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I'm trying to understand the discussion around equation (2.1) of the paper 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 eigenvalue equations $L\phi=\lambda \phi$ and $\dot{\phi}=B\phi$ are compatible, where $\lambda$ is a fixed parameter 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?


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up vote 2 down vote accepted

This answer may be a bit late, but it looks like the author is using "Lax pair" to mean a pair of equations in $\mu$, one of which is independent of $t$ and one of which is not. Here, "independent of $t$" means the same as it does in the traditional Lax pair approach - $L\phi=\lambda\phi$ where $\lambda_t=0$, so $L$ doesn't change with $t$. $B$, on the other hand, is inherently dependent on $t$, since $\phi_t=B\phi$.

Later in the paper (e.g., equation 6.4), the author returns to the traditional concept of a Lax pair (albeit in matrix form: a pair of matrices $X$ and $T$ such that $\phi_x=X\phi$, $\phi_t=T\phi$, which gives the compatibility condition $X_t-T_x+[X,T]=0$ for the usual commutator $[X,T]=XT-TX$).

If that doesn't seem to help, the group which innovated the Unified Transform Method keeps a very nice collection of results and papers related to the technique here:

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