# Innovative solutions for lossy nonlinear transmission lines model using a modified extended mapping approach with fractional effects

## TL;DR

This paper presents new soliton solutions for a lossy nonlinear transmission line model using a modified extended mapping approach with fractional effects.

## Contribution

The study introduces a novel application of conformable fractional derivatives to generate diverse soliton solutions in nonlinear systems.

## Key findings

- Exact analytical solutions including hyperbolic, trigonometric, and exponential solitons were derived.
- Parametric analysis revealed the influence of β₁ on spatial wave evolution.
- Generated 2D, 3D, and density plots illustrate the physical behavior of the solutions.

## Abstract

This study investigates soliton solutions of the lossy nonlinear electrical transmission line (Loss-NLETL) model using the Modified Extended Mapping (Mod-EM) technique. The model incorporates the effect of a conformable fractional derivative (Con-FD) with respect to the spatial variable \documentclass[12pt]{minimal}
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\begin{document}$$x$$\end{document}, enabling a more generalized and exact depiction of the system’s behavior. A set of exact analytical solutions has been obtained, including composite hyperbolic-type solutions, trigonometric periodic waves, singular periodic wave solutions, dark soliton solutions, exponential traveling wave solutions, hyperbolic soliton solutions, singular hyperbolic waveforms, mixed-type soliton structures involving kink and rational hyperbolic components, and Jacobi elliptic wave solutions. Several of these solutions are novel and extend the existing literature. To illustrate the physical behavior and structural diversity of the solutions, 2D, 3D, and density plots have been generated. Furthermore, a parametric analysis has been conducted to explore the influence of the variation in \documentclass[12pt]{minimal}
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\begin{document}$${\beta }_{1}$$\end{document}​, a key parameter affecting the spatial evolution of the waveforms. The results demonstrate the effectiveness of the method in generating diverse soliton structures in complex nonlinear systems, with potential implications for applied physics and electrical engineering applications.