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.
Definition.
A Lax pair is a pair of matrices or operators formula_1 dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation:
where formula_3 is the commutator.
Often, as in the example below, formula_4 depends on formula_5 in a prescribed way, so this is a nonlinear equation for formula_5 as a function of formula_7.
Isospectral property.
It can then be shown that the eigenvalues and more generally the spectrum of "L" are independent of "t". The matrices/operators "L" are said to be "isospectral" as formula_7 varies.
The core observation is that the matrices formula_9 are all similar by virtue of
where formula_11 is the solution of the Cauchy problem
where "I" denotes the identity matrix. Note that if "P(t)" is skew-adjoint, "U(t,s)" will be unitary.
In other words, to solve the eigenvalue problem "Lψ = λψ" at time "t", it is possible to solve the same problem at time 0 where L is generally known better, and to propagate the solution with the following formulas:
Link with the inverse scattering method.
The above property is the basis for the inverse scattering method. In this method, "L" and "P" act on a functional space (thus "ψ = ψ(t,x)"), and depend on an unknown function "u(t,x)" which is to be determined. It is generally assumed that "u(0,x)" is known, and that "P" does not depend on "u" in the scattering region where formula_15.
The method then takes the following form:
Examples.
Korteweg–de Vries equation.
The Korteweg–de Vries equation 
can be reformulated as the Lax equation
with
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 formula_30.
formula_31
Heisenberg picture.
In the Heisenberg picture of quantum mechanics, an observable without explicit time dependence satisfies
formula_32
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:
The last is remarkable, as it implies that both the Schwarzschild metric and the Kerr metric can be understood as solitons.