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Verify a solution in a PDE?

Posted 4 months ago
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I'm having trouble verifying solutions for complex PDEs including conjugate and absolute value.

For Example: I have a complex PDE where superscript * denotes conjugate of the unknown function \[Psi][x, t]. Candidate solution:

a = 1/2 (8 \[Beta] - b Subscript[A, 1]^2)

 \[Psi][x_, t_] :=E^(I (-kx + \[Theta] + 
  1/2 t (-8 k^2 \[Beta] - 2 \[Gamma] + 2 b Subscript[A, 1]^2 + 
     b k^2 Subscript[A, 1]^2))) Subscript[A, 1] Tanh[x + k t (8 \[Beta] - b Subscript[A, 1]^2)]; 

I want to check that the candidate solution satisfies the PDE or not:

  (*Checking the solution*)
            FullSimplify[
             I*D[\[Psi][x, t], t] + a*D[\[Psi][x, t], {x, 2}] + 
               b*ComplexExpand@(Abs[\[Psi][x, t]]^2)*\[Psi][x, t] -  \[Beta]/(
                ComplexExpand@(Abs[\[Psi][x, t]]^2)* 
                 ComplexExpand@
                  Conjugate[\[Psi][x, t]] )*(2*Abs[\[Psi][x, t]]^2*
                   D[ComplexExpand@(Abs[\[Psi][x, t]]^2), {x, 2}] - (D[
                    ComplexExpand@(Abs[\[Psi][x, t]]^2), x])^2) - \[Gamma]*\[Psi][
                 x, t] == 0 ]

In the definition of psi, you probably meant k*x instead of kx. With that emendment, the equality is proved immediately for the following numerical values of the variables: b = 1, k = 1, \[Beta] = 1, \[Theta] = 1, \[Gamma] = 1. Perhaps you can generalize the result gradually.

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