Application of the Hamiltonian formulation to nonlinear light-envelope propagations

TL;DR

This paper introduces a Hamiltonian-based analytical approach to approximate soliton solutions and stability analysis of the nonlocal nonlinear Schrödinger equation, relevant for nonlinear light-envelope propagation.

Contribution

It presents a novel Hamiltonian formulation for the NNLSE, enabling analytical solutions and stability criteria, which aligns with previous numerical and analytical results.

Findings

Derived approximate soliton solutions for NNLSE

Established stability conditions via Hamiltonian analysis

Proved equivalence between NNLSE and Euler-Lagrange equation

Abstract

A new approach, which is based on the new canonical equations of Hamilton found by us recently, is presented to analytically obtain the approximate solution of the nonlocal nonlinear Schr\"{o}dinger equation (NNLSE). The approximate analytical soliton solution of the NNLSE can be obtained, and the stability of the soliton can be analytically analysed in the simple way as well, all of which are consistent with the results published earlier. For the single light-envelope propagated in nonlocal nonlinear media modeled by the NNLSE, the Hamiltonian of the system can be constructed, which is the sum of the generalized kinetic energy and the generalized potential. The extreme point of the generalized potential corresponds to the soliton solution of the NNLSE. The soliton is stable when the generalized potential has the minimum, and unstable otherwise. In addition, the rigorous proof of the equivalency between the NNLSE and the Euler-Lagrange equation is given on the premise of the response function with even symmetry.