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

$$\frac{dL}{dt}=[M,L]$$\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

$$\lambda \psi =L\psi$$\lambda \psi = L \psi

$$\frac{d\psi }{dt}=M\psi$$\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.

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