Conservation laws and Calapso-Guichard deformations of equations describing pseudo-spherical surfaces. (English) Zbl 0992.53005

Reyes, Enrique G.

J. Math. Phys. 41, No. 5, 2968-2989 (2000).

From author’s abstract: The relation between the Chern and Tenenblat approach to conservation laws of equations describing pseudo-spherical surfaces (conservation laws obtained from pseudo-spherical structure) and the more familiar “Riccati equation” approach (conservation laws obtained from associated linear problem) is investigated. Two examples (cylindrical Korteweg-de Vries (KdV) and Lund-Regge equations) are presented. Chern and Tenenblat’s point of view is then connected with the theory of soliton surfaces. A generalization of the original Chern-Tenenblat construction of conservation law-results and a reasonable family of large deformations for scalar equations describing pseudo-spherical surfaces, the “equations describing Calapso-Guichard surfaces”, can be introduced. It is shown that these equations are also the integrability conditions of linear problems.

Reviewer: Raul Ibáñez (Bilbao)

MSC:

53A05	Surfaces in Euclidean and related spaces
37K25	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
35L65	Hyperbolic conservation laws
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q53	KdV equations (Korteweg-de Vries equations)

Keywords:

conservation laws; Riccati equation; pseudo-spherical surfaces; soliton surfaces

References:

[1]	DOI: 10.1002/sapm198674155 · Zbl 0605.35080
[2]	DOI: 10.1016/0550-3213(79)90517-0
[3]	DOI: 10.1006/jdeq.1998.3617
[4]	DOI: 10.1006/jdeq.1998.3617
[5]	DOI: 10.1063/1.528020 · Zbl 0695.35038
[6]	DOI: 10.1143/PTP.53.419 · Zbl 1079.35506
[7]	DOI: 10.1088/0031-8949/40/6/003 · Zbl 1063.35549
[8]	J. Krasil’shchik and A. Verbovetsky, ”Homological Methods in Equations of Mathematical Physics,” lectures given in August 1998 at the Diffiety Institute International Summer School, Levoča, Slovakia, preprint DIPS 7/98, math.DG/9808130 (1998).
[9]	DOI: 10.1063/1.525079 · Zbl 0471.35071
[10]	DOI: 10.1006/jdeq.1995.1005 · Zbl 0815.35036
[11]	Alekseev A. A., Teoret. Mat. Fiz. 91 pp 30– (1992)
[12]	DOI: 10.1103/PhysRevD.14.1524 · Zbl 0996.81509
[13]	DOI: 10.1016/0003-4916(78)90156-2 · Zbl 0394.53016
[14]	DOI: 10.1007/BF02763081
[15]	DOI: 10.1016/0375-9601(81)90924-5
[16]	DOI: 10.1103/PhysRevD.15.1540
[17]	DOI: 10.1216/RMJ-1980-10-1-105 · Zbl 0407.53002
[18]	DOI: 10.1007/BF01624787