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.
dependent on time, acting on a fixed Hilbert space, and satisfying Lax's equation: where
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
is the solution of the Cauchy problem where I denotes the identity matrix.
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: The result can also be shown using the invariants
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.
[1] 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
The method then takes the following form: If the Lax matrix additionally depends on a complex parameter
By the isospectral property, this curve is preserved under time translation.
Such curves appear in the theory of Hitchin systems.
[3] 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.
dependence is through a single scalar function
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.
When a PDE admits a zero-curvature representation but not a Lax equation representation, the connection components
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.
The previous example used an infinite-dimensional Hilbert space.
Examples are also possible with finite-dimensional Hilbert spaces.
[4] In the Heisenberg picture of quantum mechanics, an observable A without explicit time t dependence satisfies with H the Hamiltonian and ħ 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 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.