I try to solve numerically phi(x,t) to get the same fig [![enter image description here][1]][1]. Actually, this figure obtained employing pathlength continuation techniques as bundled in auto07p (or more modern pde2path) by Matlab

So any help to NDSolve by Mathematica? The linear ansatz. Which numerical methods are you employing?: shooting? pseudo-arclength continuation?

(\[PartialD]f (x,t) )/\[PartialD]t= a(\[PartialD]^2f (x,t) )/\[PartialD]x^2 [(b^4+r)f(x,t) + (f (x,t))^3+2b^2(\[PartialD]^2f (x,t) )/\[PartialD]x^2+(\[PartialD]^4f (x,t) )/\[PartialD]x^4],
     Boundary Conditions: f(-100,t)=0 and f(100,t)=0.
     the ansatz: f(x,t)= c+ d exp(\[Beta] t+i \[Pi]/50 x).
    a=b=1, -1<r<1, -0.8<c<-0.2, d=0.5 and \[Beta]= 3.7*10^-5.

 b = 1; L = 100; a0 = -0.3; b0 = 0.5; q = 1; r = -0.9;

\[Phi]0[x_] := a0 + b0 Cos[2 \[Pi] x/L];

\[Phi]1[x_, t_] := (r + q^4) \[Phi][x, t] + \[Phi][x, t]^3; \[Phi]2[ x_, t_] := D[\[Phi][x, t], {x, 4}] + 2 q^2 D[\[Phi][x, t], {x, 2}]; \[Phi]12[x_, t_] := \[Phi]1[x, t] + \[Phi]2[x, t];

SHE[x_, t_] := D[\[Phi][x, t], t] == b D[\[Phi]12[x, t], {x, 2}];

sol = NDSolve[{SHE[x, 
t], \[Phi][x, 0] == \[Phi]0[x], \[Phi][-100, t] == 
0, \[Phi][100, t] == 0 }, \[Phi], {x, -100, 100}, {t, 0, 2}]; 
P[x_, t_] := \[Phi][x, t] /. sol;
densityPlot = Plot3D[Evaluate[P[x, t]], {x, -100, 100}, {t, 0, 5}, AxesLabel -> {"x", "t", "\[Phi](x,t)"}]

Attachments:

h.png