# Nonlinear q-Stokes phenomena for q-Painlevé I

@article{Joshi2018NonlinearQP, title={Nonlinear q-Stokes phenomena for q-Painlev{\'e} I}, author={Nalini T. Joshi and Christopher J. Lustri and S. Luu}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2018}, volume={52} }

We consider the asymptotic behavior of solutions of the first -difference Painlevé equation in the limits and . 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…

## 2 Citations

### First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants

- MathematicsComptes Rendus. Mathématique
- 2022

We construct a relation between the leading pre-factor function A(z) and the singulants u0(z), u1(z), and recurrence relation of the singulants at higher levels for the solution of…

### Capturing the cascade: a transseries approach to delayed bifurcations

- MathematicsNonlinearity
- 2021

Transseries expansions build upon ordinary power series methods by including additional basis elements such as exponentials and logarithms. Alternative summation methods can then be used to ‘resum’…

## References

SHOWING 1-10 OF 95 REFERENCES

### Stokes phenomena in discrete Painlevé II

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2017

This work determines the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and finds that the behaviour of these asymPTotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.

### Stokes phenomena in discrete Painlevé I

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2015

This study considers the asymptotic behaviour of the first discrete Painlevé equation in the limit as the independent variable becomes large, and identifies two types of solutions which are pole-free within some sector of the complex plane containing the positive real axis.

### Quicksilver Solutions of a q‐Difference First Painlevé Equation

- Mathematics
- 2013

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific…

### Smoothing of the Stokes phenomenon for high-order differential equations

- MathematicsProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- 1992

We extend the class of functions for which the smooth transition of a Stokes multiplier across a Stokes line can be rigorously established to functions satisfying a certain differential equation of…

### Singular dynamics of a q-difference Painlevé equation in its initial-value space

- Mathematics
- 2014

We construct the initial-value space of a q-discrete first Painlevé equation explicitly and describe the behaviours of its solutions w(n) in this space as n → ∞ , ?> with particular attention paid to…

### Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

- Mathematics
- 2013

We study a system of discrete Painlevé V equations via the Riemann–Hilbert approach. We begin with an isomonodromy problem for dPV, which admits a discrete Riemann–Hilbert problem formulation. The…

### Continuous limit of the difference second Painleve equation and its asymptotic solutions

- Mathematics
- 2012

. The discrete second Painlev´e equation dP II is mapped to the second Painlev´e equation P II by its continuous limit, and then, as shown by Kajiwara et al., a rational solution of dP II also…

### Asymptotic behaviour around a boundary point of the q-Painlevé VI equation and its connection problem

- Mathematics
- 2010

We study analytic properties of solutions to the q-Painlevé VI equation (q-PVI), which was derived by Jimbo and Sakai as the compatibility condition for a connection preserving deformation (CPD) of a…

### Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

- MathematicsEuropean Journal of Applied Mathematics
- 2001

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to…

### The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series

- Mathematics
- 1999

Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent.…