Lax pairs for linear PDEs

Question

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has the Lax pair $\mu_x+ik\mu=q$ and $M(\partial_x,\partial_y)\mu=0$, where k is any complex number and $\mu$ is a function.

The way I'm used to thinking of Lax pairs is as operators $L$ and $B$ such that $\dot{L}+[L,B]=0$ is equivalent to the original PDE. This is equivalent to requiring that the equations $L\phi=\lambda\phi$ and $\dot{\phi}=B\phi$ are compatible, where $\lambda$ is a fixed eigenvalue and $\phi$ is any function. Can anyone explain how this connects with the discussion in the paper? What are $L$ and $B$ in the above case?

Thanks!

Answer 1

The term Lax pair is fairly often used in a broader sense as a pair of linear equations (not necessarily of the form you mention, i.e., $L\phi=\lambda\phi$ and $\dot{\phi}=B\phi$) whose compatibility condition is the system under study. In fact, this notion is occasionally stretched even further, for example, in the theory of integrable dispersionless systems one encounters nonlinear Lax pairs, see e.g. this paper and references therein.