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.
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References
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Pramanik, G.C. Non-linear secular-free solution of a model equation. Acta Physica 39, 97–99 (1975). https://doi.org/10.1007/BF03157022
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DOI: https://doi.org/10.1007/BF03157022