Bogomolny equations

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.

A Bogomolny bound rewrites a field-theory energy as a sum of nonnegative squares plus a topological term. Configurations satisfying the resulting first-order Bogomolny equations saturate the bound.

Square completion for a one-dimensional kink ( $E \geq ∣ W (ϕ_{+}) - W (ϕ_{-}) ∣$ )

For the static energy

E = \int [ϕ^{'2} + W^{'} (ϕ)^{2}] d x /2

E = \frac{1}{2} \int (ϕ^{'} \mp W^{'})^{2} d x \pm [W (ϕ_{+}) - W (ϕ_{-})] .

The Bogomolny bound is the absolute boundary difference. Equality requires the Bogomolny equation

ϕ^{'} = \pm W^{'} (ϕ)

with the sign selected by that difference. Differentiating it gives the Euler-Lagrange field equation

ϕ^{''} = W^{'} W^{''}

. A stationary solution of that second-order equation need not saturate the bound in a general field theory.

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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.