Lax pair

In mathematics, in the theory of differential equations, a Lax pair is a pair of time-dependent matrices that describe certain solutions of differential equations. They were developed by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve a variety of the so-called exactly solvable models 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 eigenvalues 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:$L=-partial^2+u,$ (a Sturm-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 cubic non-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.