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3 - Definitions and Predictions of Integrability for Difference Equations

Published online by Cambridge University Press:  05 July 2011

By
Jarmo Hietarinta
Edited by
Decio Levi ,
Peter Olver ,
Zora Thomova  and
Pavel Winternitz
Jarmo Hietarinta
Affiliation:
University of Turku
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
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Summary

Abstract

In these lectures we take a look at various meanings of integrability for difference equations, and the possibility of algorithmic methods to identify (partial) integrability. Analogies with continuum equations are used when possible, but the world of discrete equations is richer and many new things enter.

Preliminaries

Points of view on integrability

One can approach the integrability of dynamical equations from two opposite directions:

Top down: In this approach one first chooses some high level mathematical structure and then derives its consequences/manifestations for dynamical equations. The underlying mathematical structure is there from the start and therefore as a result one gets a method for generating equations with good properties.

Bottom up: In this case the equation is given, for example: “In my application I found this equation, what can you say about it?” In order to answer questions of this type we need a toolbox of (algorithmic) methods that can be applied. With them we can, hopefully, identify the equation or at least say whether it is integrable or partially integrable or chaotic. If the equation is at least partially integrable we may be able to say something about its structure, construct some solutions, conserved quantities etc. And although complete integrability itself is structurally unstable, many properties persist in nearby non-integrable systems, and one can try a perturbative expansion around the integrable one.

Type
Chapter
Information
Symmetries and Integrability of Difference Equations , pp. 83 - 114
Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] Ablowitz, M. J., Halburd, R., and Herbst, B. 2000. On the extension of the Painlevé property to difference equations. Nonlinearity, 13(3), 889–905.CrossRefGoogle Scholar
[2] Adler, V. E. 1998. Bäcklund transformation for the Krichever–Novikov equation. Internat. Math. Res. Notices, 1998(1), 1–4.CrossRefGoogle Scholar
[3] Adler, V. E. 2006. Some incidence theorems and integrable discrete equations. Discrete Comput. Geom., 36(3), 489–498.CrossRefGoogle Scholar
[4] Adler, V. E., and Suris, Yu. B. Private communication.
[5] Adler, V. E., Bobenko, A. I., and Suris, Yu. B. 2003. Classification of integrable equations on quad-graphs. The consistency approach. Comm. Math. Phys., 233(3), 513–543.CrossRefGoogle Scholar
[6] Adler, V. E., Bobenko, A. I., and Suris, Yu. B. 2009. Discrete nonlinear hyperbolic equations; classification of integrable cases. Funct. Anal. Appl., 43(1), 3–17.CrossRefGoogle Scholar
[7] Atkinson, J., Hietarinta, J., and Nijhoff, F. 2007. Seed and soliton solutions of Adler's lattice equation. J. Phys. A, 40(1), F1–F8.CrossRefGoogle Scholar
[8] Falqui, G., and Viallet, C.-M. 1993. Singularity, complexity, and quasi-integrability of rational maps. Comm. Math. Phys., 154(1), 111–125.CrossRefGoogle Scholar
[9] Grammaticos, B., Ramani, A., and Papageorgiou, V. 1991. Do integrable mappings have the Painlevé property?Phys. Rev. Lett., 67(14), 1825–1828.CrossRefGoogle ScholarPubMed
[10] Hietarinta, J. 1987. Direct methods for the search of the second invariant. Phys. Rep., 147(2), 87–154.CrossRefGoogle Scholar
[11] Hietarinta, J. 2004. A new two-dimensional lattice model that is ‘consistent around the cube’. J. Phys. A, 37(6), L67–L73.CrossRefGoogle Scholar
[12] Hietarinta, J. 2005. Searching for CAC-maps. J. Nonlinear Math. Phys., 12(suppl. 2), 223–230.CrossRefGoogle Scholar
[13] Hietarinta, J., and Viallet, C. 1998. Singularity confinement and chaos in discrete systems. Phys. Rev. Lett., 81(2), 325–328.CrossRefGoogle Scholar
[14] Hietarinta, J., and Viallet, C. 2000. Discrete Painlevé I and singularity confinement in projective space. Chaos Solitons Fractals, 11(1), 29–32.CrossRefGoogle Scholar
[15] Hietarinta, J., and Viallet, C. 2007. Searching for integrable lattice maps using factorization. J. Phys. A, 40(42), 12629–12643.CrossRefGoogle Scholar
[16] Hietarinta, J., and Zhang, D.-j. 2009. Soliton solutions for ABS lattice equations II: Casoratians and bilinearization. arXiv:0903.1717.
[17] Hirota, R. 1981. Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan, 50(11), 3785–3791.CrossRefGoogle Scholar
[18] Hirota, R., Kimura, K., and Yahagi, H. 2001. How to find the conserved quantities of nonlinear discrete equations. J. Phys. A, 34(48), 10377–10386.CrossRefGoogle Scholar
[19] J., Atkinson, J., Hietarinta, and Nijhoff, F. 2008. Soliton solutions for Q3. J. Phys. A, 41(14), 142001.Google Scholar
[20] Miwa, T. 1982. On Hirota's difference equations. Proc. Japan Acad. Ser. A Math. Sci., 58(1), 9–12.CrossRefGoogle Scholar
[21] Nijhoff, F. 2002. Lax pair for the Adler (lattice Krichever–Novikov) system. Phys. Lett. A, 297(1-2), 49–58.CrossRefGoogle Scholar
[22] Nijhoff, F., Atkinson, J., and Hietarinta, J. 2009. Soliton solutions for ABS lattice equations I: Cauchy matrix approach. arXiv:0902.4873.
[23] Nijhoff, F. W., and Walker, A. J. 2001. The discrete and continuous Painlevé VI hierarchy and the Garnier systems. Glasg. Math. J., 43A, 109–123.CrossRefGoogle Scholar
[24] Nijhoff, F. W., Ramani, A., Grammaticos, B., and Ohta, Y. 2001. On discrete Painlevé equations associated with the lattice KdV aystems and the Painlevé VI equation. Stud. Appl. Math., 106(3), 261–314.CrossRefGoogle Scholar
[25] Nimmo, J. J. C., and Schief, W. K. 1997. Superposition principles associated with the Moutard transformation: an integrable discretization of a (2 + 1)-dimensional sine-Gordon system. Proc. Roy. Soc. London Ser. A, 453(1957), 255–279.CrossRefGoogle Scholar
[26] Ohta, Y., Tamizhmani, K.M., Grammaticos, B., and Ramani, A. 1999. Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations. Phys. Lett. A, 262(2-3), 152–157.CrossRefGoogle Scholar
[27] Quispel, G. R. W., Roberts, J. A. G., and Thompson, C. J. 1988. Integrable mappings and soliton equations. Phys. Lett. A, 126(7), 419–421.CrossRefGoogle Scholar
[28] Ramani, A., Grammaticos, B., and Hietarinta, J. 1991. Discrete versions of the Painlevé equations. Phys. Rev. Lett., 67(14), 1829–1831.CrossRefGoogle ScholarPubMed
[29] Rasin, O. G., and Hydon, P. E. 2007. Conservation laws for integrable difference equations. J. Phys. A, 40(42), 12763–12773.CrossRefGoogle Scholar
[30] Sahadevan, R., and Capel, H. W. 2003. Complete integrability and singularity confinement of nonautonomous modified Korteweg–de Vries and sine-Gordon mappings. Phys. A, 330(3-4), 373–390.CrossRefGoogle Scholar
[31] Tremblay, S., Grammaticos, B., and Ramani, A. 2001. Integrable lattice equations and their growth properties. Phys. Lett. A, 278(6), 319–324.CrossRefGoogle Scholar

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