ISSN:
1572-9036
Keywords:
pseudo-spherical surfaces
;
Calapso–Guichard surfaces
;
kinematic integrability
;
geometric integrability
;
formal integrability
;
symmetries
;
conservation laws
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The relation between scalar evolution equations which are the integrability condition of sl(2,R)-valued linear problems with parameter (‘kinematic’ integrability) and those which possess recursion operators (‘formal’ integrability) is studied: using that kinematically integrable equations describe one-parameter families of pseudo-spherical surfaces and vice versa, it is shown that every second order formally integrable evolution equation is kinematically integrable, and that this result cannot be extended as proven to the third-order case. Conservation laws of kinematically integrable equations obtained from their underlying pseudo-spherical structure are compared with the ones one finds from the ‘Riccati equation’ version of their associated linear problems. Symmetries (generalized/nonlocal) for these equations are also studied, by considering infinitesimal deformations of the associated pseudo-spherical surfaces. Finally, conservation laws for equations describing pseudo-spherical surfaces immersed in a flat three-space are found, and the class of ‘equations describing Calapso–Guichard surfaces’ is introduced.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1010774630016
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