Nahm equations
In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators. Deep study of the Nahm equations was carried out by Nigel Hitchin and Simon Donaldson. Conceptually, the equations arise in the process of infinite-dimensional hyperkähler reduction. They can also be viewed as a dimensional reduction of the anti-self-dual Yang-Mills equations . Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the moduli space of monopoles; and the existence of hyperkähler structure on coadjoint orbits of complex semisimple Lie groups, proved by , , and . Equations 

Let  be three matrix-valued meromorphic functions of a complex variable . The Nahm equations are a system of matrix differential equations

together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the Levi-Civita symbol, in the form

  

More generally, instead of considering  by  matrices, one can consider Nahm's equations with values in a Lie algebra . Additional conditions 
The variable  is restricted to the open interval , and the following conditions are imposed:
 
 
  can be continued to a meromorphic function of  in a neighborhood of the closed interval , analytic outside of  and , and with simple poles at  and ; and
 At the poles, the residues of  form an irreducible representation of the group SU(2). Nahm–Hitchin description of monopoles 

There is a natural equivalence between 
 the monopoles of charge  for the group , modulo gauge transformations, and 
 the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of  by the group . Lax representation 

The Nahm equations can be written in the Lax form as follows. Set

then the system of Nahm equations is equivalent to the Lax equation

As an immediate corollary, we obtain that the spectrum of the matrix  does not depend on . Therefore, the characteristic equation 

which determines the so-called spectral curve in the twistor space is invariant under the flow in . See also 
Bogomolny equation
Yang–Mills–Higgs equations

References

External links 
Islands project – a wiki about the Nahm equations and related topics

Differential equations
Eponymous equations of physics
Mathematical physics
Integrable systems