Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 74 / 3 / b / Solution

Expanding the exponential for $|ka|\ll 1$ gives $e^{-2ka}=1-2ka+2k^2{a}^2+O((ka{)}^3)$. Substitution of the plane wave $A=e^{i(kx-\omega t)}$ in the model giveswhich is exactly the quadratic approximation to the dispersion relation.

For the slowly varying width, put $X=ϵx$ and use the WKB approximationThe derivatives areAt leading order $\Omega =ak-a^2{k}^2$. On the downstream small-wavenumber branch writeso $q$ is independent of $X$. The next order gives the slowly varying triangular-jet envelope transport equationIntegration therefore yieldsIndeed, $q/(1-2q)=\Omega +O({\Omega }^2)$. The requested power is the leading low-frequency envelope, not the exact finite-frequency exponent of the quadratic model. The corresponding leading field is this wave envelope times $\exp (i{\int }_0^{x}q/a(ϵs)ds-i\Omega t)$. Positive slowly varying width and separation from the zero of the local group velocity, $1-2q=0$, are needed for this WKB approximation. Extremely large $|\log [a(X)/a(0)]|$ can also invalidate replacing the exact exponent by $\Omega$.