Lax pairs

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs 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.

A Lax pair is a pair of matrices or operators ${\displaystyle L(t),P(t)}$ dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation:

where ${\displaystyle [P,L]=PL-LP}$ is the commutator. Often, as in the example below, ${\displaystyle P}$ depends on ${\displaystyle L}$ in a prescribed way, so this is a nonlinear equation for ${\displaystyle L}$ as a function of ${\displaystyle t}$.

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