×

zbMATH — the first resource for mathematics

Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. II: Tzitzéica equation. (English) Zbl 0946.35086
For part I cf. [M. Musette, R. Conte, ibid. 39, 5617-5630 (1998; Zbl 0932.35180)].
Summary: From the existing methods of singularity analysis only, we derive two equations which define the Bäcklund transformation of the Tzitzéica equation. This is achieved by defining a truncation in the spirit of the approach of Weiss et al., so as to preserve the Lorentz invariance of the Tzitzéica equation. If one assumes a third-order scattering problem, this truncation admits a unique solution, thus leading to a matrix Lax pair and a Darboux transformation. In order to obtain the Bäcklund transformation (BT), which is the main new result of this paper, one represents the Lax pair by an equivalent two-component Riccati pseudopotential. This yields two different BTs; the first one is a BT for the Hirota-Satsuma equation, while the second one is a BT for the Tzitzéica equation. One of the two equations defining the BT is the fifth ordinary differential equation of Gambier.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35A30 Geometric theory, characteristics, transformations in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1063/1.532554 · Zbl 0932.35180
[2] DOI: 10.1063/1.525721 · Zbl 0514.35083
[3] DOI: 10.1063/1.525875 · Zbl 0531.35069
[4] Tzitzéica G., C. R. Acad. Sci. Paris 144 pp 1257– (1907)
[5] DOI: 10.1063/1.526147 · Zbl 0598.76008
[6] DOI: 10.1016/0167-2789(87)90217-X · Zbl 0612.58050
[7] DOI: 10.1088/0266-5611/10/3/014 · Zbl 0803.35113
[8] DOI: 10.1143/JPSJ.40.611 · Zbl 1334.76016
[9] DOI: 10.1016/0375-9601(80)90578-2
[10] DOI: 10.1098/rspa.1976.0154 · Zbl 0353.35063
[11] DOI: 10.1098/rspa.1977.0012
[12] Zhiber A. V., Sov. Phys. Dokl. 24 pp 607– (1979)
[13] DOI: 10.1063/1.529620 · Zbl 0764.35088
[14] DOI: 10.1007/BF01017916 · Zbl 0541.53030
[15] DOI: 10.1016/0167-2789(81)90120-2 · Zbl 1194.37113
[16] DOI: 10.1016/0167-2789(81)90120-2 · Zbl 1194.37113
[17] DOI: 10.1088/0305-4470/27/11/036 · Zbl 0843.35109
[18] DOI: 10.1007/BF02393211 · JFM 40.0377.02
[19] DOI: 10.1007/BF03029121 · JFM 39.0685.05
[20] Tzitzéica G., C. R. Acad. Sci. Paris 150 pp 955– (1910)
[21] DOI: 10.1063/1.531576 · Zbl 0863.58074
[22] Tzitzéica G., C. R. Acad. Sci. Paris 150 pp 1227– (1910)
[23] DOI: 10.1007/BF02420225 · JFM 48.0800.02
[24] DOI: 10.1063/1.1666254 · Zbl 0268.35053
[25] Sharipov R. A., Trans. Inst. Math. BNC, URO AN SSSR, Ufa 66 pp 1– (1991)
[26] DOI: 10.1063/1.530283 · Zbl 0788.53053
[27] DOI: 10.1007/BF01015902 · Zbl 0852.35134
[28] DOI: 10.1016/0375-9601(95)00955-8 · Zbl 1072.35555
[29] DOI: 10.1016/S0375-9601(96)00703-7 · Zbl 1037.83503
[30] DOI: 10.1143/JPSJ.49.771 · Zbl 1334.35282
[31] DOI: 10.1063/1.529302 · Zbl 0734.35086
[32] DOI: 10.1088/0951-7715/7/3/012 · Zbl 0803.35111
[33] DOI: 10.1063/1.531485 · Zbl 0863.35099
[34] DOI: 10.1016/0375-9601(88)90942-5
[35] DOI: 10.1063/1.526655 · Zbl 0565.35103
[36] DOI: 10.1063/1.526009 · Zbl 0565.35094
[37] DOI: 10.1063/1.527134 · Zbl 0617.35118
[38] DOI: 10.1016/0375-9601(89)90072-8
[39] DOI: 10.1088/0305-4470/29/16/032 · Zbl 0898.35098
[40] DOI: 10.1063/1.526997 · Zbl 0629.35106
[41] DOI: 10.1007/BF01108506
[42] DOI: 10.1088/0305-4470/20/10/012 · Zbl 0663.35089
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.