I'm doing a PhD and my work so far has involved linear stability analysis. I believe I have a grasp on that. Now, however, my supervisor wants me to work with weakly nonlinear analysis, that is, keeping terms $\mathcal{O}(\epsilon^2)$ (where $\epsilon$ is my small parameter) and neglecting higher-order terms. I have a bunch of papers using weakly nonlinear analysis that are from one research team, however I'm not understanding their working. I set up my asymptotic series different to the research team's, so I'll break the discussion up into parts, mine first, then theirs.
I have two asymptotic series $$\phi(r,\theta,t) \sim \phi_0(r,t) + \epsilon\phi_1(r,\theta,t) + \epsilon^2\phi_2(r,\theta,t) + \mathcal{O}(\epsilon^3),$$ and $$r = s(\theta,t) \sim s_0(t) + \epsilon s_1(\theta,t) + \epsilon^2 s_2(\theta,t) + \mathcal{O}(\epsilon^3).$$ From my understanding, I sub $\phi$ and $s$ into my governing equations, plug $s$ into $\phi$ when required and use Taylor series expansions to make $\phi$ a function of only $s_0$ (and $\theta$ and $t$) instead of the full $s$.
In the papers by the research team however, their $r$ function is $$r = R(t) + \zeta(\theta,t),$$ where $\zeta$ is their perturbation term. They then use Fourier expansions, letting $$\zeta = \sum_{k=-\infty}^{k=\infty} \zeta_k \exp(\mathrm{i}kx),$$ where $\zeta_k$ is the Fourier amplitude. After they sub into their governing equations, they evaluate on $r = \zeta$ instead of $r = R$.
I've been at this for weeks and I'm still struggling with understanding the process of weakly nonlinear analysis. My supervisor isn't certain how to do it either, however he mentioned the first set-up that I've provided here. The rest of the work that I've done so far has been using the first set-up, so it would be immensely appreciated if help could be provided with how to use weakly nonlinear analysis with the first set-up, however any help at all would be appreciated.