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

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

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

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

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

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.

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

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

Revised on October 15, 2012 21:47:23
by Zoran Škoda
(193.198.162.13)