Variational Analysis: Collapse of the Static Soliton Wave Beams in a One-Dimensional Discrete System

A system that experiences sudden state changes at specific times is said to be discrete. The majority of systems that are studied in operations research and management science, such as transportation or communication studies, are under the application of discrete systems. This study investigates the analytical study of the static soliton for Cubic-Quintic Discrete Nonlinear Schrödinger Equation (DNLSE) in discrete system. Subsequently, static soliton, that is often used to characterize specific self-action regime in a continuous one-dimensional problem, is defined as a self-reinforcing wave packet that keeps its form and velocity while it travels in a medium. Moreover, it is well-known that the NLSE is a known integrable equation of partial differential equation. Therefore, the variational approximation method is applied to transform the partial differential equation of the main equation into ordinary differential equations, thus, to derive the equations for soliton parameters evolution during the interaction process. The method is used to qualitatively study the Discrete NLSE and characterize self-action modes. It is shown that in discrete media, both wide and narrow wave beams (relative to the grating scale) experience weakened diffraction, resulting in the “collapse” of the one-dimensional wave field when the power is greater than the critical threshold. As a result, the central fiber is able to self-channel radiation.