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

Stephane Malek

Working Paper Article Version 1 This version is not peer-reviewed

Version 1 : Received: 28 July 2021 / Approved: 29 July 2021 / Online: 29 July 2021 (10:32:20 CEST)
Version 2 : Received: 21 March 2022 / Approved: 22 March 2022 / Online: 22 March 2022 (11:41:03 CET)

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 quasi-linear 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 is exhibited. The other feature concerns the problem of confluence of these solutions as q tends to 1.

asymptotic expansion; confluence; formal power series; partial differential equation; q-difference equation

Computer Science and Mathematics, Algebra and Number Theory

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

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