# Implicit quiescent soliton perturbation in optical metamaterials with complex Ginzburg–Landau equation having nonlinear chromatic dispersion and Kudryashov’s forms of self–phase modulation structures by lie symmetry

## TL;DR

This paper uses Lie symmetry to study stable solitons in optical materials with complex dispersion and self-phase modulation, offering insights for optical communication.

## Contribution

A novel methodological approach using Lie symmetry reduction to derive soliton profiles and constraints without trial functions.

## Key findings

- Lie symmetry analysis yields unified quadratures and explicit constraints for quiescent solitons.
- Parameter sweeps show how dispersion and Kudryashov coefficients control soliton amplitude and width.
- The study identifies stable and unstable regimes for solitons, aiding optical system design.

## Abstract

The paper explores the retrieval of quiescent perturbed solitons in optical metamaterials characterized by nonlinear chromatic dispersion and Kudryashov’s forms of self-phase modulation structures. The foundational model utilized is the complex Ginzburg-Landau equation. The novelty is methodological, as we use Lie symmetry reduction to transform the stationary traveling-wave system into an invariant first-order form, yielding unified quadratures and explicit existence constraints for quiescent solitons across all Kudryashov SPM variants without imposing ad hoc trial functions. A successful implementation of Lie symmetry analysis has been conducted for this retrieval process. In particular, the symmetry-based reduction produces algebraic admissibility conditions that provide a reproducible parameter-screening rule for localized profiles. Additionally, both linear and generalized temporal evolutions are considered. Representative parameter sweeps illustrate how the Kudryashov coefficients and dispersive parameters control the soliton amplitude and width within the admissible windows predicted by the theory. These analyses offer helpful details about the stability and dynamics of the solitons, in the sense of identifying regimes where stationary localization is supported by the reduced dynamical system and where it is precluded by violation of the existence conditions, thereby highlighting potential applications in advanced optical communication systems and providing explicit design-side parameter limits that can help avoid operating regimes associated with pulse degradation.•The study models quiescent soliton perturbation in optical metamaterials using a perturbed complex Ginzburg–Landau framework that incorporates nonlinear chromatic dispersion and Kudryashov-type non-Kerr self-phase modulation structures.•Lie symmetry is applied to retrieve quiescent soliton profiles in implicit form, producing integral-form solutions together with constraint conditions on the governing parameters.•The analysis considers both linear and generalized temporal evolutions and treats multiple Kudryashov self-phase modulation variants, clarifying how evolution choice and SPM structure influence the resulting soliton behavior and potential optical applications.

The study models quiescent soliton perturbation in optical metamaterials using a perturbed complex Ginzburg–Landau framework that incorporates nonlinear chromatic dispersion and Kudryashov-type non-Kerr self-phase modulation structures.

Lie symmetry is applied to retrieve quiescent soliton profiles in implicit form, producing integral-form solutions together with constraint conditions on the governing parameters.

The analysis considers both linear and generalized temporal evolutions and treats multiple Kudryashov self-phase modulation variants, clarifying how evolution choice and SPM structure influence the resulting soliton behavior and potential optical applications.

Image, graphical abstract