Lax pair

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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.

[edit] Definition

A Lax pair is a pair of matrices or operators on a Hilbert space $L, M$ such that

$\frac{dL}{dt}=LM-ML$

It can then be shown that the eigenvalues and the continuous spectrum of L are independent of t. The matrices/operators L are said to form an isospectral series.

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) = A - 1 (t) L A (t)

Here, the motion of A can be arbitrarily complicated, yet the solution is still essentially a linear problem.

[edit] References

P. Lax, Comm. Pure Applied Math. 21 (1968) p. 467
P. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions, (1976) Princeton University Press.