Construction of New Surface Wave Solutions of the Modified KdV Equation

The major concern of this work is to propose new prototypes of surface hybrid waves, in particular waves propagating without sprawl or deformation on the surface of a fluid. The model considered for this purpose is the modified KdV (Korteweg-de Vries) equation. A peculiarity of the obtained solutions is that they form packages constituted by combinations of waves belonging to the two main families of well-known bright and dark solitary waves. This putting together creates competitions between the different components of the considered packages which, following the values assigned to the parameters of the considered system and in relation to those of the wave parameters, generate hybrid or multi-form structures. The direct method of resolution which made possible the obtained results is that of Bogning-Djeumen Tchaho-Kofane extended to the new implicit Bogning functions. The existence conditions of some solutions are obtained. The numerical simulations carried out with a view to testing the observable and applicable characters of the obtained solutions revealed their stabilities over a relatively long time, and at the same time, confirmed the recommended theoretical forecasts. We are convinced that the solutions proposed as part of this work will make it possible to detect, understand and explain some physical phenomena linked to fluid molecular interactions, former or new, which constantly occur on the fluid surfaces, mainly at the shallow water surface.