Asymptotics and Confluence for a Singular Nonlinear q -Difference-Differential Cauchy Problem

Abstract

We examine a family of nonlinear $q-$ difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilation $q-$ difference operators, recently investigated by the author, and quasilinear Kowalevski type problems that involve contraction $q-$ difference operators. We build up local holomorphic solutions to these problems. Two aspects of these solutions are explored. One facet deals with asymptotic expansions in the complex time variable for which a mixed type Gevrey and $q-$ Gevrey structure are exhibited. The other feature concerns the problem of confluence of these solutions as $q>1$ tends to 1.