Help with a 2nd order PDE involving mixed derivatives

[aurban]

I have a PDE in two variables, u and v, which takes the form

<br /> \frac{\partial\psi}{\partial u\hspace{1pt}\partial v} + \frac{1}{r}\left(\frac{\partial r}{\partial u} \frac{\partial \psi}{\partial v} + \frac{\partial r}{\partial v}\frac{\partial\psi}{\partial u}\right)

for an auxiliary field r=r(u,v). It would be nice to have this equation in a form that is amenable to the method of characteristics, as there are independent evolution equations for r in both directions. My first idea was to define some intermediary fields \rho = r\partial_v\psi and \tilde{\rho} = r\partial_u\psi, then write down the wave equation as

\partial_u(r\rho)+\partial_v(r\tilde{\rho}) = \rho\partial_u r + r\partial_u\rho + \tilde{\rho}\partial_vr+r\partial_v\tilde{\rho} = 0.

Then, making the ansatz

-\left(\rho\partial_u r + r\partial_u\rho\right) = \tilde{\rho}\partial_vr+r\partial_v\tilde{\rho} = \gamma = \text{const.}

one arrives at a system of four first-order equations along characteristics: two for \psi and one each for \rho and \tilde{\rho}. (Initial data is free for the last two fields along the characteristic for which it lacks an evolution equation.)

My question is, does this seem reasonable or is there a better method for approaching this problem?

 

[hunt_mat]

There are a number of things that you can do, the equation that you have written down is a hyperbolic equation, so characteristics are a possibility, the use of a Riemann function may also be of some use here as well.