We consider the asymptotic behavior of solutions of the first q-difference Painlevé equation in the limits |q|→1 and n→∞. Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behavior associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behavior is described by a power series expression, and find that the Stokes curves are described by curves known as q-spirals.
|Number of pages||30|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|Publication status||Published - 21 Jan 2019|
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- discrete Painlevé equations
- nonlinear discrete asymptotics
- exponential asymptotics
- Stokes phenomena
- q difference equations
- Exponential asymptotics
- Discrete Painlevé equations
- Nonlinear discrete asymptotics
- Q difference equations
Joshi, N., Lustri, C. J., & Luu, S. (2019). Nonlinear q-Stokes phenomena for q-Painlevé I. Journal of Physics A: Mathematical and Theoretical, 52(6), 1-30. . https://doi.org/10.1088/1751-8121/aaf77c