Extended reductive perturbation method and its relation to the re-normalization method

In this work, by utilizing the extended reductive perturbation method, we studied the propagation of weakly nonlinear waves in a collisionless cold plasma and formally obtained the governing evolution equations of various order terms in the perturbation expansion. The result obtained here is exactly the same with that of Kodama and Taniuti [5], who employed "so called'' the re-normalization method. Seeking a progressive wave solution to these evolution equations we obtained the speed correction terms so as to remove some possible secularities. The result obtained here is consistent with the results of Malfliet and Wieers [6], who employed the dressed solitary wave method and Demiray [7], who utilized the modified reductive perturbation method.