One Parameter Families of Bäcklund Maps

Abstract

It has been known for a hundred years that a geometrical structure may fruitfully be studied in terms of its linearizations and its invariance group. Here the question presents itself in the form: how may a Bäcklund map be deformed, and what are its symmetries? Recently, Ibragimov and Anderson [46] have studied infinitesimal diffeomorphisms of jet bundles which preserve the contact module. Leaving to the future a systematic study of symmetries of Bäcklund problems we here show how extended point transformations of jet bundles can be combined with a Bäcklund map to give a family of Bäcklund maps with the same integrability conditions. In particular we give conditions sufficient for a deformation of a Bäcklund map to be an ordinary Bäcklund map with the same integrability conditions. The deformation is constructed from suitably related 1-parameter groups of transformations of the independent variables x^a, the original dependent variables z^μ and the new variables y^A. The parameter (eigenvalue) appearing in the linear scattering equations derivable from Bäcklund maps (as in example 5.1) may be introduced by deforming Bäcklund maps in this way.