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This study focuses on revealing closed-form solutions for traveling waves in the nonlinear dynamical model of dust-acoustic waves in multispecies of dusty plasma. Plasma consists of cold dust grains, along with isothermal electrons and positive ions for balance. Fluid equations simplify through reductive perturbation, yielding the modified Korteweg–de Vries equation for analyzing weak dust-acoustic solitons. Applying this method to a specific scenario involving negatively charged dust grains and protons with all electrons bound to the dust grains shows support for rarefactive supersonic solitons, contrasting with typical compressive ion-acoustic solitons. Introducing streaming reveals a subtle soliton amplitude decrease. Using the Generalized exp(- S (φ)) expansion method effectively derives various soliton solutions, including kink, anti-kink, singular, hyperbolic functions, and diverse solitary waves. Furthermore, the equation's dynamic behaviors are explored through equilibrium point bifurcations. Employing a Galilean transformation, the model is transformed into a planar dynamical system, and qualitative inspection is performed. Phase portraits depict bifurcation outcomes. Our methodology is significant for tackling the novel problem and applying previously untested approaches in this context, resulting in the development of multiple exact and novel optical soliton solutions. This highlights the efficiency and broad application of our new strategy to addressing nonlinear challenges in astrophysics, laboratories, space, and technology. [ABSTRACT FROM AUTHOR]

This investigation aims to examine the resolution of the generalized nonlinear time fractional Harry Dym (ℕ ℍ) equation through a combined application of analytical and numerical methodologies. The primary objective is to scrutinize the equation's behavior and present proficient methodologies for its resolution. The Khater II analytical technique and numerical frameworks, specifically the Cubic-B-spline, Quantic-B-spline, and Septic-B-spline schemes, are proposed for this purpose. The outcomes of this inquiry yield substantial revelations regarding the characteristics of ℕ ℍ equation. Both the analytical approach and numerical schemes demonstrate their efficacy in yielding precise solutions. These findings carry noteworthy implications across various disciplines, encompassing physics, mathematics, and engineering. The investigated model appears in plasma physics research on soliton theory and nonlinear waves. Soliton waves, which keep their shape and velocity while propagating, are found in plasma physics and other domains. In plasma environments, the Harry Dym equation describes these solitons and their behavior. Solitons are essential for understanding plasma dynamics, including nonlinear waves and structures. They help comprehend plasma dynamics, wave interactions, and other nonlinear processes. The principal deductions drawn from this study underscore the effectiveness and viability of the proposed techniques in resolving the ℕ ℍ equation. This research introduces innovative contributions in terms of insights and methodologies pertinent to analogous nonlinear fractional equations. Its scope encompasses nonlinear dynamics, fractional calculus, and numerical analysis. [ABSTRACT FROM AUTHOR]

This paper discusses the newly discovered acoustic waves that arise from the equilibrium between the inertia of dust particles and plasma pressure. The propagation of these waves is demonstrated to be non-linear, with supersonic solitons exhibiting either a positive or negative electrostatic potential, and linear, with normal modes in a dusty plasma. Implementing the unified method and the generalized projective Ricatti equation technique, we generate a variety of optical wave structures for the governing framework. Using 3 D and 2 D dimensional illustrations, the fundamental behavior of the derived solutions is visually analyzed by choosing appropriate values for arbitrary parameters that satisfy the specified constraints. The employed methodologies are acknowledged to be a powerful and effective mechanism for creating solitary wave solutions for non-linear evolution models. [ABSTRACT FROM AUTHOR]

This study introduces innovative analytical solutions for the (2 + 1) -dimensional nonlinear Jaulent–Miodek () equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of waves. Within the realm of nonlinear fluid dynamics, the equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the (2 + 1) -dimensional equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field. [ABSTRACT FROM AUTHOR]

Understanding the intricate Korteweg–de Vries ( c K d V ) equation is paramount for comprehending various nonlinear wave phenomena, owing to its capacity to depict the propagation of nonlinear waves in media characterized by complex-valued dispersive and nonlinear coefficients. The practical ramifications of the c K d V equation are extensive, spanning disciplines such as optics, plasma physics, and other realms dealing with intricate media. This investigation employs the Khater II method and a generalized rational approach as analytical methodologies to generate novel exact solutions for the c K d V equation. The Khater II method is employed to transform the nonlinear partial differential equation into a nonlinear ordinary differential equation, thus facilitating a systematic resolution. Moreover, the generalized rational technique utilizes a rational ansatz function to derive diverse forms of solutions. Application of these methodologies leads to the discovery of fresh exact solutions, encompassing solitary and periodic wave solutions for the c K d V equation. These solutions are expressed in rational forms featuring arbitrary functions, thereby expanding the repertoire of known solutions for the model. The efficacy of the analytical methodologies employed becomes evident through the discovery of these novel exact solutions, thereby enriching our understanding of the physical interpretations and wave characteristics associated with the c K d V equation. The derived solutions augment the existing body of knowledge pertaining to the model. This research enhances our comprehension of the c K d V equation, thereby advancing nonlinear wave analysis with potential applications in physics, optics, plasma science, and allied engineering domains. Future investigations may explore the extension of these methodologies to address other nonlinear wave equations. [ABSTRACT FROM AUTHOR]