Exact amplitude-phase equation
= Exact amplitude-phase equation
{title2=$\epsilon^2A''+k^2A=C^2/A^3,\quad\sigma A^2=C$}
Substitution of $y=Ae^{i\epsilon^{-1}\int\sigma dX}$ into $\epsilon^2y''+k^2y=0$ gives the displayed real amplitude equation and conserved imaginary-part flux. Conversely these equations produce an exact complex solution wherever $A\ne0$. The <WKB approximation> is obtained by expanding this exact system where $|k|$ stays positive.