Simple looking but hard to solve nonlinear PDE

[smallphi]

I am solving the following simple looking nonlinear PDE:

$$(\partial f / \partial t)^2 - (\partial f / \partial r)^2 = 1$$

Using different tricks and ansatzs I've obtained the following analytic solutions so far:

$$f(r,t) = a\, t + b\, r + c, \,\,\,\, a^2 - b^2=1.$$

$$f(r,t) = \sqrt{(t-r-c)(t+r+d)} .$$

where a,b,c,d are constants.

Note that if one changes the variables to u = (t+r)/2, v=(t-r)/2 then the PDE looks even more ridiculously simple:

$$\frac{\partial f }{\partial u } \,\, \frac{ \partial f }{ \partial v}= 1.$$

The above two solutions were obtained by trying additive and multiplicative separation of the (u,v) variables.

Given the apparent simplicity of the equation is there a general way to approach it and generate more solutions?

 

[Defconist]

Yes, there is a general solution for any first order nonlinear equation. Method of characteristics, usually this methods requires initial condition, but you can supply something general like u(x,0) = f(x). This way you get either a general solution or something that is pretty close. I myself haven't studied the nonlinear case yet, you have to study it up, good luck.

Defconist
p.s. Try out this free book: http://www.freescience.info/go.php?id=1493&pagename=books

 

[smallphi]

Thanks for the link. Another good course is

http://ocw.mit.edu/OcwWeb/Mathematics/18-306Spring2004/LectureNotes/

 

[adroffner]

Method of Characteristics and Nonlinear PDE

The Method of Characteristics is more complex for Fully Nonlinear PDE. This Stanford http://www.stanford.edu/class/math220a/handouts/firstorder.pdf" derives the system of ODEs used to solve them. The two example PDEs are very much like the ones in your question.

I'm able to follow up to the Quasi-linear and Semi-linear cases. Burger's Equation is one of those. I can't yet solve the Fully Nonlinear PDE, with the this method.