Stokes matrices of a reducible equation with two irregular singularities of Poincaré rank 1 via monodromy matrices of a reducible Heun type equation

2020/07/30 by Tsvetana Stoyanova, Stoyanova, Tsvetana
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Numerical methods for differential equations #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2007.15291

Abstract

We consider a second order reducible equation having non-resonant irregular singularities at x=0 and x=∞. Both of them are of Poincaré rank 1. We introduce a small complex parameter ε that splits together x=0 and x=∞ into four different Fuchsian singularities x_L=-√(ε), x_R=√(ε) and x_LL=-1/√(ε), x_RR=1/√(ε), respectively. The perturbed equation is a second order reducible Fuchsian equation with 4 different singularities, i.e. a Heun type equation. Then we prove that when the perturbed equation has exactly two resonant singularities of different type, all the Stokes matrices of the initial equation are realized as a limit of the nilpotent parts of the monodromy matrices of the perturbed equation when ε → 0 in the real positive direction. To establish this result we combine a direct computation with a theoretical approach.

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