# Lukasz Stepien

# Poland

The Pedagogical Univeristy of Cracow, ul. Podchorazych 2, 30-084 Krakow

SCS Slide Set On Bogomolny equations

"*Euler-Lagrange equations of many models in physics are nonlinear partial differential equations of second order.*

*However, in some cases one can consider first-order nonlinear partial differential equations, instead of Euler-Lagrange equations. Such first-order equations were derived in 1976 by Bogomolny (and then they are called often as 'Bogomolny equations' or 'Bogomolâ€™nyi equations'). Parallelly, they were derived by Belavin, Polyakov, Schwartz and Tyupkin in 1975, for another model - SU(2) Yang-Mills theory (these equations are called also as 'BPS equations'). Similar results were obtained also by Hosoya in 1978.*

*In this entry, a derivation of Bogomolny equations (called as 'Bogomolny decomposition'), by using the so-called strong necessary conditions method, was presented. Some properties of an exact solution of these equations, have been shown, too.*"