Fractional Nonlinear Dynamics and Solitary Waves in the β- Fractional mKdV-gMEW Equation

Fractional Nonlinear Dynamics and Solitary Waves in the β- Fractional mKdV-gMEW Equation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Fractional Nonlinear Dynamics and Solitary Waves in the β- Fractional mKdV-gMEW Equation Adham Abhi, Sharmin Sultana, Pinakee Dey This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9121111/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 15 You are reading this latest preprint version Abstract This paper examines the space-time β-fractional modified Korteweg-de Vries generalized modified equal-width (mKdV-gMEW) equation, a nonlinear dispersive wave model governing ion-acoustic wave propagation in plasma, shallow-water dynamics, and nonlinear optics, extended to the β-fractional framework to encode memory effects and spatial nonlocality inaccessible to classical integer-order formulations. Exact traveling wave solutions comprising bright and dark solitons, kink and anti-kink waves, parabolic solitons, peakon solitons, and rational solutions are derived by simultaneously applying the modified extended tanh expansion method and the improved one-variable expansion method, with the effect of the β-fractional order on soliton amplitude, width, and energy localization examined in detail. A thorough dynamic analysis of the traveling-wave reduction shows periodic, quasi-periodic, and chaotic regimes using phase portraits, Poincaré sections, time series, and the largest Lyapunov exponent. This is backed up by bifurcation analysis, sensitivity analysis, multi-stability exploration, and Lyapunov stability, which shows that the solutions are stable even with small changes. In contrast to previous research limited to single-method integer-order models devoid of a cohesive dynamical framework, this study represents the inaugural dual-method β-fractional analysis of the mKdV-gMEW equation, providing a comprehensive analytical dynamical framework with significant ramifications for plasma physics, shallow-water hydrodynamics, and nonlinear optics. Physical sciences/Mathematics and computing Physical sciences/Optics and photonics Physical sciences/Physics β-fractional mKdV-gMEW equation modified extended tanh expansion method improved one variable expansion method soliton solutions bifurcation chaotic dynamics Lyapunov stability sensitivity analysis multi-stability Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 07 Apr, 2026 Reviews received at journal 06 Apr, 2026 Reviews received at journal 05 Apr, 2026 Reviews received at journal 03 Apr, 2026 Reviewers agreed at journal 03 Apr, 2026 Reviewers agreed at journal 01 Apr, 2026 Reviewers agreed at journal 01 Apr, 2026 Reviewers agreed at journal 01 Apr, 2026 Reviewers agreed at journal 31 Mar, 2026 Reviewers agreed at journal 31 Mar, 2026 Reviewers invited by journal 31 Mar, 2026 Editor assigned by journal 31 Mar, 2026 Editor invited by journal 31 Mar, 2026 Submission checks completed at journal 26 Mar, 2026 First submitted to journal 26 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Exact traveling wave solutions comprising bright and dark solitons, kink and anti-kink waves, parabolic solitons, peakon solitons, and rational solutions are derived by simultaneously applying the modified extended tanh expansion method and the improved one-variable expansion method, with the effect of the β-fractional order on soliton amplitude, width, and energy localization examined in detail. A thorough dynamic analysis of the traveling-wave reduction shows periodic, quasi-periodic, and chaotic regimes using phase portraits, Poincar\u0026eacute; sections, time series, and the largest Lyapunov exponent. This is backed up by bifurcation analysis, sensitivity analysis, multi-stability exploration, and Lyapunov stability, which shows that the solutions are stable even with small changes. 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