Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-12-01T01:08:16.677Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete equations corresponding to fourth-order differential equations of the P2 and K2 hierarchies

Published online by Cambridge University Press:  17 February 2009

N. A. Kudryashov  and
M. B. Soukharev
N. A. Kudryashov
Affiliation:
Department of Applied Mathematics, Moscow Engineering and Physics Institute, 31 Kashirskoe shosse, 115409 Moscow, Russia; e-mail: [email protected].
M. B. Soukharev
Affiliation:
Department of Applied Mathematics, Moscow Engineering and Physics Institute, 31 Kashirskoe shosse, 115409 Moscow, Russia; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using the Bäcklund transformations for the solutions of fourth-order differential equations of the P2 and K2 hierarchies, corresponding discrete equations are found.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 149 - 160
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Airault, H., “Rational solutions of Painlevé equations”, Stud. Appl. Math. 61 (1979) 31–53.CrossRefGoogle Scholar
[2]Boiti, M. and Pempinelli, F., “Similarity solutions of the Korteweg-de Vries equation”, Nuovo Cimento Soc. Ital. Fis.B 51 (1979) 70–78.CrossRefGoogle Scholar
[3]Caudrey, P. J., Dodd, R. K. and Gibbon, J. D., “A new hierarchy of Korteweg-de Vries equations”, Proc. Roy. Soc. LondonA 351 (1976) 407–422.Google Scholar
[4]Clarkson, P., Mansfield, E. L. and Webster, H. N., “On the relation between discrete and continuous Painlevé equations”, Theoret. and Math. Phys. 122 (2000) 1–16.CrossRefGoogle Scholar
[5]Dodd, R. K. and Gibbon, J. D., “The prolongation structure of a higher order Korteweg-de Vries equation”, Proc. Roy. Soc. LondonA 358 (1978) 287–296.Google Scholar
[6]Flaschka, H. and Newell, A. C., “Monodromy- and spectrum-preserving deformations. I”, Comm. Math. Phys. 76 (1980) 65–116.CrossRefGoogle Scholar
[7]Fokas, A. S., Grammaticos, B. and Ramani, A., “From continuous to discrete Painlevé equations”, J. Math. Anal. Appl. 180(1993) 342–360.CrossRefGoogle Scholar
[8]Gordoa, P. R., Joshi, N. and Pickering, A., “Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations”, Nonlinearity 12 (1999) 955–1008.CrossRefGoogle Scholar
[9]Grammaticos, B., Nijhoff, F. W. and Ramani, A., “Discrete Painlevé equations”, in The Painlevé Property. One century later (ed. Conte, R.), CRM Series in Math. Phys., (Springer, New York, 1999) 413–516.CrossRefGoogle Scholar
[10]Grammaticos, B. and Ramani, A., “From continuous Painlevé IV to the asymmetric discrete Painlevé I”, J. Phys.A 31 (1998) 5787–5798.Google Scholar
[11]Gromak, V. I., “Nonlinear evolution equations and equations of p type”, Differential equations 20 (1984) 2042–2048, (Russian).Google Scholar
[12]Gromak, V. I. and Lukashevich, N. A., Analytical Properties of Painlevé equations (Izdat. Universitetskoe, Minsk, 1990).Google Scholar
[13]Gross, D. J. and Migdal, A. A., “Nonperturbative two-dimensional quantum gravity”, Phys. Rev. Lett. 64 (1990) 127–130.CrossRefGoogle ScholarPubMed
[14]Gross, D. J. and Migdal, A. A., “A nonperturbative treatment of two-dimensional quantum gravity”, Nuclear Phys.B 340 (1990) 333–365CrossRefGoogle Scholar
[15]Hone, A. N. W., “Non-autonomous Hénon-Heils systems”, PhysicaD 118 (1998) 1–26.Google Scholar
[16]Kudryashov, N. A., “The first and the second Painlevé equations of higher order and some relations between them”, Phys. Lett.A 224 (1997) 353–360.CrossRefGoogle Scholar
[17]Kudryashov, N. A., “On new transcendents defined by nonlinear ordinary differential equations”, J. Phys. A.: Math. Gen. 31(1998) L129–L137.CrossRefGoogle Scholar
[18]Kudryashov, N. A., “Transcendents defined by nonlinear fourth-order ordinary differential equations”, J. Phys. A.: Math. Gen. 32 (1999) 999–1013.CrossRefGoogle Scholar
[19]Kudryashov, N. A., “Two hierarchies of ordinary differential equations and their properties”, Phys. Lett.A 252 (1999) 173–179.CrossRefGoogle Scholar
[20]Kudryashov, N. A., “Fourth-order nonlinear differential equations with solutions in the form of transcendents”, Theoret. and Math. Phys. 122 (2000) 58–71.CrossRefGoogle Scholar
[21]Kudryashov, N. A. and Soukharev, M. B., “Uniformization and transcendence of solutions for the first and second Painlevé hierarchies”, Phys. Lett.A 237 (1998) 206–216.CrossRefGoogle Scholar
[22]Kupershmidt, B. and Wilson, G., “Modifying Lax equations and the second Hamiltonian structure”, Invent. Math. 62 (1981) 403–436.CrossRefGoogle Scholar
[23]Lukashevich, N. A., “On the theory of the second Painlevé equation”, Differential equations 7 (1971) 1124–1125.Google Scholar
[24]Nijhoff, F. W. and Papageorgiou, V. G., “Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation”, Phys. Lett.A 153 (1991) 337–344.CrossRefGoogle Scholar
[25]Nijhoff, F. W., Satsuma, J., Kajiwara, K., Grammaticos, B. and Ramani, A., “A study of the alternate discrete Painlevé II equation”, Inverse Problems 12 (1996) 697–716.CrossRefGoogle Scholar
[26]Quispel, G. R. W., Roberts, J. A. G. and Thompson, C. J., “Integrable mappings and soliton equations”, Phys. Lett.A 126 (1988) 419–421.CrossRefGoogle Scholar
[27]Veselov, A. P., “Integrable mappings”, Russian Math. Surveys 46 (1991) 3–45.CrossRefGoogle Scholar
[28]Weiss, J., “On classes of integrable systems and the Painlevé property”, J. Math. Phys. 25 (1984) 13–24.CrossRefGoogle Scholar
[29]Weiss, J., “Bäcklund transformation and the Painlevé property”, J. Math. Phys. 27 (1986) 1293–1305.CrossRefGoogle Scholar