Discrete dressing transformations and Painlevé equations. (English) Zbl 0969.39501
Summary: The authors propose a discrete analog of the dressing transformation. The starting point is a variant of the quotient-difference algorithm which, in this case, corresponds to a linear problem with shifts in the eigenvalues. The proper periodicity conditions lead to one-dimensional systems which are discrete Painlevé equations. The authors obtain thus the alternate \(d-P_{\text{II}}\) equation and a novel form for the discrete \(P_{\text{IV}}\) equation.
MSC:
39A12 | Discrete version of topics in analysis |
35Q53 | KdV equations (Korteweg-de Vries equations) |
37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
References:
[1] | Veselov, A. P.; Shabat, A. B., Funct. Anal. Appl., 27, 1 (1993) |
[2] | Ramani, A.; Grammaticos, B.; Caurier, E., J. Math. Comp. Sim., 37, 451 (1994) · Zbl 0811.65053 |
[3] | Grammaticos, B.; Ramani, A.; Moreira, I. C., Physica A, 196, 574 (1993) · Zbl 0803.34008 |
[4] | Adler, V. E., Physica D, 73, 335 (1994) · Zbl 0812.34030 |
[5] | Ramani, A.; Grammaticos, B.; Hietarinta, J., Phys. Rev. Lett., 67, 1829 (1991) · Zbl 1050.39500 |
[6] | Grammaticos, B.; Ramani, A.; Papageorgiou, V. G., Phys. Rev. Lett., 67, 1825 (1991) · Zbl 0990.37518 |
[7] | Fokas, A. S.; Grammaticos, B.; Ramani, A., J. Math. An. Appl., 180, 342 (1993) · Zbl 0794.34013 |
[8] | Nijhoff, F.; Satsuma, J.; Kajiwara, K.; Grammaticos, B.; Ramani, A., Inv. Probl., 12, 697 (1996) · Zbl 0860.35124 |
[9] | Papageorgiou, V.; Grammaticos, B.; Ramani, A., Lett. Math. Phys., 34, 91 (1995) · Zbl 0831.58028 |
[10] | Ramani, A.; Grammaticos, B., Physica A, 228, 160 (1996) · Zbl 0912.34011 |
[11] | Ramani, A.; Grammaticos, B., The Grand Scheme for discrete Painlevé equations, (Lecture at the Toda Symp. (1996)) · Zbl 0951.39005 |
[12] | Spiridonov, V.; Vinet, L.; Zhedanov, A., Lett. Math. Phys., 29, 63 (1993) · Zbl 0790.33017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.