Manipulate[Plot3D[Evaluate[u[t, x] /. Quiet[NDSolve[{
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"2", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x] ==
d + c*u[t, x] + b*u[t, x]^2 + a*u[t, x]^3 +
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "2"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x], u[0, x] == E^-x^2,
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[0, x] == 0, u[t, -x0] == u[t, x0]},
u, {t, 0, x0}, {x, -x0, x0}]]], {x, -x0, x0}, {t, 0, x0},
Mesh -> None, ColorFunction -> "Rainbow", MaxRecursion -> 2,
PlotPoints -> 50], {{a, -0.3}, -4, 0}, {{b, -0.6}, -4,
0}, {{c, 0.8}, -4, 1}, {{d, 0}, -1, 1}, {{x0, 16, "Solution Range"},
5, 20}]