Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow: Part I

The problem formulations of models for internal solitary waves in a stratified shear flow with a free surface are presented. Solitary waves solutions are generated by deriving the nonlinear higher order of extended KdV equations for the free surface displacement. All coefficients of the nonlinear higher-order extended KdV equation are expressed in terms of integrals of the modal function for the linear long-wave theory. The electric field potential and the fluid pressure in the form of traveling wave solutions of the extended KdV equation are obtained. The stability of the obtained solutions and the movement role of the waves by making the graphs of the exact solutions are discussed and analyzed.