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In this study, we investigate the -D Jaulent-Miodek (JM) equation, which is significant due to its energy-based Schrödinger potential and applications in fields such as optics, soliton theory, signal processing, geophysics, fluid dynamics, and plasma physics. Given its broad utility, a rigorous mathematical analysis of the JM equation is essential. The primary objective of this work is to derive exact soliton solutions using the Modified Sub-Equation (MSE) and Modified Auxiliary Equation (MAE) techniques. These solutions are computed using Maple 18, and encompass a variety of wave structures, including bright solitons, kink solitons, periodic waves, and singular solitons. The potential applications of these solutions span diverse domains, such as nonlinear dynamics, fiber optics, ocean engineering, software engineering, electrical engineering, and other areas of physical science. Through numerical simulations, we visualize the physical characteristics of the obtained soliton solutions using three distinct graphical formats: 3D surface plots, 2D contour plots, and line plots, based on the selection of specific parameter values. Our results demonstrate that the MSE and MAE techniques are not only efficient but also straightforward in extracting soliton solutions for the JM equation, outperforming other existing methods. Furthermore, the solutions presented in this study are novel, representing contributions that have not been previously reported in the literature.

In this article, two most prominent models namely the Camassa–Holm nonlinear Schrödinger equation and the ( 3 + 1 ) -dimensional Boussinesq equation have been investigated for extracting new and novel solitary wave solutions. The Camassa–Holm nonlinear Schrödinger (CHNLS) equation is a mathematical model that combines the features of two important equations in physics: the Camassa–Holm equation and the nonlinear Schrödinger equation. The Camassa–Holm equation describes the propagation of shallow water waves over a flat bottom, and has soliton solutions with sharp peaks called peakons. The nonlinear Schrödinger equation describes the evolution of wave packets in nonlinear and dispersive media, and has soliton solutions with smooth profiles. The CHNLS equation has been addressed analytically to determine the exact solutions by implementing extended  G ′ / G 2 -expansion approach. The Boussinesq equation is a mathematical model that is capable of simulating weakly nonlinear and long-wave approximations is also solved analytically by applying extended  G ′ / G 2 -expansion approach. This model finds its applications in various fields such as coastal engineering, and numerical models for water wave simulation in harbors and shallow seas. The two considered equations have significant applications in mathematical physics and their exact wave solutions are essential to understand their dynamical behavior. Dark solitons, bright solitons, and periodic waves are observed from the obtained results. It is reasonable to say that our approach provides an impressive mathematical mechanism for producing traveling wave solutions for numerous mathematical and physical models. Also our proposed method improves the accuracy of the solution. The technique employed here is basic and concise. Graphs are presented to depict the behavior of some of the retrieved dynamical wave structures. All computations are done using the mathematical software Maple.

In this article, we investigate the STF modified Benjamin-Bona-Mahony (STF-mBBM) equation, which is important in understanding wave phenomena across various technical scenarios such as ocean waves, acoustic gravity waves and cold plasma physics. We describe the fundamental properties of fractional calculus and its application to the STF-mBBM equation. Utilizing beta derivatives, we enhance our understanding of the intricate wave dynamics involved. Through the modified -expansion method (M -EM), we derive periodic, and kink singular soliton solutions and represent them graphically. We present the influence of the fractional parameter on traveling wave with 2 D , 3 D , surface and contour plots, providing a thorough understanding of the physical phenomena associated with the fractional model. In addition, we utilize the Hamiltonian property to analyze the chaotic dynamics of the solutions we’ve acquired. We perform two types of analysis using the Galilean transformation: a local sensitivity examination is conducted to see how the model responds to changes in individual input factors, and a global sensitivity examination is conducted to comprehend the correlation between the variability in the results and the variability in each input variable throughout its whole range of significance. This comprehensive approach allows us to determine traveling wave solutions effectively, offering new insights into the non-linear dynamical behavior of the system. The findings from this study are unique and significant for further exploration of the equation, offering valuable insights for future researchers.

Graphene has emerged as a highly attractive nanomaterial, characterized by its distinctive thermodynamic, electrical, and mechanical features. These attributes render it ideal for a wide array of purposes across materials research, storing energy, computing, filtering water, and medication. In this study, we investigate the nonlinear thermophoretic wrinkle motion equation describing wave propagation and thermal-driven deformation in graphene sheets. Governing model is -dimensional graphene sheets (GS) equation, focusing on the component heat transfer during thermodynamic motion. The study employs modified generalized exponential differential function method and the improved Cham method. These techniques are employed to analyze the nonlinear behavior of wave propagation in GS, leading to exact wave solutions such as bright, bell, periodic, and twofold waves. The creative application of these analytical techniques provides efficient structures for tackling complex nonlinear theories in mathematical physics, thereby enhancing solution strategies for these equations. This study makes a notable contribution to computational mathematics, material research, and nanotechnology by providing exact solutions and deepening our comprehension of graphene non linear properties. The findings present significant implications, suggesting potential applications in the design of novel materials with customized properties to facilitate technological progress, thereby expanding the frontiers of nanotechnology and materials research.

This research explores the (2+1)-D nonlinear electrical transmission line equation (NLETLE), highlighting its unique localized wave solutions and the interactions that arise from them. Through the application of a novel multivariate generalized exponential differential function technique and generalized logistic equation approach, we have successfully generated a diverse array of new structures, particularly characterized by bright soliton, bright-singular soliton, kink soliton, and periodic waveforms. These solutions play a crucial role in demonstrating the complex structure and varied dynamics that are characteristic of nonlinear systems in higher dimensions. To achieve a comprehensive understanding, we depict these solutions using 3D surface density plots and line graphs. Additionally, we analyze the dynamic behavior of the system through bifurcation analysis, which is graphically represented by phase portraits. Subsequently, we incorporate periodic functions into the dynamical system to investigate the nonlinear properties of the dynamical system, in order to uncover its chaotic behavior, utilizing concepts derived from the theory of chaos. The observation and confirmation of chaotic behavior are achieved by employing a range of chaos detection tools. In addition, we conduct a sensitivity analysis to determine how minor modifications in the system affect its overall behavior, which in turn provides greater insight into its robustness and ability to respond to perturbations. By varying the initial conditions, we analyze multistability, which highlights the system’s ability to display multiple stable states influenced by choosing suitable parametric values. The results acquired from this research are new and significant for the continued exploration of the (2+1)-D NLETLE, offering direction for future scholars.

In quantales, algebraic structures that use fuzzy relations and cores of successor class, the notion of upper and lower approximation bi-ideals is introduced. In quantales with upper and lower approximations, we present illustrations of bi-ideals, interior ideals, quasi-ideals, (m, n)-ideals, and (m, n)-quasi-ideal and explore the features of these bi-ideals by expanding the concept of left and right ideals. Furthermore, we develop semi-prime bi-ideals and prime bi-ideals with upper and lower approximations, and we investigate their properties in quotient quantales. This work aims to improve our understanding of these algebraic structures to minimize uncertainty in a variety of domains.

In this paper, the exact explicit traveling wave solutions to the simplified modified Camassa–Holm (SMCH) equation. The SMCH equation is a significant nonlinear evolution equation because it can be used to describe certain physical processes in oceanography and fluid dynamics. The SMCH equation has important applications in mathematics, physics, and engineering. Using the modified auxiliary equation method and the extended G ′ G 2 -expansion method, we achieve precise solutions with traveling wave behavior. The proposed methods are applied for the first time in this work to examine the considered mathematical model. The solutions are extracted in terms of trigonometric, hyperbolic, and rational functions. The obtained results not only confirm the previously reported solutions of SMCH equation but also offer some new results. To clearly illustrate the physical implications of the examined equation, we provide graphical representations using 3 D , 2 D and contour plots for certain values of the parameters. The illustrations help to clarify the diverse characteristics and behaviors, associated with the solutions, providing useful insights for researchers as well as practitioners across a variety of scientific and technical disciplines.

This manuscript delves into the dynamic behavior of the ( 3 + 1 ) -dimensional nonlinear extended quantum Zakharov-Kuznetsov (NLEQZK) equation, exploring soliton solutions, chaotic phenomena, bifurcation, sensitivity, and stability through the lens of planar dynamical system theory. Firstly, the generalized Arnous method is used for securing some soliton solutions. Solutions obtained using the generalized Arnous method include hyperbolic, rational, and logarithmic forms. To visualize these graphically, we select appropriate parameter values and examine the graphical behavior of the obtained solutions. The visual characteristics of the solutions that were created are also evaluated in this procedure through the utilization of 3D surface graphs and line graphs representing various parameter values. The governing equation is derived using the Galilean transformation to facilitate bifurcation analysis. Chaotic behavior in the NLEQZK equation is investigated by introducing a perturbed term in the dynamical system and presenting various analyses, including Poincaré maps, time series, 2-dimensional (2 D ) phase portraits, and 3-dimensional (3 D ) phase portraits. Additionally, the Runge–Kutta method is employed for sensitivity analysis, confirming that slight adjustments to the initial conditions and check whether the system is sensitive or not. The outcomes of this study contribute valuable insights to the understanding of nonlinear dynamical systems and soliton theory.