Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 76 / 1 / ii

For the second-harmonic feedback in a long-wave convection amplitude equation, at $\mu =2$ the linear steady operator is $L_0=-(1+{\partial }_x^2{)}^2$. Its eigenvalue on $e^{inx}$ is $-(1-n^2{)}^2$, so the critical Fourier modes are $n=\pm 1$. More generally the linear dispersion relation is $\lambda (k)=-1+\mu k^2-k^4$, giving first onset at $\mu =2,k=1$.

At order ${ϵ}^2$, the weakly nonlinear expansion givesWith ${\Theta }_1=A{e}^{ix}+\overline{A}{e}^{-ix}$,The constant part disappears under differentiation. Inverting $L_0$ on the second harmonic, whose eigenvalue is $-9$, givesup to a critical-harmonic correction absorbed into the definition of $A$. At order ${ϵ}^3$,The coefficient of $e^{ix}$ in the last three terms is respectivelyThe Fredholm solvability condition requires their sum to vanish, because $L_0$ annihilates the critical Fourier mode. For a nonzero amplitude,This requires ${\mu }_2/(3-8s^2/9)>0$. The cubic amplitude coefficient is positive for $s^2<27/8$, yielding a small-amplitude branch in a supercritical bifurcation; it is negative for $s^2>27/8$, yielding a subcritical bifurcation branch in this leading approximation. At $s^2=27/8$, the cubic coefficient vanishes and the quoted relation is singular: higher-order nonlinear terms and a different detuning balance are needed. One must not assert a finite $p$ there.

If a term $ϵ{\mu }_1$ were included in $\mu$, order ${ϵ}^2$ would also contain $-{\mu }_1{\Theta }_{1xx}={\mu }_1{\Theta }_1$. The quadratic nonlinearity produces only the zeroth and second harmonics, with the zeroth differentiated away, so it cannot balance a first-harmonic contribution at that order. Solvability would give ${\mu }_{1}A=0$. Hence a nontrivial critical-mode expansion forces ${\mu }_1=0$ and first balances detuning against cubic amplitude effects at order ${ϵ}^3$.