# Looking for what this type of PDE is generally called

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• masaakim
In summary, the conversation discusses a famous, symmetric nonlinear PDE that no one knows how to properly handle. The participants suggest looking for information using keywords like "bilinear," "dual," and "double." They also mention that the PDE involves functions of x and y, with a given function for ρ. The equation is a second-order PDE, and the idea is to find a way to control the solution for ϕ. The method of symmetry group analysis, particularly the work of author Olver, may be helpful in solving the PDE due to its scaling and rotational symmetries. The expert also mentions the possibility of looking into form-invariance under these transformations.
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.

## 1. What is a PDE?

A PDE, or partial differential equation, is a type of mathematical equation that involves multiple variables and their partial derivatives. It is commonly used to describe physical phenomena in fields such as physics, engineering, and economics.

## 2. What is the purpose of a PDE?

The purpose of a PDE is to model and describe the behavior of a system or process in terms of its changing variables. It allows for a more accurate and comprehensive understanding of the system's behavior compared to simpler mathematical equations.

## 3. How is a PDE different from an ordinary differential equation (ODE)?

A PDE involves multiple variables and their partial derivatives, while an ODE only involves one variable and its derivatives. This means that a PDE can describe systems with more complex and interdependent variables, while an ODE is limited to systems with only one independent variable.

## 4. What are some common types of PDEs?

Some common types of PDEs include the heat equation, wave equation, and diffusion equation. These are used to model physical phenomena such as heat transfer, wave propagation, and diffusion processes.

## 5. What is the general name for the type of PDE I am looking for?

The general name for the type of PDE you are looking for is a reaction-diffusion equation. This type of PDE combines both diffusion and reaction processes and is commonly used to model chemical reactions and biological systems.

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