The Kadomtsev-Petviashvili equation is defined as:
\begin{equation} \partial_x (u_t-6uu_x-u_{xxx})=-3 \alpha^2 u_{yy} \end{equation}
If we define two operators as
$$ A = -4 \partial_x^3 - 6u \partial_x - 3(u_x - \alpha w) $$ $$ H = - ( \partial_x^2 + u) / \alpha $$
The Lax representation is
$$ H_t - A_y = [A,H] $$
Now I need to find an associated linear system. I am given that we can consider the KP equation as the condition that the following system of differential equations are compatible.
\begin{equation} \alpha \Psi_y = - \Psi_{xx} - u \Psi \end{equation}
\begin{equation} \Psi_t = -4 \Psi_{xxx} - 6u \Psi_x - 3(u_x - \alpha w)\Psi \end{equation}
But I do not understand why this true. Why does the compatibility of the system imply that $u$ is a solution and vice-versa?
In the context of the Inverse Scattering Transform (IST), it's been shown (Zhou, Xin, Direct and inverse scattering transforms with arbitrary spectral singularities, Commun. Pure Appl. Math. 42, No. 7, 895-938 (1989). ZBL0714.34022.), at least in a weighted Sobolev space, that every step of the IST is well-defined and reversible. This implies that if you can solve the associated ODE's related to the Lax pair, you can recover the solution to the original PDE via the usual Gelfand-Levitan-Marchenko integral equation. Note that sometimes the steps of the IST fail, in which case you cannot obtain a solution. However, if the steps work, you get a solution.