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

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.