## Intrinsic formulation of geometric integrability and associated Riccati system generating conservation laws.(English)Zbl 1188.53090

The aim of the paper is to study, firstly, the formulation of Bäcklund transformations based on a Pfaffian system for the case of nonlinear evolution equations which describe pseudospherical surfaces, this is, surfaces with negative constant Gauss curvature, and secondly the determination of conservation laws for such equations.
Starting from the structure equations of a surface with Gauss curvature equal to $$-1$$, the author is able to transform them into an associated system of differential equations in a Riccati form and to formulate the equivalent linear problem. All this has been done in an intrinsic way.
Finally, it is shown that geometrical properties of a pseudospherical surface provide a systematic method for obtaining an infinite number of conservation laws.

### MSC:

 53C80 Applications of global differential geometry to the sciences 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35Q53 KdV equations (Korteweg-de Vries equations) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text:

### References:

 [1] DOI: 10.1017/CBO9780511623998 [2] DOI: 10.1063/1.1337796 · Zbl 1016.53008 [3] DOI: 10.1016/0550-3213(79)90517-0 [4] DOI: 10.1143/PTP.53.419 · Zbl 1079.35506 [5] DOI: 10.1103/PhysRevLett.30.1262 [6] DOI: 10.1103/PhysRevLett.33.925 · Zbl 1329.35347 [7] DOI: 10.1002/sapm198674155 · Zbl 0605.35080 [8] DOI: 10.1063/1.528020 · Zbl 0695.35038 [9] DOI: 10.1063/1.533284 · Zbl 0992.53005 [10] DOI: 10.1023/A:1010774630016 · Zbl 0995.35054 [11] DOI: 10.1017/CBO9780511606359 [12] DOI: 10.1142/3812 [13] DOI: 10.1002/sapm1988783227 · Zbl 0681.35087 [14] DOI: 10.1002/sapm1989812125 · Zbl 0697.58059
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.