article · AIMS Mathematics
This research investigates the complex dynamics and solitary wave solutions of the two-plus-one-dimensional complex modified Korteweg-de Vries system, a mathematical framework relevant to fluid dynamics, plasma physics, optics, and nonlinear dynamics. Two analytical approaches, the auxiliary equation method and the Hirota bilinear method, were employed to construct a wide variety of novel solitons. These include dark, bright, singular, periodic, W-shaped, exponential, hyperbolic, and rational wave formats. The study also derives previously undocumented lump solutions, such as homoclinic breathers, periodic cross rational waves, and M-shaped interactions with kink waves. In addition, Galilean transformation is applied to examine the dynamic framework, revealing bifurcations, chaotic flows, and related dynamic features confirmed through multi-dimensional visualisations.
Understanding nonlinear wave phenomena is essential for modelling complex behaviours in natural and engineered environments. By expanding the mathematical toolkit for the complex modified Korteweg-de Vries system, this work improves theoretical comprehension of wave interactions, turbulence, and chaotic patterns found across fluid mechanics, optical devices, and plasma systems.
This work represents early-stage fundamental research in mathematical physics and nonlinear wave theory. It provides theoretical models that could eventually assist researchers and engineers working in optical communications, plasma device design, or fluid mechanics simulation tools. However, because the study is entirely theoretical and analytical, direct real-world commercialisation or technology transfer pathways remain very distant.
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<abstract><p>This research focuses on the fascinating exploration of the $ (2+1) $-dimensional complex modified Korteweg-de Vries (CmKdV) system, exhibiting its complex dynamics and solitary wave solutions. This system is a versatile mathematical model that finds applications in various branches of physics, including fluid dynamics, plasma physics, optics, and nonlinear dynamics. Two newly developed methodologies, namely the auxiliary equation (AE) method and the Hirota bilinear (HB) method, are implemented for the construction of novel solitons in various formats. Numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, singular, periodic, combo, $ W $-shape, mixed trigonometric, exponential, hyperbolic, and rational, based on the proposed methods. Furthermore, we also find some lump solutions, including the periodic cross rational wave, the homoclinic breather (HB) wave solution, the periodic wave solution, the $ M $-shaped rational wave solution, the $ M $-shaped interaction with one kink wave, and the multiwave solution, which are not documented in the literature. In addition, we employ the Galilean transformation to derive the dynamic framework for the presented equation. Our inquiry includes a wide range of topics, including bifurcations, chaotic flows, and other intriguing dynamic properties. Also, for the physical demonstration of the acquired solutions, 3D, 2D, and contour plots are provided. The resulting structure of the acquired results can enrich the nonlinear dynamical behaviors of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.</p></abstract>
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DOI: 10.3934/math.2024780
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