On the integrability of scalar partial differential equations in two independent variables : some geometric aspects

The problem of integrability of scalar partial differential equations in two independent variables is investigated from a geometric viewpoint. The structure of "equations describing pseudo-spherical surfaces" introduced by S. S. Chern and Keti Tenenblat is taken as the starting point, and the fact that every equation which describes pseudo-spherical surfaces is the integrability condition of a sl(2, R)--linear problem is exploited throughout.A classification of evolution equations of the form ut = F(x, t, u, ..., uxm) which describe one-parameter families of pseudo-spherical surfaces ("kinematic integrability") is performed, under a natural a priori assumption on the form of the associated family of linear problems. The relationship between the class of equations which results, and the class of equations which are formally integrable in the sense of Mikhailov, Shabat and Sokolov, is studied. It is shown that every second order formally integrable evolution equation is kinematically integrable, and it is also shown that this result cannot be extended as proven to third order formally integrable evolution equations. A special case is proven, however, and moreover, the Harry-Dym, cylindrical KdV, and a family of equations solved by inverse scattering by Calogero and Degasperis, are shown to be kinematically integrable.The theory of coverings due to Krasil'shchik and Vinogradov is introduced, and (local, nonlocal) conservation laws and (generalized, nonlocal) symmetries of kinematically integrable equations are investigated within this framework: theorems on the existence of generalized and/or nonlocal symmetries are proven, and several sequences of local and/or nonlocal conservation laws are constructed. The relationship between the Cavalcante, Chern, and Tenenblat approach (conservation laws from pseudo-spherical structure) and the more familiar "Riccati equation" approach (conservation laws from the associated linear problem) is analysed. An appendix on nonlocal Hamiltonians for evolution equations is also included.The relationship between Chern and Tenenblat "intrinsic" point of view, and the "extrinsic" approach due to Sym is studied, and more conservation laws for equations describing pseudo-spherical surfaces are found. It is also shown that within the "extrinsic" framework, a new class of equations, interpretable as two-parameter deformations of the equations in the Chern-Tenenblat class, can be introduced. It is pointed out that these deformations are themselves the integrability condition of sl(2, R)--linear problems, and conservation laws for them are briefly considered