- Lax pair
In
mathematics , in the theory ofdifferential equations , a Lax pair is a pair of time-dependent matrices that describe certain solutions of differential equations. They were developed byPeter Lax to discusssoliton s incontinuous media . Theinverse scattering transform makes use of the Lax equations to solve a variety of the so-calledexactly solvable model s of physics.Definition
A Lax pair is a pair of matrices or operators dependent on time and acting on a fixed Hilbert space, such that
:
where 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
eigenvalue s and the continuous spectrumof "L" are independent of "t". The matrices/operators "L" are said to be "isospectral " as varies.The core observation is that the above equation is the infinitesimal form of a family of matrices all having the same spectrum, by virtue of being given by
:
Here, the motion of "g" can be arbitrarily complicated.Conversely suppose for an arbitrary once differentiable family of invertible operators .Then differentiaing we see : with .
Example
The
KdV equation is:It can be reformulated as the Lax equation:with: (aSturm-Liouville operator ):and 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:
*
Benjamin–Ono equation
* One dimensional cubicnon-linear Schrödinger equation
*Davey-Stewartson system
*Kadomtsev–Petviashvili equation
*Korteweg–de Vries equation
*KdV hierarchy
*Modified Korteweg-de Vries equation
*Sine-Gordon equation References
*
* P. Lax and R.S. Phillips, "Scattering Theory for Automorphic Functions", (1976) Princeton University Press.
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