The Maxwell equations are the special case of the Yang-Mills equations when the gauge group is just the abelian circle group U(1). These equations take the form
where F is the curvature of a U(1) connection :
Note that the Bianchi identity also gives the equation
which is also traditionally given as one of the Maxwell equations.
In the presence of a current , the Maxwell equations now take the form
(after normalizing various physical constants).
The Maxwell equations are linear. However, by coupling the Maxwell equations to a scalar field one obtains the nonlinear Maxwell-Klein-Gordon equations.