It can then be shown that the eigenvalues and more generally the spectrumofL are independent of t. The matrices/operators L are said to be isospectralas varies.
The core observation is that the matrices are all similar by virtue of
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:
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:
Compute the spectrum of , giving and
In the scattering region where is known, propagate in time by using with initial condition
If the Lax matrix additionally depends on a complex parameter (as is the case for, say, sine-Gordon), the equation
defines an algebraic curvein with coordinates By the isospectral property, this curve is preserved under time translation. This is the spectral curve. Such curves appear in the theory of Hitchin systems.[2]
Any PDE which admits a Lax-pair representation also admits a zero-curvature representation.[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. 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.
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.
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 .[4]
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.
Lax, P. (1968), "Integrals of nonlinear equations of evolution and solitary waves", Communications on Pure and Applied Mathematics, 21 (5): 467–490, doi:10.1002/cpa.3160210503archive
P. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions[1], (1976) Princeton University Press.