Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 331 / 2 / ii / Solution

At criticality choose the fundamental modeIts quadratic self-interaction satisfiesThe critical linear operator has eigenvalues $-64{\pi }^6$ on $\sin (2\pi z)$ and $-60(a^2+{\pi }^2{)}^3$ on $\cos (2ax)\sin (2\pi z)$. The slaved second-order correction is thereforewhere

Introduce a slow time $\tau$ and project the next-order equation onto the fundamental mode. Detuning $R-R_c$ contributes $a^2(R-R_c)A$, while the interaction of $w_1$ with $w_2$ has fundamental componentThe solvability condition is the Landau amplitude equation