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 ]