Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 2

Remove the common nonzero factor $e^{-i\omega t}$ and expand in the small parameter $ϵ$. To first order, the modified Laplacian contributes only its $y$ derivatives: its $x$ derivative term has the extra factor ${ϵ}^2$. The regular perturbation expansion therefore givesThus the two residuals are $P+ϵQ+O({ϵ}^2)$ and $R+ϵS+O({ϵ}^2)$, respectively. Subscripts after a comma denote partial derivatives. We retain $\omega \ne 0$, as appropriate to the oscillatory wave and the elimination below.

Insert the proposed leading relation into the two zeroth-order residuals. It givesand, since $b_{0,y}=-i{\psi }_{0,yy}$,Hence $\boxed{b_0=-i{\psi }_{0,y}⟹P=R=0}$. The same relation is forced by $P=0$ when $\omega \ne 0$; its $y$ derivative then automatically gives $R=0$.

The first-order condition $Q=0$, combined with $b_0=-i{\psi }_{0,y}$, givesDifferentiate this expression with respect to $y$ and insert it into $S=0$. The ${\psi }_1$ terms cancel, leavingThus the compatibility partial differential equation isIt is exactly ${\partial }_y(2{\psi }_{0,x}+i{\psi }_{0,yyy})=0$. Integrating with respect to $y$ therefore giveswhere the integration constant may be any function of $x$. Conversely, differentiating this last equation with respect to $y$ recovers the compatibility equation.

For separation of variables matching the boundary Fourier mode, take ${\psi }_0=X(x)e^{iky}$ with $X(0)=1$. Because ${\partial }_y^3{e}^{iky}=-i{k}^3{e}^{iky}$, the reduced ordinary differential equation isThereforeThe decay of a Fourier mode in a third-order spatial evolution equation requires a sign check: this satisfies the requested limit at positive infinity precisely when $\boxed{k>0}$. For $k=0$ it remains identically one, while for $k<0$ its amplitude grows exponentially. Hence for those nonpositive values there is no separated solution meeting both prescribed boundary conditions; the printed assumption that $k$ is merely real needs this additional restriction for decay.