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Concept
Lax pair
Summary
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.
A Lax pair is a pair of matrices or operators dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation:
where is the commutator.
Often, as in the example below, depends on in a prescribed way, so this is a nonlinear equation for as a function of .
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 varies.
The core observation is that the matrices are all similar by virtue of
where 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:
(no change in spectrum)
Invariants of tensors
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.
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 ,
Knowing in the scattering region, compute and/or .
Official source
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