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We use both linear and nonlinear stability analysis methods to sort these equilibrium sites into groups based on their features, showing the differences between stable nodes, saddle points, and limit cycles. We demonstrate local and global stability through a comprehensive analysis of phase pictures, highlighting the impact of parameter variations on the system\u2019s dynamic behavior. Second, we analyze the chaotic behavior of the system under different initial conditions and perturbation terms using tools such as two- and three-dimensional phase portraits, time series, Poincar\u00e9 diagrams, and multi-stability. The incorporation of Poincar\u00e9 maps offers a powerful tool for analyzing the topology of chaotic orbits, enabling a deeper understanding of the system\u2019s underlying dynamical structure. Additionally, we originate the exact traveling wave for the tM-fRLW-Burger model employing the modified simple equation method. For specific values of the free parameters, we conduct a comprehensive analysis of the obtained solutions, accompanied by some unique diagrams. These solutions come in the form of trigonometric, exponential, and hyperbolic functions. For certain parameter values, we identify many wave phenomena, including kink waves, anti-kink waves, interactions between anti-kink and periodic waves, periodic lump and kink interactions, dark bell-shaped waves, and periodic waves, illustrating the model\u2019s intricate solution structure. Finally, we observe the modulation instability of the proposed model.<\/jats:p>","DOI":"10.1142\/s0218127425501822","type":"journal-article","created":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T12:37:59Z","timestamp":1760704679000},"source":"Crossref","is-referenced-by-count":10,"title":["Exploring Exact Traveling Wave Solutions of Fractional Nonlinear Model Emerging from Surface Water Waves in a Channel: Bifurcation, Chaos, and Stability"],"prefix":"10.1142","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0070-728X","authenticated-orcid":false,"given":"Md. Mamunur","family":"Roshid","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hamdard University Bangladesh, Hamdard Nagar, Gazaria, Munshiganj 1510, Bangladesh"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-6472-3365","authenticated-orcid":false,"given":"Md. 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