Hello Bryan. Do that, and I appreciate if you would tell me what they say.

Hello Bryan,

I forwarded my notebook to a friend who has a more recent version of Mathematica, and he too, doesn't get the same pictures as I do. Seems there indeed is something wrong. However, he pointed out that your differential equation in the NDSolve-statement might be overdetermined. In principle, you have twice the statement containing xmax.....

You should delete one of them and try your code again.

I have no idea why my version (Mathematica 7) works properly and also with one of these conditions deleted.

Your original form was

Params:

m = 1;
k = 1;
\[Omega] = Sqrt[m^2 + k^2];
xmax = 1000;

giving the function f0

sol = \[Phi] /. NDSolve[
     {-D[\[Phi][t, x], {t, 2}] + D[\[Phi][t, x], {x, 2}] - 
        m^2*\[Phi][t, x] == 0, \[Phi][t, -xmax] == 
       Exp[-I*k*xmax - I*\[Omega]*t],
      \[Phi][t, xmax] == Exp[I*k*xmax - I*\[Omega]*t],
      \[Phi][0, x] == Exp[I*k*x],
      Derivative[1, 0][\[Phi]][0, x] == -I*\[Omega]*Exp[I*k*x]},
     \[Phi], {t, 0, 2}, {x, -xmax, xmax}][[1, 1]];
f0 = sol

In shorter form (one of the xmax-conditions deleted)

Clear[sol];
sol = \[Phi] /. NDSolve[
     {-D[\[Phi][t, x], {t, 2}] + D[\[Phi][t, x], {x, 2}] - 
        m^2*\[Phi][t, x] == 0,
      \[Phi][t, xmax] == Exp[I*k*xmax - I*\[Omega]*t],
      \[Phi][0, x] == Exp[I*k*x],
      Derivative[1, 0][\[Phi]][0, x] == -I*\[Omega]*Exp[I*k*x]},
     \[Phi], {t, 0, 2}, {x, -xmax, xmax}][[1, 1]];
f1 = sol

with function f1 as a result.

However, there exists a symbolic solution. Your differential equation reads

deq[f_] := -D[f, t, t] + D[f, x, x] - m^2 f

I found a symbolic solution ff

ff = (1/2 (E^(-I Sqrt[a^2 + m^2] t) + E^(I Sqrt[a^2 + m^2] t)) C[1] + 
     1/2 I (E^(-I Sqrt[a^2 + m^2] t) - E^(I Sqrt[a^2 + m^2] t)) C[
       2]) (E^(I a x) K[1] + E^(-I a x) K[2]);

Indeed

deq[ff] // FullSimplify
(* 0 *)

Adjusting the constants, you get function ff1, which fulfills (almost) all your conditions:

ff /. t -> 0
rr1 = {K[1] -> 1, K[2] -> 0, C[1] -> 1, 
  C[2] -> -I \[Omega] /Sqrt[k^2 + m^2], a -> k}
ff1 = ff /. rr1
ff1 /. t -> 0
\[Phi][0, x] == Exp[I*k*x]

and

D[ff1, t] /. t -> 0
-I*\[Omega]*Exp[I*k*x]

Finally, fix k and m

ff1 /. {k -> 1, m -> 1}

and you will see that the two numerical solutions obtained above and the symbolic solution are identical. Plot all three functions together for different tmes and regions (try the Im-parts as well)

Manipulate[
 Plot[
  {Re@f0[tt, xx], Re@f1[tt, xx], Re@ff1 /. {t -> tt, x -> xx}},
  {xx, -10, 10}, 
  PlotStyle -> {{Dashing[.1], Thickness[.01], Black}, 
    Blue, {Dashing[.05], Thick, Red}}
  ],
 {tt, 0, 2}
 ]

I take this as a sign that NDSolve (on my system) works properly. What is happening with yours?

I attach the latest notebook as well if you consider sending it to Wolfram support.

Attachments:

Untitled-wcom.nb

Thank you. I wonder if this is a bug with my version of Mathematica, as your images look correct but if I try to run the notebook, I get different images.

EDIT: Could you try running your code on a Mathematica Online notebook? I get the same issue there as in my desktop version.

Could you possibly attach a few images of what you get? I don't seem to be getting any plots with the correct wavenumber, regardless of what x range I plot over.