nLab Lax equation

Idea

The Lax equation is a general kind of equation controlling integrable systems.

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.

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

References

• Peter Lax, Integrals of nonlinear equation of evolution and solitary waves, Commun. on Pure and Applied Mathematics 21 5 (1968) 467-490 [doi:10.1002/cpa.3160210503]