Lax equation

Lax equation is used in integrable systems; namely some systems are equivalent to the Lax equation.

**Lax equation** is a linear ordinary differential equation of the form

$\frac{d L}{d t} = [M, L]$

for $n\times n$-matrix-valued function $L = L(t)$, where $M$ is also a $n\times n$ matrix. The pair $(L,M)$ is also called a **Lax pair**.

The Lax equation is the compatibility condition for the system

$\lambda \psi = L \psi$

$\frac{d \psi}{d t} = M\psi$

where $\psi = \psi(t)$ is a vector function which is an eigenvector for $L$ with eigenvalue $\lambda$. To see this make a derivative of $L\psi$ and use the Leibniz rule.

$M L \psi = M \lambda \psi = \lambda M \psi = \lambda \frac{d \psi}{d t} = \frac{d (\lambda\psi)}{d t} = \frac{d (L\psi)}{d t} = \frac{d L}{d t} \psi + L \frac{d \psi}{d t} = \frac{d L}{d t} \psi + L M\psi$

- Peter Lax,
*Integrals of nonlinear equation of evolution and solitary waves*, Commun. on Pure and Applied Mathematics**21**:5, 467–490, 1968 doi