Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 67 / 2

The method of multiple scales introduces independent variables for the fast oscillation and its slow modulation, for example $t_0=t$, $T=ϵt$, and, when needed, $T_2={ϵ}^{2}t$. The derivative becomes $d/dt={\partial }_{t_0}+ϵ{\partial }_T+{ϵ}^2{\partial }_{T_2}+\cdots$. One expands the solution in $ϵ$ while allowing its leading amplitudes and phases to depend on the slow times.

A solvability condition in the method of multiple scales removes resonant forcing of each fast homogeneous mode. Otherwise a correction contains a secular term, such as $ϵt\sin t$, and an ordinary asymptotic expansion fails on long times. The resulting slow amplitude equations incorporate that accumulated effect in the leading solution. One must still check that the amplitudes remain within the assumed weak-coupling regime and that omitted slower effects have not accumulated.

Assume $0<ϵ\ll 1$ and $X,Y,\alpha =O(1)$. For $n=2$, use the method of multiple scales with $T=ϵt$ and complex amplitudesAt first order the forcing equations areThe product has frequencies $\pm 1,\pm 3$. Its $e^{it}$ coefficient is $\overline{A}B$, while it has no $e^{2it}$ coefficient. Removing the secular terms givesThe leading initial conditions give $A(0)=X/2$ and $B=Y/2$. Put $b=Y/2$, $A=p+iq$. The primary parametric resonance of a two-to-one oscillator pair is governed byFor $b^2>{\alpha }^2$, choose $\lambda >0$. Solving with $p(0)=X/2,q(0)=0$ yieldsIf $b^2<{\alpha }^2$, let $\nu =\sqrt{{\alpha }^2-b^2}/2$; replace $\cosh (\lambda T)$ by $\cos (\nu T)$ and $\sinh (\lambda T)/\lambda$ by $\sin (\nu T)/\nu$. On the boundary $b^2={\alpha }^2$, take the continuous limit:Thus exponential growth occurs when $|Y|>2|\alpha |$ and $X\ne 0$, on the scale $t=O({ϵ}^{-1})$. Equality is a marginal case that can produce linear slow growth: for $b=\alpha \ne 0$ the displayed initial phase excites it, whereas for $b=-\alpha$ it does not. If $X=0$, $x$ stays identically zero and the other oscillator is exactly uncoupled. The leading $y$ amplitude is constant; its backreaction is a higher-order effect on this time scale.

These are leading approximations for bounded slow time $T$, hence $t=O({ϵ}^{-1})$, while the amplitudes stay $O(1)$. First-order corrections can enforce the initial velocities exactly; the leading formula alone has an $O(ϵ)$ initial-velocity mismatch. In the unstable case the expansion must not be extrapolated through arbitrarily large amplification, where weak coupling and omitted terms cease to be uniform.

For $n=1$, both leading oscillations have frequencies $\pm 1$, so their quadratic product contains only frequencies $0,\pm 2$. There is no first-order resonant coupling: the solvability condition in the method of multiple scales gives $A_T=i\alpha A/2$ and $B_T=0$, a frequency shift without first-order amplitude growth. The constant and second-harmonic corrections can feed back into the fundamental at second order. Consequently the first possible growth scale from this second-order resonance from quadratic oscillator coupling isThis identifies the possible slow scale, not a guaranteed instability for every parameter or initial state. In particular, the first-order $\alpha$ frequency shift may detune the second-order resonance unless appropriately tuned; a second-order amplitude analysis would decide actual growth.

Let $K(X)=|k(X)|>0$ on a smooth interval, with $X=ϵx$. For the WKB approximation for a slowly varying oscillator, writeSubstitution into ${ϵ}^2{y}_{XX}+K(X{)}^{2}y=0$ gives, at the first two orders,Thus $S_0^{\prime }=\pm iK$ and $S_1^{\prime }=-K^{\prime }/(2K)$. The leading WKB approximation isThe $K^{-1/2}$ amplitude is the transport correction accompanying the rapid wave phase. Real solutions are equivalent real sine and cosine combinations.

For validity, $K$ must be smooth, nonzero and slowly varying compared with the local wavelength. In physical $x$ derivatives, sufficient local checks are $|K_x|/K^2\ll 1$ and $|K_{xx}|/K^3\ll 1$. Indeed each displayed branch has relative residualSmooth positive $K(X)$ bounded away from zero has these properties on fixed slow intervals. At a zero of $k$ the WKB approximation fails and a classical turning point requires a different local analysis, often an Airy turning-point connection formula. Rapid variations, coefficient singularities and excessively long accumulation of wave phase error also lie outside this leading approximation.