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.
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Definition
A Lax pair is a pair of matrices or operators
where
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
The core observation is that the matrices
where
where I denotes the identity matrix. Note that if L(t) is self-adjoint and P(t) is skew-adjoint, then 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
- Compute the spectrum of
L ( 0 ) , givingλ andψ ( 0 , x ) , - In the scattering region where
P is known, propagateψ in time by using∂ ψ ∂ t ( t , x ) = P ψ ( t , x ) with initial conditionψ ( 0 , x ) , - Knowing
ψ in the scattering region, computeL ( t ) and/oru ( t , x ) .
Example
The Korteweg–de Vries equation is
It 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.
Equations with a Lax pair
Further examples of systems of equations that can be formulated as a Lax pair include: