Quasilinear PDE of Second Order - Some Analytical Direction Needed I'm looking at this second order quasilinear PDE:
$\alpha_{xx} - \alpha_{yy} - m\alpha^2 = 0$
Attempted Strategies:


*

*Fourier Transform resulted in convolution due to the $\alpha^2$ term, I don't think that would be pretty to work on.

*Separation of Variables resulted in not being able to separate the variables.

*I tried substituting $u=\alpha_x$ and $v=\alpha_y$ but I'm left with $(\int u_x dx)^2$ or $(\int u_y dy)^2$ in my equation afterwords due to substituting $u$ and $v$ back in, which seems like a dead end.


Do you guys have any suggestions for an analytical solution method?
 A: I think it's pretty hopeless to get an explicit formula for solutions for this equation (although specific examples of solutions might be known). Your equation is called the nonlinear wave equation, and it is an active area of research (to get a glimpse of the variety of research results, find the DispersiveWiki website, although this will not help in your case). In general it is not even true that an initial value problem (that is, prescribing the solution for $x=0$, here I'm interpreting your $x$ as time) has a global solution; often one has local solutions (that is, for small $x$) which can blow up in finite time.
A: Letting $t=x+y$ and $s=x-y$ as in the standard treatment of the 1-D wave equation, the equation $$\alpha_{st}=\frac{m}{4} \alpha^2.$$ If $m>0$, trying the trick of separation of variables does, indeed, lead to an explicit solution (i.e., assume $\alpha(s,t)=A(s)B(t)$).
As Willie Wong mentions below, this method had no hope from the start of producing a set of particular solutions which could be used to write a general solution, since a linear combination of particular solutions will not be a solution, in general.
