Boundary conditions for integrable discrete chains (Q2773067)

It is known since the works of Birkhoff that (simple) recurrences (finite difference equations) of one integer index define a larger class of functions than the analogous ordinary differential equations (ODEs). Birkhoff has shown that the sequence of transformations from an integer index to a rational, and a fortiori to a continuous one is generally divergent. This proof is similar to that of Poincaré to the non-integrability of most ODEs. The proof of Birkhoff applies a fortiori, unless there is some convincing counter-evidence, to double-index recurrences, regardless of whether they appear in applications or not (properties of crystals, quantum mechanics, cell structures, etc.)NEWLINENEWLINENEWLINEThe paper describes a few particular `compatibility' relations for boundary and initial conditions for some double-index recurrences analogous to hyperbolic partial differential equations. Whether these relations guarantee `correctly posed' formulations in the sense of Hadamard, is an open question. The allusion to the Korteweg-de Vries equation suggests the possibility of solitary waves. -- The notion of integrability does not assure the existence of an explicit solution in terms of the two independent indices, but merely the absence of `indetermination'. -- The ideas contribute nevertheless to the `classification' of types of double-index recurrences.