Numerical realization of the variational method for generating self-trapped beams

TL;DR

This paper presents a numerical variational method based on Rayleigh-Ritz optimization for predicting self-trapped beams in nonlinear media, overcoming analytical limitations and enabling robust soliton solutions.

Contribution

It introduces a numerical approach that improves upon traditional variational methods by avoiding analytical Lagrangian calculations, allowing for more complex soliton profile predictions.

Findings

Successfully predicts various types of solitons including vortices and azimuthons

Demonstrates robustness of self-trapped beams during propagation

Enables generation of sophisticated soliton profiles in nonlinear models

Abstract

We introduce a numerical variational method based on the Rayleigh-Ritz optimization principle for predicting two-dimensional self-trapped beams in nonlinear media. This technique overcomes the limitation of the traditional variational approximation in performing analytical Lagrangian integration and differentiation. Approximate soliton solutions of a generalized nonlinear Schr\"odinger equation are obtained, demonstrating robustness of the beams of various types (fundamental, vortices, multipoles, azimuthons) in the course of their propagation. The algorithm offers possibilities to produce more sophisticated soliton profiles in general nonlinear models.