Hamiltonians of the Generalized Nonlinear Schrödinger Equations

Abstract

Some types of the generalized nonlinear Schrödinger equation of the second, fourth and sixth order are considered. The Cauchy problem for equations in the general case cannot be solved by the inverse scattering transform. The main objective of this paper is to find the conservation laws of the equations using their transformations. The algorithmic method for finding Hamiltonians of some equations is presented. This approach allows us to look for Hamiltonians without the derivative operator and it can be applied with the aid of programmes of symbolic calculations. The Hamiltonians of three types of the generalized nonlinear Schrödinger equation are found. Examples of Hamiltonians for some equations are presented.