Resolution des edp non lineaires: application a l’equation de benjamin-ono moyennant la methode de fan

Abstract

In this thesis, the nonlinear Benjamin-Ono PDE was studied using Fan's method, which is recognised for its ability to generate exact solutions in a variety of forms. This equation, which models physical phenomena, is of major interest from both the theoretical and practical points of view. It belongs to a class of integrable equations with stable localised soliton solutions, the study of which is crucial in several areas of mathematical physics. The idea is based on an appropriate transformation enabling the initial PDE to be reduced to an algebraic system of equations by introducing an auxiliary equation, the solutions of which are given in trigonometric, hyperbolic or rational form. In order to validate the effectiveness of this approach in the specific case of the Benjamin-Ono equation, a numerical implementation was carried out.