"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 'Bogomol’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."