Suppose I have the wave equation:
$u_{tt}=c^2u_{xx}$
I factorize the Laplacian operators:
$\partial_{tt}-c^2\partial_{xx}=(\partial_{t}-c\partial_{x})(\partial_{t}+c\partial_{x})$
From there I can use the change of variable $u(t,x)=v(r,s)$ where $r=x+ct$ and $s=x-ct$
What change of variable do I choose if the Laplacian operators factor into a $(a-b)^2$ identity?
Assuming your question is, what change of variables would you use for solving $(\partial_t−\partial_x)^2u = 0$...
One quickly sees that this equation is parabolic and thus only has one characteristic. One could use the one characteristic direction for one new variable, and any direction that is not parallel to the characteristic direction for the other.
To elaborate, for $u_{tt} - 2u_{xt} + u_{xx} = 0,$ one sees that $$\frac{dx}{dt} = \frac{-(-2) \pm \sqrt{4-4}}{2} = 1,$$ so you have $x = t + \xi.$ Thus, use $\xi(x,t) = x - t$ as one variable. The other one can be anything not parallel, such as $\eta(x,t) = t.$
In the case of repeated roots when factorizing Laplacian operators, use the following change of variables:
$u(t,x)=v(t,s)$ where $s=x+\frac{t}{2}$