How to solve the following equation ?

[mark gazi]

It is a nonliear Partial differential equation . you can find it in the appendix.
If you have some good idea about this equation ,ok ,please reply to me .thanks!

 

[HallsofIvy]

The first thing you would do, of course, is separate into "u> 0", "u< 0" cases. After that it is pretty much standard.

 

[mark gazi]

My question

First, thank you .but if the solution of the equation (U) is a complex number, how to separate the "u" into the two condition:(1) U>0; (2) U<0.

 

[HallsofIvy]

If u is a complex valued function, then separate it into real and imaginary parts: u(x)= p(x)+ i q(x). |u|²= p²+ q² and you have two non-linear differential equations for p and q. They will still be rather nasty I am afraid. MOST non-linear equations can be solved in closed form.