On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities

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• 作者：Alberto Lastra^a ; Sté ; phane Malek^b ; Javier Sanz^a ; ^{jsanzg@am.uva.es}
• 关键词：34K25 ; 34M25 ; 34M30 ; 33E30
• 刊名：Journal of Differential Equations
• 出版年：2012
• 出版时间：15 May, 2012
• 年：2012
• 卷：252
• 期：10
• 页码：5185-5216
• 全文大小：343 K

文摘

We consider a Cauchy problem for some family of linear q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables for given formal power series initial conditions. Under suitable conditions and by the application of certain q-Borel and Laplace transforms (introduced by J.-P. Ramis and C. Zhang), we are able to deal with the small divisors phenomenon caused by the Fuchsian singularity, and to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the compact sets of , is . The small divisors? effect is an increase in the order of q-exponential growth and the appearance of a power of the factorial in the corresponding q-Gevrey bounds in the asymptotics.