Conservation laws and Hamiltonians of the mathematical model with unrestricted dispersion and polynomial nonlinearity

Abstract

Mathematical model of the generalized nonlinear Schrödinger equations is considered. The main feature of the family of equations is that it contains equations of arbitrary order and nonlinearity in the form of a polynomial. Equations of the family are not integrable by the inverse scattering transform but all equations have the bright and embedded optical solitons. It is shown that all equations of the family have three conservation laws. Conservation laws corresponding to the equations of this mathematical model are found taking into account the direct calculations without the differential operator. These three obtained integrals of motion correspond to the power, the momentum and the energy of optical solitons. Conservative quantities corresponding to the bright and embedded solitons are calculated. The Hamiltonian is obtained in general form for the entire family of equations. Detailed formulas of the conserved quantities are given for equations of the fourth and eighth orders.