I am trying to numerically simulate (using NDSolve) the following differential equation with the following boundary conditions:

$$partial_x^2psi +cpartial_x^4psi – b(apsi + psi^4)= lambdapartial_tpsi$$

$$partial_tpsi(x,t=infty)= 0$$

$$psi(x,t=0)=f(x)$$

where $a,b,c,lambda$ are all constants. I have been encountering a couple of problems. For example, it seems like Mathematica cannot simulate fourth order partial differential equations(?) Also when working with simpler versions of the problem, I was running into trouble with the boundary condition at $tto infty$. The code I’ve been working with is something along the lines of this:

```
a = 1;
b = 1;
c = 1;
l = 1;
NDSolveValue({D((Psi)(x, t), {x, 2}) + D((Psi)(x, t), {x, 4}) -
b*(a*(Psi)(x, t) + (Psi)(x, t)^3) ==
l*D((Psi)(x, t), t), (Psi)(x, 0) ==
Tanh(x), (D((Psi)(x, t), t) /. t -> 1000) == 0}, (Psi), {x, -20,
20}, {t, 0, 1000}
)
```

I tried making the time boundary condition some large number, rather than putting in Infinity. But this entire scheme seems to be too simple.