- #1

- 91

- 0

## Main Question or Discussion Point

Greetings to all. In a physics problem, I have come across a system of coupled PDEs

for 2 functions B(r,t) and V(r,t) on E^3 equipped with polar spherical coordinates (r,t,p).

(I write t for theta and p for phi.) With a comma denoting partial derivation and

D^2 denoting the Laplacian, the PDEs read

Equation 1

D^2 B = B*r^2*(sin^2 t)*( (V,r)^2 + (V,t)^2/r^2 ),

Equation 2

D^2 V + ( (2/r) + (B,r/B) )*V,r + ( 2*(cot t) + (B,t/B) )*V,t/r^2 = 0.

A trivial solution of this system is V = 0, B = 1 - m/r, where m is a constant.

However, I would want to find a nontrivial solution where V is not constant

and where said trivial solution represents a limit. Does anybody know how to

find such a solution?

for 2 functions B(r,t) and V(r,t) on E^3 equipped with polar spherical coordinates (r,t,p).

(I write t for theta and p for phi.) With a comma denoting partial derivation and

D^2 denoting the Laplacian, the PDEs read

Equation 1

D^2 B = B*r^2*(sin^2 t)*( (V,r)^2 + (V,t)^2/r^2 ),

Equation 2

D^2 V + ( (2/r) + (B,r/B) )*V,r + ( 2*(cot t) + (B,t/B) )*V,t/r^2 = 0.

A trivial solution of this system is V = 0, B = 1 - m/r, where m is a constant.

However, I would want to find a nontrivial solution where V is not constant

and where said trivial solution represents a limit. Does anybody know how to

find such a solution?