Bogomolny equations are first-order field equations obtained by setting every nonnegative square in a Bogomolny bound to zero. Their solutions automatically satisfy the corresponding second-order Euler-Lagrange equations.
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The Bogomolny equations are a set of partial differential equations that arise in the context of supersymmetric field theories and are particularly significant in the study of solitons, such as magnetic monopoles. Named after the physicist E.B. Bogomolny, these equations provide a way to find solutions that satisfy certain stability conditions. In the context of gauge theory, the Bogomolny equations generally involve a relationship between a gauge field and scalar fields.