Ukrainian Mathematical Bulletin

Volume 2, Issue 4, 2005 pp. 557-589.

Generalized de-Rham--Hodge--Skrypnyk Theory: Differential-Geometric and Spectral Aspects and Some Applications

Authors: Yarema A. Prykarpats'kyi, Anatolii M. Samoilenko, and Anatolii K. Prykarpats'kyi
Author institution: Yarema A. Prykarpats'kyi, Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs'ka Str., Kyiv, 01601, Ukraine AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow Poland, yarchyk@imath.kiev.ua Anatolii M. Samoilenko, Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs'ka Str., Kyiv, 01601, Ukraine, sam@imath.kiev.ua Anatolii K. Prykarpats'kyi, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland, Institute for Applied Problems in Mechanics and Mathematics of NAS of Ukraine, Lviv, 79601, Ukraine, pryk.anat@ua.fm

Summary: We study differential-geometric and topological structures of the Delsarte transmutation operators and the Gelfand--Levitan--Marchenko-type equations associated with these operators by using the generalized de-Rham--Hodge--Skrypnyk differential complexes. The relationships between the spectral theory and special properties of Berezans'kyi-type congruence for operators permutable in Delsarte's sense are established. Some applications to special multidimensional differential operators, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine expansion associated with the self-dual Yang--Mills equations are presented. The soliton solutions of the associated family of dynamic systems are discussed.

Contents Full-Text PDF