= Solution
Expanding the exponential for $|ka|\ll1$ gives $e^{-2ka}=1-2ka+2k^2a^2+O((ka)^3)$. Substitution of the <plane wave> $A=e^{i(kx-\omega t)}$ in the model gives
$$
-i\omega=-iak+ia^2k^2,\qquad \boxed{\omega=ak-a^2k^2},
$$
which is exactly the quadratic approximation to the <dispersion relation>.
For the slowly varying width, put $X=\epsilon x$ and use the <WKB approximation>
$$
A=B(X)\exp\left(\frac{i}{\epsilon}S(X)-i\Omega t\right),\qquad k(X)=S_X.
$$
The derivatives are
$$
A_x=(ikB+\epsilon B_X)e^{iS/\epsilon-i\Omega t},\quad
A_{xx}=[-k^2B+i\epsilon(k_XB+2kB_X)+\epsilon^2B_{XX}]e^{iS/\epsilon-i\Omega t}.
$$
At leading order $\Omega=ak-a^2k^2$. On the downstream small-wavenumber branch write
$$
k=\frac q{a(X)},\qquad q=\frac{1-\sqrt{1-4\Omega}}2,
$$
so $q$ is independent of $X$. The next order gives the <slowly varying triangular-jet envelope> transport equation
$$
(a-2a^2k)B_X=a^2k_XB,\qquad
\frac{B_X}{B}=-\frac{q}{1-2q}\frac{a_X}{a}.
$$
Integration therefore yields
$$
\boxed{B(X)=B(0)\left[\frac{a(X)}{a(0)}\right]^{-q/(1-2q)}},\qquad
\boxed{B(X)\simeq B(0)\left[\frac{a(X)}{a(0)}\right]^{-\Omega}}\quad(\Omega\ll1).
$$
Indeed, $q/(1-2q)=\Omega+O(\Omega^2)$. 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 $\exp(i\int_0^x q/a(\epsilon s)\,ds-i\Omega t)$. 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 $\Omega$.
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