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.

Parametric asymptotic expansions and confluence for Banach valued solutions to some singularly perturbed nonlinear q-difference-differential Cauchy problem

“…In comparison to our previous work [8], we are not able to construct analytic solutions to (1), ( 2) in all arguments t, z and ϵ but only analytic in ϵ whose values are located in the second embedding introduced in [8]. However, for some special type of nonlinear q-difference and differential Cauchy problem, analytic solutions both in complex time and space could be exhibited in a recent contribution of the author, see [10]. These problems are expressed as a coupling of a nonperturbative version of the linear Cauchy problem (3), ( 4) and a classical Cauchy-Kowaleski type partial differential equation with quadratic nonlinearity which involves the action of the contractive q-difference operator t → q −l t for some integers l ≥ 1.…”

“…No Gevrey type expansions with mixed order are reached in the present work. Notice that such double scales expansions were obtained for the holomorphic solutions to the special nonlinear q-difference and differential Cauchy problems investigated in [10].…”

“…The next proposition rephrases Proposition 1 from [10] for C−valued holomorphic maps. for all u ∈ S d,δ .…”

“…In this trend, we can mention the recent major work [3] on the confluence of some discret solutions for the q-Painlevé VI equations as q tends to 1 to analytic solutions for the famous Painleve VI equations using Hamiltonian systems representations. This last work has been strongly influential for the investigation of confluence properties for the special type of nonlinear q-difference and differential Cauchy problems mentioned earlier in this introduction and undertaken in [10].…”

“…for all u ∈ U . Eventually, paying heed to (10) and teaming up (73) with (75), we reach the next fitting estimates for the term P 3 (u, n, ϵ). Namely, whenever u ∈ U and ϵ ∈ D ϵ 0 .…”