{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T01:57:03Z","timestamp":1780970223521,"version":"3.54.1"},"reference-count":64,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2024,9,19]],"date-time":"2024-09-19T00:00:00Z","timestamp":1726704000000},"content-version":"vor","delay-in-days":262,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2024,1]]},"abstract":"<jats:p>In this work, we study the modulation instability (MI) and closed\u2010form soliton solution of the modified Korteweg\u2013de Vries\u2013Zakharov\u2013Kuznetsov (mKdV\u2010ZK) equation with a truncated <jats:italic>M<\/jats:italic>\u2010fractional derivative. The mKdV\u2010ZK equation can be used to describe the behavior of ion\u2010acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved <jats:italic>F<\/jats:italic>\u2010expansion scheme and unified solver techniques for the mKdV\u2010ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three\u2010dimensional (3D) plots with density plots and two\u2010dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed\u2010form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved <jats:italic>F<\/jats:italic>\u2010expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell\u2010shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena.<\/jats:p>","DOI":"10.1155\/2024\/1754782","type":"journal-article","created":{"date-parts":[[2024,9,19]],"date-time":"2024-09-19T11:11:29Z","timestamp":1726744289000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Modulation Instability and Dynamical Analysis of New Abundant Closed\u2010Form Solutions of the Modified Korteweg\u2013de Vries\u2013Zakharov\u2013Kuznetsov Model With Truncated <i>M<\/i>\u2010Fractional Derivative"],"prefix":"10.1155","volume":"2024","author":[{"given":"Md. 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