Slowly varying triangular-jet envelope ($B\propto a^{-q/(1-2q)},q=(1-\sqrt{1-4\Omega })/2$)

For $A_t=-a(ϵx)A_x-ia(ϵx{)}^2{A}_{xx}$, a downstream WKB approximation at fixed frequency $\Omega <1/4$ has local wavenumber $k=q/a$. The next-order amplitude equation is $(a-2a^{2}k)B_X=a^2{k}_{X}B$, giving the stated envelope. For small $\Omega$, it reduces to $B\propto a^{-\Omega }$ to first order in frequency. The approximation requires positive slowly varying width and separation from the double root at $\Omega =1/4$.