# Does this PDE have a name?

I'm looking for any and all information that might be known about the following second-order PDE for one function $u(x,y)$:

$u_{xy} = u_x e^u + u_y e^{-u}$

e.g., Does it have a name? Is it known to be integrable (or not)? Anything else interesting about it?

• It might help to explain how you ran across this in your research. Why are you interested in it? It might give us a place to start looking. – user69208 Jul 18 '16 at 2:24
• Looks like some generalized version of Liouville's equation. – Igor Khavkine Jul 18 '16 at 3:31
• A collaborator and I are attempting to classify second-order PDEs of a certain form that have Backlund transformations. Most of the examples we're finding are well-known, but we didn't recognize this one. – Jeanne Clelland Jul 18 '16 at 13:03

By changing the variables as $x=t+z$ and $y=t-z$ you get $$u_{tt} -u_{zz} =2u_t \cosh(u) +2u_z \sinh(u)$$which is a nonlinear wave equation.