Darboux-integrable equations with non-Abelian nonlinearities

Abstract

We introduce a new class of nonlinear equations admitting a representation in terms of Darboux-covariant compatibility conditions. Their special cases are, in particular, (i) the "general" von Neumann equation $i\dot{\rho }=[H,f(\rho )]$, with $[f(\rho ),\rho ]=0$, (ii) its generalization involving certain functions $f(\rho )$ which are non-Abelian in the sense that $[f(\rho ),\rho ]\ne 0$, and (iii) the Nahm equations.

LaTeX, 19 pages