Secular-free solution up to third order of a model nonlinear equation for dispersive waves

Abstract

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation

$$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$

where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.