Painleve Differential Equations in the Complex Plane

V. I. Gromak, Ilpo Laine, Shun Shimomura9783110173796, 3-11-017379-4

Focusing on five of the 50 Painlevé equations (two of which are modified to make the meromorphic like the others), Gromak (mathematics and mechanics, Belarussian State U., Belarus), Laine (mathematics, U. of Joensuu, Finland), and Shimomura (mathematics, Keio U., Japan) investigate their growth and value distributions. The core of the book is devoted to the integration of the equations in terms of elementary functions or some classical transcendental functions such as the Airy, Bessel, or hypergeometric functions, e.g. the Bläcklund transformations. A final chapter describes some applications of the equations relevant to such areas as hydrodynamics, plasma physics, nonlinear optics, and solid state physics”