# Looking for what this type of PDE is generally called

• I
• masaakim

#### masaakim

TL;DR Summary
(since I'm completely outside this community, please delete this post if someone sees it is inappropriate) We have this type of nicely symmetric very famous nonlinear pde in our area. But no one knows how to handle it properly in general. Suggestions on how it is called in general would be a great help.
We have this type of very famous nicely symmetric pde in our area. However, no one knows how to handle it properly since it is a nonlinear pde.
Suggestions on how it is called in general would help us further googling. I already tried keywords like "bilinear", "dual", "double", but by far could not find any relevant material on the internet.
$$\frac{\partial^2 \phi}{\partial x\partial x}\frac{\partial^2 z}{\partial y\partial y} -2\frac{\partial^2 \phi}{\partial x\partial y}\frac{\partial^2 z}{\partial x\partial y} +\frac{\partial^2 \phi}{\partial y\partial y}\frac{\partial^2 z}{\partial x\partial x} =\rho.$$

##\phi,z## and ##\rho## are functions of ##x## and ##y##. ##\rho## is given (let's say it is simply ##\rho=1##). When ##\phi## is given, then this equation is a second-order pde. With the determinant of the second derivatives of ##\phi## be positive, the second-order pde is elliptic (e.g. Laplace equation). With the determinant of the second derivatives of ##\phi## be negative, the second-order pde is hyperbolic (wave equation).
The idea is to unlock ##\phi## so that we can have more control over the second-order pde.

We are not mathematicians, please be tolerant of inaccurate word choices.
Thank you!

Last edited:
There is a method for non-linear PDEs which may help called symmetry group analysis. Look for references by the author Olver from Springer publishing. Here it looks like there are some definite scaling symmetries as well as rotational symmetry in the x-y plane. (Not that the solutions will have these symmetries but the equation will be form-invariant under these transformations.)

If I have time I'll look at it but I'm going to be quite busy for the next couple of weeks so no promises.