Lax pair

Remarks

Example: for $u(x, t)$ smooth, put $L = -\partial_x^2 + u$ and $P = -4\partial_x^3 + 6u\partial_x + 3u_x$, operators acting on functions of $x$. A direct computation gives $[P, L] = 6 u u_x - u_{xxx}$, as a multiplication operator, while $\frac{dL}{dt} = u_t$. So $(L, P)$ is a Lax pair for the Korteweg–De Vries equation $u_t - 6 u u_x + u_{xxx} = 0$.

Non-example: if $P = 0$, the Lax equation says that $L$ is constant in $t$. When $L$ determines $u$, as $L = -\partial_x^2 + u$ does, a pair $(L, 0)$ is a Lax pair only for the trivial equation $u_t = 0$.

If $P(t)$ is skew-adjoint and $U(t)$ solves $U' = P U$ with $U(0) = I$, then $L(t) = U(t) L(0) U(t)^{-1}$. So the eigenvalues of $L(t)$ do not depend on $t$: the flow is isospectral. For matrices, $\tr L(t)^k$ is then constant for each $k$.