Lax pair

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.

The result can also be shown using the invariants for any These satisfy

due to the Lax equation, and since the characteristic polynomial can be written in terms of these traces, the spectrum is preserved by the flow.

Any PDE which admits a Lax pair representation also admits a zero curvature representation. In fact, the zero curvature representation is more general and for other integrable PDEs, such as the sine Gordon equation, the Lax pair refers to matrices that satisfy the zero curvature equation rather than the Lax equation. Furthermore, the zero curvature representation makes the link between integrable systems and geometry manifest, culminating in Ward's programme to formulate known integrable systems as solutions to the anti self dual Yang–Mills (ASDYM) equations.

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.

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 .