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<p>The nonlinear coupled Riemann wave equation serves as a mathematical framework for analyzing the interaction between short and long waves in various physical phenomena. Its importance lies in capturing both soliton-like behaviors and complex instabilities that arise in nonlinear media. Despite this significance, exact solutions for this equation, particularly in fractional-order forms, remain limited. In this article, numerous soliton solutions of the NLCRW equation are derived by applying novel modified (<i>G</i>′/<i>G</i><sup>2</sup>)-expansion method, thereby advancing the state of the art in analytical wave modeling. By employing several fractional derivatives, including the M-Truncated, β, and Conformable operators, the method produces diverse solution families such as hyperbolic, trigonometric and rational forms. Comparative 2D and 3D visualizations further highlight M-type and singular periodic solitary wave structures across these derivatives. In addition, the dynamic behavior of a perturbed nonlinear Hamiltonian system is investigated using bifurcation analysis, Poincaré sections, and Lyapunov exponents. The bifurcation diagrams and phase portraits reveal regime transitions and underscore the critical role of system parameters. Sensitivity and multistability analyses confirm the influence of initial conditions on long-term dynamics. These results provide insights relevant to ion-acoustic waves in plasma, shallow-water wave propagation, and the transmission of optical pulses, where nonlocal interactions and memory effects play an essential role.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月4日

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  • Amna Mumtaz,
  • Khalid Masood,
  • Muhammad Shakeel,
  • Nehad Ali Shah

摘要

The nonlinear coupled Riemann wave equation serves as a mathematical framework for analyzing the interaction between short and long waves in various physical phenomena. Its importance lies in capturing both soliton-like behaviors and complex instabilities that arise in nonlinear media. Despite this significance, exact solutions for this equation, particularly in fractional-order forms, remain limited. In this article, numerous soliton solutions of the NLCRW equation are derived by applying novel modified (G′/G2)-expansion method, thereby advancing the state of the art in analytical wave modeling. By employing several fractional derivatives, including the M-Truncated, β, and Conformable operators, the method produces diverse solution families such as hyperbolic, trigonometric and rational forms. Comparative 2D and 3D visualizations further highlight M-type and singular periodic solitary wave structures across these derivatives. In addition, the dynamic behavior of a perturbed nonlinear Hamiltonian system is investigated using bifurcation analysis, Poincaré sections, and Lyapunov exponents. The bifurcation diagrams and phase portraits reveal regime transitions and underscore the critical role of system parameters. Sensitivity and multistability analyses confirm the influence of initial conditions on long-term dynamics. These results provide insights relevant to ion-acoustic waves in plasma, shallow-water wave propagation, and the transmission of optical pulses, where nonlocal interactions and memory effects play an essential role.