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Exploring complex dynamics in nonlinear Riemann wave models using fractional calculus-based expansion.

Amna Mumtaz1, Khalid Masood2, Muhammad Shakeel1

  • 1Department of Mathematics, Faculty of Basic Sciences, University of Wah, Wah Cantt, 47040, Pakistan.

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|December 4, 2025
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Summary
This summary is machine-generated.

Researchers derived new exact solutions for the nonlinear coupled Riemann wave equation using a modified expansion method. This advances analytical wave modeling for phenomena like plasma and optical pulses.

Keywords:
Bifurcation analysisConformable derivativeLyapunov exponentsM-truncated derivativeNon-linear coupled Riemann wave equation (NLCRW)Novel modified (G′/G 2)-expansion approachSensitivity analysisβ-derivative

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Area of Science:

  • Nonlinear Dynamics
  • Mathematical Physics
  • Wave Phenomena

Background:

  • The nonlinear coupled Riemann wave equation (NLCRW) models interactions between short and long waves.
  • It is crucial for understanding soliton behaviors and instabilities in nonlinear media.
  • Exact solutions, especially for fractional-order NLCRW, are scarce.

Purpose of the Study:

  • To derive novel soliton solutions for the NLCRW equation.
  • To explore solutions across different fractional derivative types.
  • To analyze the dynamics of a perturbed nonlinear Hamiltonian system.

Main Methods:

  • Application of the modified (G'/G²) -expansion method.
  • Utilizing M-Truncated, β, and Conformable fractional derivatives.
  • Employing bifurcation analysis, Poincaré sections, and Lyapunov exponents for dynamical analysis.

Main Results:

  • Numerous soliton solutions (hyperbolic, trigonometric, rational) were obtained for the NLCRW equation.
  • M-type and singular periodic solitary wave structures were visualized.
  • Bifurcation analysis revealed regime transitions and parameter influences in the Hamiltonian system.

Conclusions:

  • The study provides a significant advancement in analytical wave modeling for the NLCRW equation.
  • The findings offer insights into ion-acoustic waves, shallow-water propagation, and optical pulse transmission.
  • Nonlocal interactions and memory effects are critical in these physical systems.