**I try to solve f(x,t) numerically. 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?**
$$ \frac{\partial f (x,t)}{\partial t}= a \left[(b^4+r)f(x,t) + (f(x,t))^3+2b^2\frac{\partial^2f(x,t)}{\partial x^2}+\frac{\partial^4f(x,t)}{\partial x^4}\right], $$ 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 $\beta= 3.7*10^-5$.
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b = 1; L = 100; a0 = -0.3; b0 = 0.5; q = 1; r = -0.9;
f0[x_] := a0 + b0 Cos[2 Pi x/L];
f1[x_, t_] := (r + q^4) f[x, t] + f[x, t]^3; f2[x_, t_] :=D[f[x, t], {x, 4}] + 2 q^2 D[f[x, t], {x, 2}];
f12[x_,t_] := f1[x, t] + f2[x, t];
SHE[x_, t_] := D[f[x, t], t] == b D[f12[x, t], {x, 2}];
sol = NDSolve[{SHE[x, t], f[x, 0] == f0[x], f[-100, t] ==
0, f[100, t] == 0 }, f, {x, -100, 100}, {t, 0, 2}];
P[x_, t_] := f[x, t] /. sol;
densityPlot =
Plot3D[Evaluate[P[x, t]],{x,-1,1}, {t, 0, 5},AxesLabel -> {"x", "t", "f(x,t)"}]
