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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. (English) Zbl 1058.34123
The author considers the particular integral of the differential equation $d^2w/dz^2+u^2z^2w=e^{iu((1/2)z^2-4z)}$, where $u$ is a large parameter. He determines the complete Stokes geometry, i.e., the active and inactive Stokes curves and the higher-order Stokes curve. (A Stokes curve to be inactive or a Stokes multiplier to change its value are events typical for higher-order Stokes phenomena.) He presents numerical examples and suggests how the study can be extended to more complicated problems.
34M40Stokes phenomena and connection problems (ODE in the complex domain)
34E05Asymptotic expansions (ODE)
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
Full Text: DOI
[1] T. Aoki, T. Kawai, Y. Takei, New turning points in the exact WKB analysis for higher-order ordinary differential equations, in: Analyse algébrique des perturbations singulières, I, Marseille-Luminy, 1991; Travaux en Cours 47 (1994) 69--84.
[2] Aoki, T.; Kawai, T.; Takei, Y.: On the exact steepest descent methoda new method for the description of Stokes curves. J. math. Phys. 42, 3691-3713 (2001) · Zbl 1063.34047
[3] Berk, H. L.; Nevins, W. M.; Roberts, K. V.: New Stokes lines in WKB theory. J. math. Phys. 23, 988-1002 (1982) · Zbl 0488.34050
[4] C.J. Howls, P.J. Langman, A.B. Olde Daalhuis, On the higher-order Stokes phenomenon, Proc. Roy. Soc. London Ser. A, to appear.
[5] C.J. Howls, A.B. Olde Daalhuis, Hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one, Proc. Roy. Soc. London, Ser. A 459 (2003), no 2038, 2599--2612. · Zbl 1063.34084
[6] Daalhuis, A. B. Olde: Hyperasymptotic solutions of higher-order linear differential equations with a singularity of rank one. Proc. roy. Soc. London, ser. A 454, 1-29 (1998) · Zbl 0919.34012
[7] Daalhuis, A. B. Olde: On the computation of Stokes multipliers via hyperasymptotics. Sūrikaisekikenkyūsho kōkyūroku 1088, 68-78 (1999) · Zbl 0951.34501
[8] Paris, R. B.; Wood, A. D.: Stokes phenomenon demystified. Bull. inst. Math. appl. 31, 21-28 (1995) · Zbl 0827.34002
[9] Temme, N. M.: Special functions. An introduction to the classical functions of mathematical physics. (1996) · Zbl 0856.33001