- 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 L(t), A(t) dependent on time and acting on a fixed Hilbert space, such that
:frac{dL}{dt}= [L,A]
where L,A] =LA-AL.Often, as in the example below, A depends on L in a prescribed way, so this is a nonlinear equation for L as a function of t.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 t varies.The core observation is that the above equation is the infinitesimal form of a family of matrices L(t) all having the same spectrum, by virtue of being given by
:L(t)=g^{-1}(t) L(0) g(t),
Here, the motion of "g" can be arbitrarily complicated.Conversely suppose L(t)=g^{-1}(t) L(0) g(t) for an arbitrary once differentiable family of invertible operators g(t).Then differentiaing we see :frac{dL}{dt}= -g^{-1} frac{dg}{dt} g^{-1} L(0) g + g^{-1} L(0) frac{dg}{dt} = LA-AL with A= g^{-1} frac{dg}{dt}.
Example
The
KdV equation is:u_t=6uu_x-u_{xxx},It can be reformulated as the Lax equation:L_t= [L,A] ,with:L=-partial^2+u, (aSturm-Liouville operator ):A=4partial^3-3(upartial+partial u),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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