Non-linear secular-free solution of a model equation

Abstract

The perturbation technique of Krylov—Bogoliubov—Mitropolsky is used to derive a secular-free solution up to third order of a model equation $$C^2 \frac{{\partial ^2 \Phi }}{{\partial x^2 }} + \frac{{\partial ^2 \Phi }}{{\partial t^2 }} + \omega _0^2 \Phi + C^2 \Phi ^3 = 0$$ whereC and ω0 are constants in space and time. Expressions are obtained for amplitude dependent frequency shifts and wave number shifts.