1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding Constants to Solution Given 2nd Order, Nonlinear DE

  1. Mar 24, 2015 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    The differential equation is as follows:


    The soliton solution is:


    3. 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.

    Please help.
  2. jcsd
  3. Mar 24, 2015 #2


    User Avatar
    Science Advisor

    I think you're making this harder than it is. The solution is just a 2-3 lines. Can your write down d^2phi/dx^2? Do you remember that 1-tanh[x]^2 = sech[x]^2. Also remember that you are trying to find an A and B that solve the equation, so you are free to choose A and B to make the problem simplify.
  4. Mar 26, 2015 #3
    d^2phi/dx^2= -2AB^2tanh(Bx), correct?

    Now I have gotten the equation


    I set 2AB^2-A+A^3tanh^2(Bx)=0

    tanh(Bx)= sqrt[2B-1]/A

    But now what? Was I supposed to change the tanh^2(Bx) into 1-sech^2(Bx)?
  5. Mar 26, 2015 #4


    User Avatar
    Science Advisor

    No. You're not differentiating this correctly. The first derivative of tanh(x) is sech^2(x). The second derivative is more complicated than just tanh[x]. Try again.
  6. Mar 26, 2015 #5
    The second derivative is

    phi''=-2AB^2tanh(Bx)sech^2(Bx)= -2AB^2tanh(Bx)+2AB^2tanh^3(Bx)

    and so

  7. Mar 26, 2015 #6
    I keep getting that phi is zero.
  8. Mar 26, 2015 #7


    User Avatar
    Science Advisor

    OK, better. Of course phi = 0 is a solution, but there is another solution. Suppose those two coefficients (2AB^2 -A), and (-2AB^2+A^3) are both zero. Is there a set of A and B that can make that happen?
  9. Mar 26, 2015 #8
    Ahhh yes. I understand!
    Thank you very much :)
  10. Mar 26, 2015 #9
    Alright, I ended up with A=+/- 1 and B=1/sqrt(2).
  11. Mar 26, 2015 #10


    User Avatar
    Science Advisor

    Sounds good to me. See, it wasn't that hard after all!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted