If, just before your second Plot, you insert
Table[u[x,t],{x,-10,10,5},{t,0,20,5}]
to see some of the expression you are trying to plot
then you will see that most of it consists of complex numbers, which do not plot.
In your definition of u[x,t] there is an extra \ which should probably be removed.
If I remove that and in your Plot3D I replace u[x,t] with Re[u[x,t]] then I see your plot but I am not certain that this is exactly what you want to plot.