Fifth-order nonlinear Schrödinger equation as Routhian reduction of the nonlinear Klein–Gordon model

We consider the nonlinear Klein–Gordon model with polynomial nonlinearity involving odd and even powers. We elaborate a hybrid Hamiltonian and Lagrangian approach, which is generally referred to as Routh’s procedure, to describe the nonlinear modulation of wave packets assuming the wave field envelope to be a slow function of time and coordinate. An extended fifth-order nonlinear Schrödinger equation is obtained from the Routhian equation for a complex canonical variable that couples the wave field and its momentum. This canonical variable is expressed in terms of the amplitude of fundamental harmonic and its derivatives with respect to coordinate. The resulting fifth-order nonlinear Schrödinger equation is demonstrated to be a Hamiltonian system under a certain constraint imposed on its model parameters. When compared with other techniques used to derive high-order nonlinear Schrödinger models, our approach preserves the Hamiltonian structure of the governing equations for the fundamental harmonic, while the input of higher harmonics is taken into account by means of variational principle applied to the averaged Routhian. In this way, it represents the most general tool for the reduction of nonlinear wave equations to high-order envelope models for slowly modulated wave packets.