the anti-self dual Yang–Mills (ASDYM) equations. Zero-curvature equation The zero-curvature equations are described by a pair of matrix-valued functions , where the subscripts denote coordinate indices rather than derivatives. Often the dependence is through a single scalar function and its derivatives. The zero-curvature equation is then It is so called as it corresponds to the vanishing of the curvature tensor, which in this case is . This differs from the conventional expression by some minus signs, which are ultimately unimportant. Lax pair to zero-curvature For an eigensolution to the Lax operator , one has If we instead enforce these, together with time independence of , instead the Lax equation arises as a consistency equation for an overdetermined system. The Lax pair can be used to define the connection components . When a PDE admits a zero-curvature representation but not a Lax equation representation, the connection components are referred to as the Lax pair, and the connection as a Lax connection. Examples Korteweg–de Vries equation The Korteweg–de Vries equation can be reformulated as the Lax equation with (a Sturm–Liouville operator) where all derivatives act on all objects to the right. This accounts for the infinite number of first integrals of the KdV equation. Kovalevskaya top The previous example used an infinite dimensional Hilbert space. Examples are also possible with finite dimensional Hilbert spaces. These include Kovalevskaya top and the generalization to include an electric Field . Heisenberg picture In the Heisenberg picture of quantum mechanics, an observable without explicit time dependence satisfies with the Hamiltonian and the reduced Planck constant. Aside from a factor, observables (without explicit time dependence) in this picture can thus be seen to form Lax pairs together with the Hamiltonian. The Schrödinger picture is then interpreted as the alternative expression in terms of isospectral evolution of these observables. Further examples Further examples of systems of equations that can be formulated as a Lax pair include: Benjamin–Ono equation One-dimensional cubic non-linear Schrödinger equation Davey–Stewartson system Integrable systems with contact Lax pairs Kadomtsev–Petviashvili equation Korteweg–de Vries equation KdV hierarchy Marchenko equation Modified Korteweg–de Vries equation Sine-Gordon equation Toda lattice Lagrange, Euler, and Kovalevskaya tops Belinski–Zakharov transform, in general relativity. The last is remarkable, as it implies that both the Schwarzschild metric and the Kerr metric can be understood as solitons. References archive P. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions, (1976) Princeton University Press. Differential equations Automorphic