# nLab 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

$\frac{dL}{dt}=\left[M,L\right]$\frac{d L}{d t} = [M, L]

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

The Lax equation is the compatibility condition for the system

$\lambda \psi =L\psi$\lambda \psi = L \psi
$\frac{d\psi }{dt}=M\psi$\frac{d \psi}{d t} = M\psi

where $\psi =\psi \left(t\right)$ 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.

$ML\psi =M\lambda \psi =\lambda M\psi =\lambda \frac{d\psi }{dt}=\frac{d\left(\lambda \psi \right)}{dt}=\frac{d\left(L\psi \right)}{dt}=\frac{dL}{dt}\psi +L\frac{d\psi }{dt}=\frac{dL}{dt}\psi +LM\psi$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