The High Order Solitary Wave Solution Of FNFD Model

Water waves are the most common dynamic factor of the ocean. Due to water depth and terrain, shallow water waves show nonlinear characteristics gradually that impact the coastal engineering. Therefore, the study of higher-order nonlinear approximate to shallow water waves enriches the water wave dynamics and is crucial to engineering practice.This study is based on an exact fully nonlinear and fully dispersive (FNFD) model. This FNFD model is first based on two exact equations is accomplished involving three variables all pertaining to their values at the water surface. Closure of the system of model equations is accomplished in differential form, by attaining a series expansion of velocity potential. A reductive perturbation method for deriving asymptotic theory for high-order solitary waves is developed using the differential closure equation of the FNFD wave theory. Using this method, we obtained the leading 17th-order solitary wave solutions. Using these solutions, we compare the horizontal velocity on the free surface, the depth-average velocity and the velocity on the bottom.Capillary-gravity solitary waves are investigated theoretically. The theoretical study based on the reductive perturbation method provides asymptotic theories for the 6th-order capillary-gravity solitary waves. As Bond numberτapproaches l/3, waves exist only for smaller and smaller values of the amplitude of the solitary waves. We calculated the diverge range of the capillary-gravity solitary wave solutions.