New solitons and bifurcation dynamics in the fractional Huxley model: Numerical and analytical approaches

Huiqin Chu; Muhammad Abuzar; Mohammed Ahmed Alomair; Abdulaziz Khalid Alsharidi; Huiqin Chu; Muhammad Abuzar; Mohammed Ahmed Alomair; Abdulaziz Khalid Alsharidi

doi:10.3934/math.2026083

AIMS Mathematics

2026, Volume 11, Issue 1: 1998-2026. doi: 10.3934/math.2026083

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New solitons and bifurcation dynamics in the fractional Huxley model: Numerical and analytical approaches

1.
School of Mathematics-Physics and Finance, Anhui Polytechnic University, Wuhu, Anhui, 241000, China
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Yunyan 550003, China
3.
Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia
4.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 20 November 2025 Revised: 10 January 2026 Accepted: 15 January 2026 Published: 21 January 2026
MSC : 35C08, 35Q55, 37K40

This research explores the higher-order nonlinear fractional Huxley equation formulated with the $ \beta $, and $ M $-truncated fractional derivatives to account for memory and hereditary effects present in nonlinear diffusion and excitation wave dynamics. The fractional formulation expands on the standard Huxley model by incorporating nonlocal temporal and spatial correlations, providing a more realistic description of finite-amplitude wave propagation and spectral energy transfer in complex media. Two advanced analytical techniques are used to derive exact solutions: the enhanced modified extended tanh expansion method (EMETEM) and the improved $ F $-expansion technique. Compared to classic perturbation and variational techniques, these methods offer greater algebraic freedom and faster convergence, resulting in a diverse family of closed-form traveling-wave solutions expressed in trigonometric, hyperbolic, exponential, and rational forms. The analytical results are further validated, and the spatiotemporal evolution of the resulting wave structures is investigated using a finite-difference numerical scheme. The accuracy and robustness of the suggested framework are confirmed by the numerical findings, which show good agreement with the analytical results. Phase-plane and bifurcation analyses demonstrate transitions between periodic, quasi-periodic, and chaotic regimes by revealing both stable and unstable spiral formations. The findings show that fractional derivatives significantly improve the dynamical characteristics of the Huxley system by allowing for finer control of diffusion, dispersion, and localized energy concentration, advancing our understanding of nonlinear fractional wave behavior in excitable and dispersive media.
- fractional Huxley model,
- computational approaches,
- exact solutions,
- finite-difference scheme,
- phase portraits and bifurcation analysis
Citation: Huiqin Chu, Muhammad Abuzar, Mohammed Ahmed Alomair, Abdulaziz Khalid Alsharidi. New solitons and bifurcation dynamics in the fractional Huxley model: Numerical and analytical approaches[J]. AIMS Mathematics, 2026, 11(1): 1998-2026. doi: 10.3934/math.2026083

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Abstract

This research explores the higher-order nonlinear fractional Huxley equation formulated with the $ \beta $, and $ M $-truncated fractional derivatives to account for memory and hereditary effects present in nonlinear diffusion and excitation wave dynamics. The fractional formulation expands on the standard Huxley model by incorporating nonlocal temporal and spatial correlations, providing a more realistic description of finite-amplitude wave propagation and spectral energy transfer in complex media. Two advanced analytical techniques are used to derive exact solutions: the enhanced modified extended tanh expansion method (EMETEM) and the improved $ F $-expansion technique. Compared to classic perturbation and variational techniques, these methods offer greater algebraic freedom and faster convergence, resulting in a diverse family of closed-form traveling-wave solutions expressed in trigonometric, hyperbolic, exponential, and rational forms. The analytical results are further validated, and the spatiotemporal evolution of the resulting wave structures is investigated using a finite-difference numerical scheme. The accuracy and robustness of the suggested framework are confirmed by the numerical findings, which show good agreement with the analytical results. Phase-plane and bifurcation analyses demonstrate transitions between periodic, quasi-periodic, and chaotic regimes by revealing both stable and unstable spiral formations. The findings show that fractional derivatives significantly improve the dynamical characteristics of the Huxley system by allowing for finer control of diffusion, dispersion, and localized energy concentration, advancing our understanding of nonlinear fractional wave behavior in excitable and dispersive media.

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