Proving a a pitchfork bifurcation: modified swift-hohenberg

[onanox]

I'm trying to solve question 4.12 from Cross and Greenside "pattern formation and dynamics in nonequilibrium systems".

the question is about the equation

$\partial_t u = r u - (\partial_x ^2 +1)^2 u - g_2 u - u^3$
Part A: with the ansatz $u=\sum_{n=0}^\infty a_n cos(nx)$ show that the bifurcation retains its pitchfork character and calculate $a_1$ to lowest order in $r$.

Part B: Find a condition on $g_2$ for the bifurcation to be supercritical

Background: The swift-hohenberg equation ($g_2=0$) has a uniform solution for $r<0$ and undergoes a pitchfork bifurcation when $r>0$ to a stationary nonlinear striped state. The question is asking about generalizing this for $g_2\neq 0$.

Attempt at solution: I tried plugging in the ansatz into the equation and requiring that $\partial_t=0$ to find a stationary state, but got caught up in evaluating $\left[\sum_{n=0}^\infty a_n cos(nx) \right]^2$. I tried looking back to see how the bifurcation was analyzed in the $g_2=0$ case, and there, we assume a single Fourier mode, find an amplitude equation and its easy to deduce that $|A|\propto |r|^{1/2}$ giving the pitchfork bifurcation.

I think I'm supposed to derive a similar relationship for $a_1$, but I don't know how to evaluate the square and cube of that infinite sum.

Any ideas?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[onanox]

Yes, coming back to it I worded in a confusing way. Let me try to clear it up.

The equation has the trivial solution $u=0$. If you linearize the equation, you find that for $r<0$ this solution is linearly stable, its linearly unstable for $r>0$. The question concerns the form of this bifurcation. For reference: http://en.wikipedia.org/wiki/Pitchfork_bifurcationI think I need to find some set of equations for the $a_i$, and show that there are two solutions (pitchfork) only when the trivial solution is unstable. But I don't know how to solve that. Effectively, the problem is reduced to evaluating $\left[ \sum_{n=0} ^\infty a_n cos(nx)\right]^M$ for $M=2,3$