Recherche des solutions exactes des équations du type Ginzburg-landau complexes avec termes de renormalisation et saturation

Abstract

In this thesis, we propose to study the modified complex Ginzburg-Landau equation. For this, we use mathematical methods proposing solutions by passing the singularities in the type CGL equation, at first, we consider the modified CGL equation with the renormalization term. We use three different methods namely : alternative (G′/G)-expansion method, sech function method and tanh function method to have some exact solutions mathematically acceptable. Afterwards, considering the same modified complex Ginzburg-Landau equation with renormalization term, we combine with the Method of soliton ansatz, the Painleve truncated approach to obtain soliton solutions, therefore physically acceptable and finally we study the influence of the renormalization term in the propagation of the solitary wave. In the second time, it is MCGL equation with both a renormalization term and a saturation term. As in the first case, we combine with method of soliton ansatz, the Painleve truncated approach to obtain soliton solutions, so physically acceptable. But except that in this case, it is the influence of the saturation term that we study in the propagation of the solitary wave. Then we apply two methods which are the arbitrary nonlinear parameters and the exponential rational function method to construct many new exact solutions of the higher order nonlinear partial differential equations namely, the higher order nonlinear Schrödinger (HNLS) equation. The solutions obtained by the current methods are generalized periodic solutions. Finally, new exact analytical solutions of a nonlinear Schrödinger equation with a cubic-quintic nonlinearity and in presence of a couple of perturbation terms. This equation describes the dynamics of soliton propagation through an optical fiber. Several solutions are found without applying the computer codes and by considering the integration constant.