˆkyuˆroku BessatsuB13 (2009), 75–84Constructing two-dimensional integrable mappingsthat possess invariants of high degreeByHironori Tanaka∗, Junta Matsukidaira∗∗, Atsushi Nobe∗∗∗and Teruhisa Tsuda†AbstractWe propose a method for constructing two-dimensional integrable mappings that possessinvariants with degree higher than two. Such integrable mappings are obtained by making acomposition of a QRT mapping and a mapping that preserves the invariant curve of the QRTmapping except for changing the integration constant involved. We show several concreteexamples whose integration constants change with period 2 and 3.§ 1. IntroductionIntegrable mappings have attracted much attention, and many studies have exam-ined them from various viewpoints, such as integrability criteria (singularity confine-ment property [1], algebraic entropy [2]) and geometric or algebraic description of theequations [3–8].In particular, second-order integrable mappings have been extensively studied, anda number of significant properties have been obtained.The QRT mapping introduced by Quispel, Roberts and Thompson in 1989 [9, 10]is the 12-parameter family of second-order integrable mappings and is given by(1.1) xn+1 =f1(xn)− xn−1f2(xn)f2(xn)− xn−1f3(xn) ,Received March 17, 2009. Accepted April 17, 2009.2000 Mathematics Subject Classification(s): 39A10∗Ryukoku University, Seta, Otsu, Shiga 520-2194, Japan.e-mail: t08m005@mail.ryukoku.ac.jp∗∗Ryukoku University, Seta, Otsu, Shiga 520-2194, Japan.e-mail: junta@rins.ryukoku.ac.jp∗∗∗Osaka University, 1-3 Machikaneyama-cho Toyonaka, Osaka 560-8531, Japan.e-mail: nobe@sigmath.es.osaka-u.ac.jp†Kyushu University, Hakozaki, Fukuoka 812-8581, Japan.e-mail: tudateru@math.kyushu-u.ac.jpc© 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.76 Hironori Tanaka et al.where fj(x) is defined by(1.2)f1(x)f2(x)f3(x) = A0x2x1×A1x2x1with arbitrary symmetric 3×3 matricesAi =αi βi γiβi $i ζiγi ζi µi , i = 0, 1.(1.3)Mapping (1.1) possesses an invariantK = K(xn−1, xn) satisfyingK(xn−1, xn) = K(xn, xn+1),where K is a ratio of biquadratic polynomials of the formK(x, y) =α0x2y2 + β0xy(x+ y) + γ0(x2 + y2) + $0xy + ζ0(x+ y) + µ0α1x2y2 + β1xy(x+ y) + γ1(x2 + y2) + $1xy + ζ1(x+ y) + µ1.(1.4)A generalization of mapping (1.1) has been proposed by Quispel, Roberts andThompson under the name “asymmetric.” It is an 18-parameter family of two-dimensionalmappings of the forms(1.5) xn+1 =f1(yn)− xnf2(yn)f2(yn)− xnf3(yn) , yn+1 =g1(xn+1)− yng2(xn+1)g2(xn+1)− yng3(xn+1) ,where fj(x), gj(x) are defined byf1(x)f2(x)f3(x) = Ax2x1×Bx2x1 ,(1.6)g1(x)g2(x)g3(x) = tAx2x1× tBx2x1 ,(1.7)with arbitrary 3×3 matrices A and B. Mapping (1.5) possesses an invariant K =K(xn−1, yn−1) satisfying K(xn−1, yn−1) = K(xn, yn−1) = K(xn, yn), where K is aConstructing two-dimensional integrable mappings 77ratio of biquadratic polynomials of the formK(x, y) =(x2 x 1)Ay2y1(x2 x 1)By2y1(1.8)=a00x2y2 + a01x2y + a02x2 + a10xy2 + a11xy + a12x+ a20y2 + a21y + a22b00x2y2 + b01x2y + b02x2 + b10xy2 + b11xy + b12x+ b20y2 + b21y + b22.(1.9)Since both symmetric and asymmetric QRT mappings constitute a multi-parameterfamily of integrable systems and meet the criteria for integrability, such as singularityconfinement and nonzero algebraic entropy, they are the prototypical second-order inte-grable mappings. However, whether the QRT mapping is the most general second-orderintegrable mapping has not yet been clarified.In [11], Hirota et al. investigated the integrability of third-order mappings andproposed nine integrable cases. In a recent work [12], most of them turned out to becompositions of two particular QRT mappings.While investigating third-order mappings in [11], Hirota et al. have found a newsecond-order integrable mapping [13](1.10) (xnxn+1 − 1)(xnxn−1 − 1) = (xn − a)(xn − 1/a)(x2n − 1)p2x2n − 1,which possesses a biquartic invariant(1.11)K(xn−1, xn) =((xn − xn−1)2 − p2(xnxn−1 − 1)2)((xn + xn−1 − a− 1/a)2 − p2(xnxn−1 − 1)2)(xnxn−1 − 1)2 .As the invariant is not biquadratic but biquartic, this is not a QRT-type mapping. Afterthis discovery, Joshi et al. derived this type of mapping as reductions from integrablelattice equations called ABS lattices and revealed that biquadratic invariants of QRTtype are in fact building blocks of the biquartic invariants [14]. Integrable mappingsthat possess invariants of much higher degree have also been obtained [15].§ 2. A discrete system that possesses a biquartic invariantIn this section, we start with the following second-order mapping:(2.1) xn+1 =1− x2nxn−1.78 Hironori Tanaka et al.!1.5 !1.0 !0.5 0.5 1.0 1.5 xn!1!1.5!1.0!0.50.51.01.5xnFigure 1. A trajectory of a solution of Eq. (2.1) for x0 = 1.2, x1 = 1.2Fig. 1 shows a solution of (2.1) in the phase plane. From this figure, we expect thatmapping (2.1) possesses an invariant that is the product of two invariants of certainmappings. In fact, this is true. The invariant of (2.1) is given by(2.2) L(xn−1, xn) =(x2n−1 + x2n − 1xn−1xn)2,which is biquartic and is also the square of a biquadratic,(2.3) K(xn−1, xn) =x2n−1 + x2n − 1xn−1xn.Two closed smooth curves in the phase plane are given by K(xn−1, xn) = κ andK(xn−1, xn) = −κ, where κ is a constant determined by an initial condition of (2.1).From the conservation of K(xn−1, xn) = K(xn, xn+1), we find the QRT mapping(2.4) xn+1 =−1 + x2nxn−1.Also, from K(xn−1, xn) = −K(xn, xn+1), we find (2.1). This kind of conservationhas been studied by Joshi et al. in [14]. They have found that integrable mappingspossessing invariants of high degree are derived from QRT invariants using this kindof conservation, although explicit direct relations between the newly derived mappingsand the QRT mappings were not given.Comparing (2.1) with (2.4), we notice a similarity between the two systems, whichreflects a symmetry of K(xn−1, xn).Constructing two-dimensional integrable mappings 79Let us consider the involution σ : (x, y) #→ (−x, y) and the QRT mapping ϕ :(x, y) #→(y,−1 + y2x). From the composition of σ and ϕ, we obtain(2.5) ϕ ◦ σ : (x, y) #→(y,1− y2x),which is identical to (2.1). By the involution σ, K(x, y) is transformed as follows.(2.6) K(σ(x, y)) = −K(x, y),which shows the symmetry of the invariant of L(x, y). From the fact above, we obtain(2.7) K(xn, xn+1) = K(ϕ ◦ σ(xn−1, xn)) = K(σ(xn−1, xn)) = −K(xn−1, xn),which is the conservation of (2.1) where we use the conservation of the QRT mappingK(ϕ(x, y)) = K(x, y).To summarize, the mapping (2.1) is a composition of the QRT mapping (2.4) andthe involution σ, which transforms the invariant of the QRT mapping to itself, changingthe integration constant.On the basis of what we explained above, we propose a method for constructingtwo-dimensional integrable mappings that possess invariants with degree higher thantwo in the next section.§ 3. Constructing two-dimensional integrable mappings that possessinvariants of high degreeLet us start with invariant curves of QRT mappings(3.1)tx(A+κB)y = (a00+κb00)x2y2+(a01+κb01)x2y+(a02+κb02)x2+(a10+κb10)xy2+ (a11 + κb11)xy+ (a12 + κb12)x+ (a20 + κb20)y2 + (a21 + κb21)y+ (a22 + κb22) = 0,where x =x2x1 , y =y2y1 and κ is an integration constant. We define themapping ϕ, ψ byϕ : (x, y) #→(f1(y)− xf2(y)f2(y)− xf3(y) , y),(3.2)ψ : (x, y) #→(x,g1(x)− yg2(x)g2(x)− yg3(x)),(3.3)80 Hironori Tanaka et al.where fj(x), gj(x) are given by (1.6) and (1.7).Next we introduce a mapping σ : (x, y) #→ (u(x, y), v(x, y)) and require that rela-tionsσ(tx(A+ κB)y) = σ(tx)(A+ κB)σ(y) = α(x, y,κ)tx(A+ β(κ)B)y,(3.4)βn(κ) = β ◦ · · · ◦ β︸ ︷︷ ︸n(κ) = κ,(3.5)hold with certain matrices A and B, a factor α(x, y,κ), and an n-unipotent functionβ(κ). The mapping σ transforms the invariant curve (3.1) into itself, changing theintegration constant κ into β(κ).By constructing the compositions of mappings ϕ, ψ, and σ, we obtain a new map-pingψ ◦ ϕ ◦ σ : (x, y) #→ (x¯, y¯),(3.6)x¯ =σ(f1(y))− σ(xf2(y))σ(f2(y))− σ(xf3(y)) ,(3.7)y¯ =g1(x¯)− σ(y)g2(x¯)g2(x¯)− σ(y)g3(x¯) ,(3.8)which possesses the invariant curve given by(3.9) (tx(A+ κB)y)(tx(A+ β(κ)B)y) · · · (tx(A+ βn−1(κ)B)y) = 0.In the next section, we show some examples of such mappings using the procedurewe have described above.§ 4. examplesExample 4.1.Let us take a mapping σ and matrices A,B as follows.σ : (x, y) #→ (−x, y),(4.1)A =a00 a01 a020 0 0a20 a21 a22 , B = 0 0 0b10 b11 b120 0 0 .(4.2)The invariant curve of the QRT mapping is given by(4.3) a00x2y2 + a01x2y + a02x2 + a20y2 + a21y + a22 + κ(x(b10y2 + b11y + b12)) = 0.Constructing two-dimensional integrable mappings 81It is easy to see that β(κ) = −κ and β2(κ) = κ. The form of the composed mappingψ ◦ ϕ ◦ σ is given byψ ◦ ϕ ◦ σ : (x, y) #→ (x¯, y¯),(4.4)x¯ = − a20y2 + a21y + a22x(a00y2 + a01y + a02),(4.5)y¯ =b12(a01x¯2 + a21)− b11(a02x¯2 + a22)− y(b10(a02x¯2 + a22)− b12(a00x¯2 + a20))b10(a02x¯2 + a22)− b12(a00x¯2 + a20)− y(b11(a00x¯2 + a20)− b10(a01x¯2 + a21)) .(4.6)Fig. 2 shows a solution of (4.5) and (4.6) in the phase plane, where(4.7) A =−1 1−10 0 0−1 1 1 , B =0 0 00 1 00 0 0 , (x0, y0) = (1.4, 1.4).!2 !1 1 2 x!0.50.51.01.5yFigure 2.82 Hironori Tanaka et al.Example 4.2.Let us take a mapping σ and matrices A,B as follows.σ : (x, y) #→(−1 + xx,−1 + yy),(4.8)A =−b10 + 12b11 + b20 − b21,−b10 − 12b11 + b12 + 2b20 − 2b22, −12b11 + b21 − b22−b10 + b12 − 2b22, −2b11 + 2b12 + 2b21 − 4b22, b10 − b11 − 2b20 + 2b21 − 2b22−12b11 + b12 − b22, b10 − 32b11 + b12 + b21 − 2b22, b10 − 12b11 − b20 + b21 − b22 ,(4.9)B =b10 − 12b11 − b20 + b21 − b22, b10 + 12b11 − b12 − 2b20 + b21, 12b11 − b20b10 b11 b12b20 b21 b22 .(4.10)A calculation shows that β(κ) =11− κ and β3(κ) = κ.In the case of(4.11) A =−2 0 10 4 31 3 2 , B =2 1 01 0 10 1 0 ,the mapping ψ ◦ ϕ ◦ σ is given byx¯ =y4 + 6y3 + 8y2 + y − 2− x (y3 − 4y2 − 2y + 1)y4 + 3y3 − y2 + 2 + xy (2y3 + 9y2 + 7y + 1) ,(4.12)y¯ =−2x¯4 − 9x¯3 − 7x¯2 − x¯− y (x¯4 + 6x¯3 + 8x¯2 + x¯− 2)2x¯4 + 8x¯3 + 11x¯2 + 3x¯− 1− y (x¯3 − 4x¯2 − 2x¯+ 1) ,(4.13)and a solution for (x0, y0) = (−0.2611,−0.2611) is shown in Fig. 3.The invariant curve is given by the union of three QRT invariant curves,(4.14)(tx(A+ κB)y)(tx(A+11− κB)y)(tx(A+−1 + κκB)y)= 0,which is illustrated in Fig. 3.A composition ψ ◦ ϕ gives the QRT mappingx¯ =−y4 − 3y3 + y2 − 2− x (2y3 + 9y2 + 7y + 1)y (2y3 + 9y2 + 7y + 1) + x (2y4 + 8y3 + 11y2 + 3y − 1) ,(4.15)y¯ =−x¯4 − 3x¯3 + x¯2 − 2− y (2x¯3 + 9x¯2 + 7x¯+ 1)x¯ (2x¯3 + 9x¯2 + 7x¯+ 1) + y (2x¯4 + 8x¯3 + 11x¯2 + 3x¯− 1) ,(4.16)Constructing two-dimensional integrable mappings 83!6 !4 !2 2 4 x!6!4!2246yFigure 3.!6 !4 !2 2 4 x!6!4!2246yFigure 4.84 Hironori Tanaka et al.and its trajectory with initial values (x0, y0) = (−0.2611,−0.2611) is illustrated inFig. 4, which is identical to the invariant curve (3.1).§ 5. conclusionIn this paper, we have proposed a method for constructing two-dimensional inte-grable mappings that possess invariants with degree higher than two. By constructingthe composition of the QRT mapping and the mapping that transforms the invariant ofthe QRT mapping to itself, changing an integration constant, we can obtain integrablemappings with invariants of higher degree. We have shown several concrete examplesof cases in which integration constants changed by mappings with period 2 and 3.Once we have a mapping σ and QRT matrices A,B that satisfy relation (3.4),we can construct a mapping with invariants of higher degree in general. However, wehaven’t investigated all possible cases. Since the form of σ reflects the symmetries ofthe QRT invariants, we may need some algebraic theories to classify them. We hope toreturn to this problem in near future.References[1] Grammaticos, B., Ramani, A. and Papageorgiou, V., Phys. Rev. Lett., 67 (1991), 1825–1828.[2] Hietarinta, J. and Viallet, C., Phys. Rev. Lett., 81 (1998), 325–328.[3] Sakai, H., Commun. Math. Phys., 220 (2001), 165–229.[4] Takenawa, T., J. Phys. A, 34 (2001), L95–L102.[5] Ramani, A., Grammaticos, B. and Ohta, Y., Commun. Math. Phys., 217 (2001), 315–329.[6] Ramani, A., Grammaticos, B. and Ohta, Y., J. Phys. A, 34 (2001), 2505–2513.[7] Tsuda, T., J. Phys. A, 37 (2004), 2721–2730.[8] Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y. and Yamada, Y., J. Phys. A, 36 (2003),L263–L272.[9] Quispel, G., Roberts, J. and Thompson, C., Phys. Lett. A, 126 (1988), 419–421.[10] Quispel, G., Roberts, J. and Thompson, C., Physica D, 34 (1989), 183–192.[11] Hirota, R., Kimura, K. and Yahagi, H., J. Phys. A, 34 (2001), 10377–10386.[12] Matsukidaira, J. and Takahashi, D., J. Phys. A, 39 (2006), 1151–1161.[13] Kimura, K., Yahagi, H., Hirota, R., Ramani, A., Grammaticos, B. and Ohta, Y., J. Phys.A, 35 (2002), 9205–9212.[14] Joshi, N., Grammaticos, B., Tamizhmani, T. and Ramani, A., Lett. Math. Phys., 78(2006), 27–37.[15] Tsuda, T., Grammaticos, B., Ramani, A. and Takenawa, T., Lett. Math. Phys., 82 (2007),39–49.

Constructing two-dimensional integrable mappings that possess invariants of high degree (Expansion of Integrable Systems)

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/176807/1/B13-06.pdf

Constructing two-dimensional integrable mappings that possess invariants of high degree (Expansion of Integrable Systems)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository