Exact amplitude-phase equation (${ϵ}^2{A}^{\prime \prime }+k^{2}A=C^2/A^3,\sigma A^2=C$)

Substitution of $y=A{e}^{i{ϵ}^{-1}\int \sigma dX}$ into ${ϵ}^2{y}^{\prime \prime }+k^{2}y=0$ gives the displayed real amplitude equation and conserved imaginary-part flux. Conversely these equations produce an exact complex solution wherever $A\ne 0$. The WKB approximation is obtained by expanding this exact system where $|k|$ stays positive.