LAUSR

Floquet’s theorem applied to Hill’s equation, is translated to its Ermakov pair, namely, the nonlinear amplitude differential equation with periodic parameter. The nonlinear version states that if $$\rho \left( z\right) $$ρz is a solution within one period d, to the nonlinear differential equation $$d^{2}\rho /dz^{2}+\rho \varOmega ^{2}-Q^{2}\rho ^{-3}=0$$d2ρ/dz2+ρΩ2-Q2ρ-3=0, with periodic parameter $$\varOmega ^{2}\left( z+Nd\right) =\varOmega ^{2}\left( z\right) $$Ω2z+Nd=Ω2z, the solution after N periods is given by $$\rho \left( z+Nd\right) =\rho \left( z\right) \rho _{d}^{-N}\left[ 1+\left( \rho _{d}^{4N}-1\right) \cos ^{2}\left( \int Q/\rho ^{2}\left( z\right) dz+N\phi _{d}\right) \right] ^{\frac{1}{2}}$$ρz+Nd=ρzρd-N1+ρd4N-1cos2∫Q/ρ2zdz+Nϕd12. This proposition is proved and a physical interpretation to the Floquet solution is given in terms of counter-propagating waves when the formalism describes one dimensional wave propagation.