Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II

Abstract

The degenerate third Painleve' equation, $u"(t)=(u^{\prime }(t){)}^2/u(t)-u^{\prime }(t)/t+1/t(-8c{u}^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions $u(t)$ as $t->+/-\infty$ and $t->+/-i\infty$ are derived and parametrized in terms of the monodromy data of the associated 2X2 linear auxiliary problem introduced in the first part of this work [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotically located along the real and imaginary axes are distinguished: asymptotics of these zeroes and poles are also obtained.

50 pages

References in corpus (2)

• Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II
• Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III

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• Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study
• Elizabethan vortices