Difficulty to create the 3D plot of the Linear Schrödinger equation (Fourier Transform)

Question

I am trying to solve the linear Schrödinger equation with Fourier transform as follows. Although analytically I was able to solve it and create the graphs I wanted. I have difficulty making the corresponding graph when I solve the problem numerically. Can you explain to me what is my mistake in the following code?

Analytical solution

Clear["Global`*"]
f[x_] = Exp[-x^2]
u[x_, t_, v_] = 
 Integrate[(Exp[-(I*Pi)/(4) + I*(x - y)^2/(4*v*t)])*
    f[y], {y, -Infinity, Infinity}, 
   Assumptions -> {v > 0, t > 0}]/(Sqrt[4 Pi*v*t])
eik5 = Plot3D[Abs[u[x, t, 0.25]]^2, {x, -25, 25}, {t, 0.05, 30}, 
  PlotRange -> All, PlotPoints -> 250]

Numerical solution

(*  Fourier tranform for the equation  Subscript[u, \
t](x,t)=i*v*Subscript[u, xx](x,t) *)
FourierSinTransform[ D[u[x, t], {t, 1}], x, \[Omega]] == 
 FourierSinTransform[ I*v* D[u[x, t], {x, 2}], x, \[Omega], 
  FourierParameters -> {1, -1}]
ode = D[
\!\(\*OverscriptBox[\(\(\ \ \)\(u\)\), \(^\)]\)[\[Omega], t], {t, 
     1}] == I*v*\[Omega]*(-\[Omega]*
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t] - 2*T0);
sol = DSolve[ode, 
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t], t]
FourierSinTransform[u[x, 0], x, \[Omega]] == 
 FourierSinTransform[f[x], x, \[Omega], FourierParameters -> {1, -1}]
ic = 
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], 0] == 
  FourierSinTransform[f[x], x, \[Omega], FourierParameters -> {1, -1}]
sol = DSolve[{ode, ic}, 
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t], t]

\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega]_, t_] = -((
  2 E^(-I t v \[Omega]^2) (-T0 + 
     E^(I t v \[Omega]^2)
       T0 + \[Omega] DawsonF[\[Omega]/2]))/\[Omega])
u[x_, t_] = InverseFourierSinTransform[
\!\(\*OverscriptBox[\(u\), \(^\)]\)[\[Omega], t], \[Omega], x, 
  FourierParameters -> {1, -1}, Assumptions -> {x > 0}]
T0 = 1;
Plot[Abs[u[x, t]]^2, {x, 0, 20}, PlotRange -> All]

You can download the notebook of my project from Wolfram Community

Answer 1

Too large in comment:

It is very simple actually. The InverseFourierSinTransform in your notebook did not evaluate. Then you are trying to plot it. Here is screen shot from your notebook near the end

You should really always make sure each command completes OK before using its output in the next command.

V 13.2.1 on windows 10