Lax pair

Summary

A Lax pair is a pair of operators $L(t)$ and $P(t)$ such that an evolution equation is equivalent to $\frac{dL}{dt} = [P, L] = PL - LP$. Then $L(t)$ stays similar to $L(0)$, so its spectrum does not change in time. The eigenvalues of $L$ are thus conserved quantities. Lax pairs are the starting point of the inverse scattering transform and a sign that an equation is integrable.