The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions

The higher order nonlinear Schrödinger (NLS) equation describes ultra-short pluse propagation in optical fibres. The generalized nonlinear fifth-order of KdV equations derived from the higher order NLS equation by using multiple scales methods. We obtained the traveling wave solutions for some different kinds of the generalized nonlinear fifth-order of KdV (fifth-order Lax; fifth-order Ito; fifth-order Sawada-Kotera; fifth-order Kaup-Kupershmidt; fifth-order Caudrey-Dodd-Gibbon) equations by applying the auxiliary equation of the direct algebraic method. These solutions for the generalized fifth order KdV equations are obtained precisely and efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves by making the graphs of the exact solutions are analyzed. All solutions are exact and stable.