Lax pair

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In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that describe the corresponding differential equations. They 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.

Definition

A Lax pair is a pair of matrices or operators <math>L(t), P(t)</math> dependent on time and acting on a fixed Hilbert space, and verifying the Lax's equation:

<math>frac=</math>


where <math>=PL-LP</math> is the commutator.Often, as in the example below, <math>P</math> depends on <math>L</math> in a prescribed way, so this is a nonlinear equation for <math>L</math> as a function of <math>t</math>.

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 <math>t</math> varies.

The core observation is that the matrices <math>L(t)</math> are all similar by virtue of

<math>L(t)=U(t,s) L(s) U(t,s)^</math>


where <math>U(t,s)</math> is the solution of the Cauchy problem

<math> frac U(t,s) = P(t) U(t,s), qquad U(s,s) = I,</math>


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...
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