Embedded Solitons of the Generalized Nonlinear Schrödinger Equation with High Dispersion

The family of generalized Schrödinger equations is considered with the Kerr nonlinearity. The partial differential equations are not integrable by the inverse scattering transform and new solutions of this family are sought taking into account the traveling wave reduction. The compatibility of the overdetermined system of equations is analyzed and constraints for parameters of equations are obtained. A modification of the simplest equation method for finding embedded solitons is presented. A block diagram for finding a solution to the nonlinear ordinary differential equation is given. The theorem on the existence of bright solitons for differential equations of any order with Kerr nonlinearity of the family considered is proved. Exact solutions of embedded solitons described by fourth-, sixth-, eighth and tenth-order equations are found using the modified algorithm of the simplest equation method. New solutions for embedded solitons of generalized nonlinear Schrödinger equations with several extremes are obtained.