# Need help solving Darboux equation

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• xdrgnh
In summary, the speaker is working on a personal math project involving a system of differential equations. They have found references stating the solutions are in terms of Hermite modular elliptic functions, but they do not know what those functions are. They are struggling to find material in English and are looking for suggestions on how to approach the problem. The project is in geometry and the speaker did not expect to encounter these complicated ODE's with solutions in unfamiliar functions. They have found a reference to a darboux frame on Wikipedia.
xdrgnh
I'm working on a personal math project and I'm running into this system of differential equations.

I have seen references which state the solutions are in terms of Hermite modular elliptic functions, but I do not know what those functions are. All of the references I can find on this equation are in french and I do not read french at all. Can anyone point me to any material in English or provide any suggestion of how to tackle this system of nonlinear ode's? I'm doing a project in geometry and did not expect to get these complicated ODE's that I never seen before and have solutions in terms of functions I never heard of.

Thanks

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## 1. What is the Darboux equation?

The Darboux equation is a partial differential equation that is used to describe the motion of a fluid in a two-dimensional space. It is named after the French mathematician Gaston Darboux, who first studied it in the late 19th century.

## 2. Why is solving the Darboux equation difficult?

The Darboux equation is difficult to solve because it is a nonlinear partial differential equation, meaning that the solution depends on the partial derivatives of the unknown function. Additionally, there is no general method for solving nonlinear partial differential equations, so each case must be approached individually.

## 3. What are some applications of the Darboux equation?

The Darboux equation has various applications in fluid mechanics, such as in the study of vortex motion, irrotational flow, and potential flow. It is also used in other areas of physics, such as in the study of electromagnetic fields and nonlinear optics.

## 4. How can I solve the Darboux equation?

There is no one-size-fits-all method for solving the Darboux equation, as it depends on the specific form and boundary conditions of the equation. However, some common approaches include using separation of variables, the method of characteristics, and numerical methods such as finite difference or finite element methods.

## 5. Are there any resources available to help with solving the Darboux equation?

Yes, there are various resources available to help with solving the Darboux equation. These include textbooks, online tutorials, and software packages that can assist with numerical solutions. Additionally, consulting with other mathematicians or attending conferences and workshops can also be helpful in finding solutions to this challenging equation.

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