# Dynamics of soliton propagation: bifurcation, chaos, and quantitative insights into the modified Camassa–Holm equation

## TL;DR

This paper explores the modified Camassa–Holm equation using a new method to find diverse wave solutions and their dynamics.

## Contribution

The study introduces the modified (G′/G)-expansion method for generating exact traveling wave solutions in nonlinear systems.

## Key findings

- The MG′/GE method successfully constructs various exact solutions for the MCH equation.
- The method is more accurate and adaptable compared to existing techniques like the sine–cosine and tanh methods.
- The study highlights the method's applicability to both classical and fractional nonlinear evolution equations.

## Abstract

The modified Camassa–Holm (MCH) equation is a significant mathematical model for describing nonlinear wave phenomena, especially in shallow water dynamics and related physical systems. Although various analytical techniques have been applied to such nonlinear equations, many difficulties have arisen in producing a wide variety of exact and structurally rich solutions. This study addresses this gap by employing the modified (G′/G)-expansion (MG′/GE) method to construct an extensive range of exact traveling wave solutions for the MCH framework, such as trigonometric, hyperbolic, and rational solutions. Numerous waveforms, including single singular, double singular, multiple bright, multiple dark, multiple singular, and singular solitons, have been found to have solutions for the MCH framework. These waveforms have numerous applications in applied sciences and engineering. The structural properties and propagation dynamics of the resulting solutions are successfully depicted by graphics such as 3D, contour, density, 2D time-evolution, and 3D revolving plots. Compared to other existing approaches, such as the sine–cosine method and the tanh method, the MG’/GE approach is substantially more accurate and adaptable. The MG’/GE technique’s durability and computing efficiency allow it to generate precise findings straightforwardly. Its broad variety of applications in nonlinear system analysis is further highlighted by its expansion to fractional-order equations. In addition to laying the foundation for future research on traveling wave phenomena in many scientific domains, the current study presents an analytical scheme for both classical and fractional nonlinear evolution equations (NLEEs).