Okay, here's the deal:
I have been given a second order nonlinear differential equation, and I have also been given the general solution with constants A and B. I am supposed to find the constants A and B. The solution represents a fermion at rest, since the solution does not vary with time.
The differential equation is as follows:
The soliton solution is:
The Attempt at a Solution
I have five pages of attempts to solve this. One professor told me it was impossible, but the professor that I am doing research with told me that it is possible. First, I found the first and second derivatives of phi, then, I input them into the differential equation. The first term goes out because the particle is at rest. Then, I tried different methods.
I tried using hyperbolic trig identities.
I tried writing the equation in terms of sinh and cosh and then tried to eliminate some things. He told me that I need to end up with a polynomial that has A and B and no x, and that I must have hyperbolic tangent on both sides of the equation to find the constants.
I also tried getting a polynomial with tanh(Bx) representing a variable p and then I got factors A, p, and the polynomial. If the equation equals zero, then at least one of those factors must be zero, so I set them equal to zero, but I still have x in there. I tried setting tanh (Bx) (aka p) equal to zero and setting the polynomial to zero and then I solved the polynomial for p and made the p equation equal to zero, however, it looked very complicated and the professor said it was a simple calculation, and even then, that would mean that phi is zero, and that is not the case because I am supposed to use phi to get energy density.