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This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. [ABSTRACT FROM AUTHOR]

This work addresses the significant issue of plasma waves interacting with non-linear dynamical systems in both perturbed and unperturbed states, as modeled by the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBK-BK) Equations. We investigate analytical solutions and the subsequent emergence of chaos within these systems. Initially, we apply advanced mathematical techniques, including the transform method and the G ′ G 2 method. These methods allow us to derive new precise solutions and enhance our understanding of the non-linear processes dominating plasma wave dynamics. Through a systematic analysis, we identify the conditions under which the system transitions from orderly patterns to chaotic behavior. This investigation provides valuable insights into the fundamental mechanisms of non-linear wave propagation in plasmas. Our results highlight the dynamic interplay between non-linearity and variation, leading to chaos, which may be useful in predicting and potentially controlling similar phenomena in practical applications. [ABSTRACT FROM AUTHOR]

In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: β , and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark–bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations. [ABSTRACT FROM AUTHOR]

In this investigation, the analytical behavior of two prominent nonlinear wave equations, namely the doubly dispersive equation (DDE) and the Ablowitz-Kaup-Newell-Segur equation (AKNSE), have been scrutinized. Bifurcation analysis, sensitivity as well as chaotic phenomena are also performed for the earlier-mentioned dynamical systems. These analyzes have profuse applications in the field of electrical circuits and control systems, phase transitions in materials, climate patterns, chemical reaction networks, forecasting market trends, signal processing, and quantum mechanics. Using an advanced mathematical technique, the exact solutions of the mentioned two-wave equations with singular bell-shaped soliton, bell-shaped soliton, anti-bell-shaped soliton, singular soliton, and singular periodic soliton have been studied. The technique utilized in the study is dependable for solving complex nonlinear problems in various natural science and engineering disciplines employed by many researchers. This study identified ten general solutions and ten particular solutions for the two mentioned equations that are novel and precise wave solutions. The solutions obtained in our study are significant not only for understanding the mentioned field but also may be used in revealing other interesting phenomena, such as the analysis of seismology, the study of compulsive collapse, and the study of the material effects and inner construction of solids. [ABSTRACT FROM AUTHOR]

In physical domains, Beta derivatives are necessary to comprehend wave propagation across various nonlinear models. In this research work, the modified Sardar sub-equation approach is employed to find the soliton solutions of (1+1)-dimensional time-fractional coupled nonlinear Schrödinger model with Beta fractional derivative. These models are fundamental in real-world applications such as control systems, processing of signals, and fiber optic networks. By using this strategy, we are able to obtain various unique optical solutions, including combo, dark, bright, periodic, singular, and rational wave solutions. In addition, We address the sensitivity analysis of the proposed model to investigate the truth that it is extremely sensitive. These studies are novel and have not been performed before in relation to the nonlinear dynamic features of these solutions. We show these behaviors in 2-D, contour 3-D structures across the associated physical characteristics. Our results demonstrate that the proposed approach offers useful results for producing solutions of nonlinear fractional models in application of mathematics and wave propagation in fiber optics. [ABSTRACT FROM AUTHOR]

We investigated the extended sixth-order Korteweg–de Vries (KdV6) equation which describes many nonlinear phenomena. The improved modified (IM) extended tanh-function method is used to conduct the study. Many exact solutions, like dark, combo singular-dark and singular soliton are achieved. Moreover, hyperbolic solutions, singular periodic solutions, rational solutions, exponential solutions and Jacobi elliptic function (JEF) solutions are derived. Graphical illustrations are presented to visually depict the dynamics of selected solutions. [ABSTRACT FROM AUTHOR]

This research explores some modernistic soliton solutions to the (3+1)-dimensional periodic potential the Gross–Pitaevskii equation with a truncated M-fractional derivative plays a significant role in Bose–Einstein condensation, which describes the dynamics of the condensate wave function. The obtained results include trigonometric, hyperbolic trigonometric and exponential function solutions. Three techniques named: the expafunction method, the Sardar sub-equation method, and the extended (G'/G)-expansion approach are employed to achieve a variety of new solutions for the governing model. More comprehensive information about the dynamical representation of some of the solutions is being presented by visualizing the 2D, 3D and contour plots. This work reveals a number of new types of traveling-wave solutions, such as the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular, and the singular soliton solutions. These novel solutions are not the same as those that were previously studied for this governing equation. The presented techniques demonstrate clarity, efficacy, and simplicity, revealing their relevance to diverse sets of dynamic and static nonlinear equations pertaining to evolutionary events in computational physics, in addition to other real-world applications and a wide range of study fields for addressing a variety of other nonlinear fractional models that hold significance in the fields of applied science and engineering. [ABSTRACT FROM AUTHOR]

Our objective is to explore the intricacies of a nonlinear nonlocal fifth-order scalar Sasa-Satsuma equation in reverse spacetime which is rooted in a nonlocal 5 × 5 matrix AKNS spectral problem. Starting with this spectral problem, we derive both local and nonlocal symmetry relations through rotations within a defined group. We then formulate a specific type of Riemann-Hilbert problem, facilitating the generation of soliton solutions. These solutions are generated by utilizing vectors that reside in the kernel of the matrix Jost solutions. Under the condition where reflection coefficients are null, the jump matrix reduces to the identity, leading to soliton solutions via the corresponding Riemann-Hilbert problem. The explicit formulas of these soliton solutions enable a comprehensive exploration of their dynamics. [ABSTRACT FROM AUTHOR]

The fractional regularized long wave equation and the fractional nonlinear shallow-water wave equation are the noteworthy models in the domains of fluid dynamics, ocean engineering, plasma physics, and microtubules in living cells. In this study, a reliable and efficient improved F-expansion technique, along with the fractional beta derivative, has been utilized to explore novel soliton solutions to the stated wave equations. Consequently, the study establishes a variety of reliable and novel soliton solutions involving trigonometric, hyperbolic, rational, and algebraic functions. By setting appropriate values for the parameters, we obtained peakons, anti-peakon, kink, bell, anti-bell, singular periodic, and flat kink solitons. The physical behavior of these solitons is demonstrated in detail through three-dimensional, two-dimensional, and contour representations. The impact of the fractional-order derivative on the wave profile is notable and is illustrated through two-dimensional graphs. It can be stated that the newly established solutions might be further useful for the aforementioned domains. [ABSTRACT FROM AUTHOR]

Abstract In this study, we examine the third-order fractional nonlinear Schrödinger equation (FNLSE) in $$(1+1)$$ -dimensional, by employing the analytical methodology of the new extended direct algebraic method (NEDAM) alongside optical soliton solutions. In order to better understand high-order nonlinear wave behaviors in such systems, the researched model captures the physical and mathematical properties of nonlinear dispersive waves, with applications in plasma physics and optics. With the aid of above mentioned approach, we rigorously assess the novel optical soliton solutions in the form of dark, bright–dark, dark–bright, periodic, singular, rational, mixed trigonometric and hyperbolic forms. Additionally, stability assessments using conserved quantities, such as Hamiltonian property, and consistency checks were used to validate the solutions. The dynamic structure of the governing model is further examined using chaos, bifurcation, and sensitivity analysis. With the appropriate parameter values, 2D, 3D, and contour plots can all be utilized to graphically show the data. This work advances our knowledge of nonlinear wave propagation in Bose–Einstein condensates, ultrafast fibre optics, and plasma physics, among other areas with higher-order chromatic effects.

Abstract In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: $$\beta$$ β , and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark–bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations.

In this investigation, the analytical behavior of two prominent nonlinear wave equations, namely the doubly dispersive equation (DDE) and the Ablowitz-Kaup-Newell-Segur equation (AKNSE), have been scrutinized. Bifurcation analysis, sensitivity as well as chaotic phenomena are also performed for the earlier-mentioned dynamical systems. These analyzes have profuse applications in the field of electrical circuits and control systems, phase transitions in materials, climate patterns, chemical reaction networks, forecasting market trends, signal processing, and quantum mechanics. Using an advanced mathematical technique, the exact solutions of the mentioned two-wave equations with singular bell-shaped soliton, bell-shaped soliton, anti-bell-shaped soliton, singular soliton, and singular periodic soliton have been studied. The technique utilized in the study is dependable for solving complex nonlinear problems in various natural science and engineering disciplines employed by many researchers. This study identified ten general solutions and ten particular solutions for the two mentioned equations that are novel and precise wave solutions. The solutions obtained in our study are significant not only for understanding the mentioned field but also may be used in revealing other interesting phenomena, such as the analysis of seismology, the study of compulsive collapse, and the study of the material effects and inner construction of solids.

We investigate the two-component BKP hierarchy equation for its Lie point symmetries. To obtain a complete classification of the group-invariant solution, we derive the one-dimensional optimal system of subalgebras of A3,3(D⨂sT2). By employing those subalgebras, we construct the invariant solutions which are represented through Weierstrass elliptic functions, Jacobi elliptic functions, and soliton solutions. Also, we analyse the existence of a bounded travelling-wave solution for the equation. Moreover, through the utilization of scale-invariant symmetry, we derive a self-similar solution. Additionally, by conducting singularity analysis, the solution can be expressed in the form of a right Painlevè series.

This research explores some modernistic soliton solutions to the (3+1)-dimensional periodic potential the Gross–Pitaevskii equation with a truncated M-fractional derivative plays a significant role in Bose–Einstein condensation, which describes the dynamics of the condensate wave function. The obtained results include trigonometric, hyperbolic trigonometric and exponential function solutions. Three techniques named: the $ \exp_a $ function method, the Sardar sub-equation method, and the extended $ (G'/G) $-expansion approach are employed to achieve a variety of new solutions for the governing model. More comprehensive information about the dynamical representation of some of the solutions is being presented by visualizing the 2D, 3D and contour plots. This work reveals a number of new types of traveling-wave solutions, such as the double periodic singular, the periodic singular, the dark singular, the dark kink singular, the periodic solitary singular, and the singular soliton solutions. These novel solutions are not the same as those that were previously studied for this governing equation. The presented techniques demonstrate clarity, efficacy, and simplicity, revealing their relevance to diverse sets of dynamic and static nonlinear equations pertaining to evolutionary events in computational physics, in addition to other real-world applications and a wide range of study fields for addressing a variety of other nonlinear fractional models that hold significance in the fields of applied science and engineering.

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water.

Abstract The study of the Kadomtsev–Petviashvili (KP) model is widely used for simulating several scientific phenomena, including the evolution of water wave surfaces, the processes of soliton diffusion, and the electromagnetic field of transmission. In current study, we explore some multiple soliton solutions of the (3+1)-dimensional generalized KP model via applying modified Sardar sub-equation approach (MSSEA). By extracting the novel soliton solutions, we can effectively obtain singular, dark, combo, periodic and plane wave solutions through a multiple physical regions. We also investigate the chaotic structure of governing model using the chaos theory. The behavior of the collected solutions is visually depicted to demonstrate the physical properties of the proposed model. The solutions obtained in this paper can expand the existing solutions of the (3+1)-dimensional KP model and enhance our understanding of the nonlinear dynamic behaviors. This approach allows for consistent and effective treatment of the computation process for nonlinear KP model.

In this study, the underlying traits of the new wave equation in extended (3+1) dimensions, utilized in the field of plasma physics and fluids to comprehend nonlinear wave scenarios in various physical systems, were explored. Furthermore, this investigation enhanced comprehension of the characteristics of nonlinear waves present in seas and oceans. The analytical solutions of models under consideration were retrieved using the sub-equation approach and Sardar sub-equation approach. A diverse range of solitons, including bright, dark, combined dark-bright, and periodic singular solitons, was made available through the proposed methods. These solutions were illustrated through visual depictions utilizing 2D, 3D, and density plots with carefully chosen parameters. Subsequently, an analysis of the dynamical nature of the model was undertaken, encompassing various aspects such as bifurcation, chaos, and sensitivity. Bifurcation analysis was conducted via phase portraits at critical points, revealing the system's transition dynamics. Introducing an external periodic force induced chaotic phenomena in the dynamical system, which were visualized through time plots, two-dimensional plots, three-dimensional plots, and the presentation of Lyapunov exponents. Furthermore, the sensitivity analysis of the investigated model was executed utilizing the Runge-Kutta method. The obtained findings indicated the efficacy of the presented approaches for analyzing phase portraits and solitons over a wider range of nonlinear systems.

Abstract Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective–diffusive Cahn–Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe–Cr–Mo) and (Fe–X–Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term ’new,’ the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings.

This study aims to employ the extended direct algebraic method (EDAM) to generate and evaluate soliton solutions to the nonlinear, space-time conformable Estevez Mansfield-Clarkson equation (CEMCE), which is utilized to simulate shallow water waves. The proposed method entails transforming nonlinear fractional partial differential equations (NFPDEs) into nonlinear ordinary differential equations (NODEs) under the assumption of a finite series solution by utilizing Riccati ordinary differential equations. Various mathematical structures/solutions for the current model are derived in the form of rational, exponential, trigonometric, and hyperbolic functions. The wide range of obtained solutions allows for a thorough analysis of their actual wave characteristics. The 3D and 2D graphs are used to illustrate that these behaviors consistently manifest as periodic, dark, and bright kink solitons. Notably, the produced soliton solutions offer new and critical insights into the intricate behaviors of the CEMCE by illuminating the basic mechanics of the wave's interaction and propagation. By analyzing these solutions, academics can better understand the model's behavior in various settings. These solutions shed light on complicated issues such as configuration dispersion in liquid drops and wave behavior in shallow water. [ABSTRACT FROM AUTHOR]

In this study, the underlying traits of the new wave equation in extended (3+1) dimensions, utilized in the field of plasma physics and fluids to comprehend nonlinear wave scenarios in various physical systems, were explored. Furthermore, this investigation enhanced comprehension of the characteristics of nonlinear waves present in seas and oceans. The analytical solutions of models under consideration were retrieved using the sub-equation approach and Sardar sub-equation approach. A diverse range of solitons, including bright, dark, combined dark-bright, and periodic singular solitons, was made available through the proposed methods. These solutions were illustrated through visual depictions utilizing 2D, 3D, and density plots with carefully chosen parameters. Subsequently, an analysis of the dynamical nature of the model was undertaken, encompassing various aspects such as bifurcation, chaos, and sensitivity. Bifurcation analysis was conducted via phase portraits at critical points, revealing the system's transition dynamics. Introducing an external periodic force induced chaotic phenomena in the dynamical system, which were visualized through time plots, twodimensional plots, three-dimensional plots, and the presentation of Lyapunov exponents. Furthermore, the sensitivity analysis of the investigated model was executed utilizing the Runge-Kutta method. The obtained findings indicated the efficacy of the presented approaches for analyzing phase portraits and solitons over a wider range of nonlinear systems. [ABSTRACT FROM AUTHOR]

In this research article, we demonstrate the generalized expansion method to investigate nonlinear integro-partial differential equations via an efficient mathematical method for generating abundant exact solutions for two types of applicable nonlinear models. Moreover, stability analysis and modulation instability are also studied for two types of nonlinear models in this present investigation. These analyses have several applications including analyzing control systems, engineering, biomedical engineering, neural networks, optical fiber communications, signal processing, nonlinear imaging techniques, oceanography, and astrophysical phenomena. To study nonlinear PDEs analytically, exact traveling wave solutions are in high demand. In this paper, the (1 + 1)-dimensional integro-differential Ito equation (IDIE), relevant in various branches of physics, statistical mechanics, condensed matter physics, quantum field theory, the dynamics of complex systems, etc., and also the (2 + 1)-dimensional integro-differential Sawda–Kotera equation (IDSKE), providing insights into the several physical fields, especially quantum gravity field theory, conformal field theory, neural networks, signal processing, control systems, etc., are investigated to obtain a variety of wave solutions in modern physics by using the mentioned method. Since abundant exact wave solutions give us vast information about the physical phenomena of the mentioned models, our analysis aims to determine various types of traveling wave solutions via a different integrable ordinary differential equation. Furthermore, the characteristics of the obtained new exact solutions have been illustrated by some figures. The method used here is candid, convenient, proficient, and overwhelming compared to other existing computational techniques in solving other current world physical problems. This article provides an exemplary practice of finding new types of analytical equations. [ABSTRACT FROM AUTHOR]

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie's modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots. [ABSTRACT FROM AUTHOR]

Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective–diffusive Cahn–Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe–Cr–Mo) and (Fe–X–Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term 'new,' the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings. [ABSTRACT FROM AUTHOR]

The time-fractional coupled Drinfel'd–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the Ω ′ Ω ,   1 Ω method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model's behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation's underlying dynamics and offer potential applications across various scientific fields. [ABSTRACT FROM AUTHOR]

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This manuscript is dedicated to the comprehensive exploration of solitary wave solutions for the fractional couple Drinfeld-Sokolov-Wilson equation, which is a versatile mathematical model that finds applications in various branches of physics, including nonlinear acoustics and fluid mechanics. The new extended direct algebraic method is employed as a powerful analytical tool throughout the study. A general algorithm that is essential for the analysis of the models stated is introduced in the manuscript. The travelling wave transformation is used to convert these models into ordinary differential equations, which makes the analysis easier to handle. The study yields a diverse set of solitary wave solutions in the form of dark, dark-bright, bright-dark, singular, periodic, mixed trigonometric, and rational forms. Also, by using the Hamiltonian property, validation of the solutions is conducted, which confirms the accuracy and stability of segregated solitary wave solutions. The discovered results are provided not only in numerical form but also with insightful physical interpretations, which contribute to a deeper comprehension of the complex dynamics these mathematical models depict. The utilization of the new extended direct algebraic method and the broad spectrum of obtained solutions contribute to the depth and significance of this research in the field of nonlinear wave equations.

This research article investigates the dissipative long wave system in (2+1) dimensions. To analyze this system, three separate integrating strategies were used: the single manifold approach, the unified method, and the generalized projective Ricatti equation method. Novel soliton solutions were effectively obtained by applying these strategies to the suggested model. The resulting solutions are various, including mixed soliton-like solutions, dark solitons, rational solutions, triangular periodic solutions, and combined Jacobi elliptic wave function solutions. Using the aforementioned approaches, an extensive variety of solution types has been discovered for the proposed model. The dynamic character of these closed-form solutions has been adequately demonstrated using both three dimensional and two dimensional graphical representations. This visualization helps to better understand the behavior of the acquired solutions.

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.

Abstract In this manuscript, a mathematical model known as the Heimburg model is investigated analytically to get the soliton solutions. Both biomembranes and nerves can be studied using this model. The cell membrane’s lipid bilayer is regarded by the model as a substance that experiences phase transitions. It implies that the membrane responds to electrical disruptions in a nonlinear way. The importance of ionic conductance in nerve impulse propagation is shown by Heimburg’s model. The dynamics of the electromechanical pulse in a nerve are analytically investigated using the Hirota Bilinear method. The various types of solitons are investigates, such as homoclinic breather waves, interaction via double exponents, lump waves, multi-wave, mixed type solutions, and periodic cross kink solutions. The electromechanical pulse’s ensuing three-dimensional and contour shapes offer crucial insight into how nerves function and may one day be used in medicine and the biological sciences. Our grasp of soliton dynamics is improved by this research, which also opens up new directions for biomedical investigation and medical developments. A few 3D and contour profiles have also been created for new solutions, and interaction behaviors have also been shown.

This research considers the Kraenkel-Manna-Merle system with an M-truncated derivative (K-M-M-S-M-T-D) that defines the magnetic field propagation (M-F-P) in ferromagnetic materials with zero conductivity (F-M-Z-C) and uses the Sardar sub-equation method (S-S-E-M). Our goal is to acquire soliton solutions (SSs) of K-M-M-S-M-T-D via the S-S-E-M. To our knowledge, no one has considered the SSs to the K-M-M-S-MTD with or without a damping effect (DE) via the S-S-E-M. The SSs are achieved as the M-shape, periodic wave shape, W-shape, kink, anti-parabolic, and singular kink solitons in terms of free parameters. We utilize Maple to expose pictures in three-dimensional (3-D), contour and two-dimensional (2-D) for different values of fractional order (FO) of the got SSs, and we discuss the effect of the FO of the K-M-M-S-MTD via the S-S-E-M, which has not been discussed in the previous literature. All wave phenomena are applied to optical fiber communication, signal transmission, porous mediums, magneto-acoustic waves in plasma, electromagnetism, fluid dynamics, chaotic systems, coastal engineering, and so on. The achieved SSs prove that the S-S-E-M is very simple and effective for nonlinear science and engineering for examining nonlinear fractional differential equations (N-L-F-D-Es).

Abstract In physical domains, Beta derivatives are necessary to comprehend wave propagation across various nonlinear models. In this research work, the modified Sardar sub-equation approach is employed to find the soliton solutions of (1+1)-dimensional time-fractional coupled nonlinear Schrödinger model with Beta fractional derivative. These models are fundamental in real-world applications such as control systems, processing of signals, and fiber optic networks. By using this strategy, we are able to obtain various unique optical solutions, including combo, dark, bright, periodic, singular, and rational wave solutions. In addition, We address the sensitivity analysis of the proposed model to investigate the truth that it is extremely sensitive. These studies are novel and have not been performed before in relation to the nonlinear dynamic features of these solutions. We show these behaviors in 2-D, contour 3-D structures across the associated physical characteristics. Our results demonstrate that the proposed approach offers useful results for producing solutions of nonlinear fractional models in application of mathematics and wave propagation in fiber optics.

The ZK–mZK–BBM equation plays a crucial role in actually depicting the gravity water waves with the long wave region. In this article, the bilinear forms of the (2 + 1)-dimensional ZK–mZK–BBM equation were derived using variable transformation. Then, the multiple soliton solutions of the ZK–mZK–BBM equation are obtained by bilinear forms and symbolic computation. Under complex conjugate transformations, quasi-soliton solutions and mixed solutions composed of one-soliton and one-quasi-soliton are derived from soliton solutions. These solutions are further studied graphically to observe the propagation characteristics of gravity water waves. The results enrich the research of gravity water wave in fluid mechanics.

The current work amalgamates the fascinating exploration of the Radhakrishnan–Kundu–Lakshmanan equation, which is a sophisticated mathematical framework in physics and finds applications in elucidating diverse physical phenomena. By leveraging the two recently developed computational approaches, namely the new extended direct algebraic method and the modified Sardar sub-equation method, we rigorously assess the novel soliton solutions including dark, bright-dark, dark-bright, periodic, singular, rational and mixed trigonometric forms. Furthermore, we also segregated W-shape, M-shape, bell shape, exponential, as well as hyperbolic soliton, which are not documented in the literature. We validate the stability and accuracy of extracted soliton wave solutions using the Hamiltonian property. Additionally, the Galilean transformation is applied and numerous standard types of results, including bifurcations, chaotic flows, and sensitivity analysis are presented. The obtained results are tested both numerically and with illuminating physical interpretations, which shows a better demonstration of the intricate dynamics of these models.

The examination of new (3 + 1)-dimensional wave equations is undertaken in this study. Initially, the identification of the Lie symmetries of the model is carried out through the utilization of the Lie symmetry approach. The commutator and adjoint table of the symmetries are presented. Subsequently, the model under discussion is transformed into an ordinary differential equation using these symmetries. The construction of several bright, kink, and dark solitons for the suggested equation is then achieved through the utilization of the new auxiliary equation method. Subsequently, an analysis of the dynamical nature of the model is conducted, encompassing various angles such as bifurcation, chaos, and sensitivity. Bifurcation occurs at critical points within a dynamical system, accompanied by the application of an outward force, which unveils the emergence of chaotic phenomena. Two-dimensional plots, time plots, multistability, and Lyapunov exponents are presented to illustrate these chaotic behaviors. Furthermore, the sensitivity of the investigated model is executed utilizing the Runge–Kutta method. This analysis confirms that the stability of the solution is minimally affected by small changes in initial conditions. The attained outcomes show the effectiveness of the presented methods in evaluating solitons of multiple nonlinear models. [ABSTRACT FROM AUTHOR]

This research considers the Kraenkel-Manna-Merle system with an M-truncated derivative (K-M-M-S-M-T-D) that defines the magnetic field propagation (M-F-P) in ferromagnetic materials with zero conductivity (F-M-Z-C) and uses the Sardar subequation method (S-S-E-M). Our goal is to acquire soliton solutions (SSs) of K-M-M-S-M-T-D via the S-S-E-M. To our knowledge, no one has considered the SSs to the K-M-M-S-MTD with or without a damping effect (DE) via the S-S-E-M. The SSs are achieved as the M-shape, periodic wave shape, W-shape, kink, anti-parabolic, and singular kink solitons in terms of free parameters. We utilize Maple to expose pictures in three-dimensional (3-D), contour and two-dimensional (2-D) for different values of fractional order (FO) of the got SSs, and we discuss the effect of the FO of the K-M-M-S-MTD via the S-S-E-M, which has not been discussed in the previous literature. All wave phenomena are applied to optical fiber communication, signal transmission, porous mediums, magneto-acoustic waves in plasma, electromagnetism, fluid dynamics, chaotic systems, coastal engineering, and so on. The achieved SSs prove that the S-S-E-M is very simple and effective for nonlinear science and engineering for examining nonlinear fractional differential equations (N-L-F-D-Es). [ABSTRACT FROM AUTHOR]

Our study analyzes the two models of the nonlinear Schrödinger equation (NLSE) with polynomial law nonlinearity by powerful and comprehensible techniques, such as the variational principle method and the amplitude ansatz method. We will derive the functional integral and the Lagrangian of these equations, which illustrate the system's dynamic. The solutions of these models will be extracted by selecting the trial ansatz functions based on the Jost linear functions, which are continuous at all intervals. We start with the Jost function that has been approximated by a piecewise linear function with a single nontrivial variational parameter in three cases from a region of a rectangular box, then use this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by piecewise linear ansatz function in two cases of the two-box potential, then approximate it by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function, and finally, will be approximated by the tanh function. Also, we utilize the amplitude ansatz method to extract the new solitary wave solutions of the proposed equations that contain bright soliton, dark soliton, bright-dark solitary wave solutions, rational dark-bright solutions, and periodic solitary wave solutions. Furthermore, conditions for the stability of the solutions will be submitted. These answers are crucial in applied science and engineering and will be introduced through various graphs such as 2D, 3D, and contour plots. [ABSTRACT FROM AUTHOR]

In this research, we present a new auxiliary equation approach, which uses two distinct fractional derivatives: β- and M-truncated fractional derivatives to explore the space-time fractional Peyrard-Bishop DNA dynamic model equation. This examines the nonlinear interplay between neighboring displacements and hydrogen bonds through mathematical modeling of DNA vibration dynamics. The solutions are tasked with examining the nonlinear interaction among neighboring displacements of the DNA strand. The generated solutions exhibit various wave patterns under varying fractional values and parametric conditions: w-shape, bright, combined periodic wave solutions, dark-bright, bell shaped, m-shaped, w-shaped with two bright solutions, and m-shape with two dark solutions. Graphical representations provide a complete analysis of these physical features. The results demonstrate the successful implementation of the proposed approach, which will be advantageous for locating analytical remedies to more nonlinear challenges.

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

This innovative study simulates the (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) equation, which has anomalous applications in the field of mathematical physics. The current research has two basic pillars. Firstly, numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, periodic, combo, W-shape, mixed trigonometric, exponential, and rational, based on the modified sardar sub-equation (MSSE) method and the improved F-expansion method. Secondly, the stability analysis of the selected model is manifested to study modulation instability (MI) gain. Furthermore, for the physical demonstration of the acquired solutions in 3D and 2D, contour and density plots are depicted. The discovered results have a distinctive feature that has not been developed before and indicate a good balance between the nonlinear physical components. Also, the resulting structure of the acquired results can enrich the nonlinear dynamical behaviours 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.

Our study analyzes the two models of the nonlinear Schrödinger equation (NLSE) with polynomial law nonlinearity by powerful and comprehensible techniques, such as the variational principle method and the amplitude ansatz method. We will derive the functional integral and the Lagrangian of these equations, which illustrate the system's dynamic. The solutions of these models will be extracted by selecting the trial ansatz functions based on the Jost linear functions, which are continuous at all intervals. We start with the Jost function that has been approximated by a piecewise linear function with a single nontrivial variational parameter in three cases from a region of a rectangular box, then use this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by piecewise linear ansatz function in two cases of the two-box potential, then approximate it by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function, and finally, will be approximated by the tanh function. Also, we utilize the amplitude ansatz method to extract the new solitary wave solutions of the proposed equations that contain bright soliton, dark soliton, bright-dark solitary wave solutions, rational dark-bright solutions, and periodic solitary wave solutions. Furthermore, conditions for the stability of the solutions will be submitted. These answers are crucial in applied science and engineering and will be introduced through various graphs such as 2D, 3D, and contour plots.

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations.

This work addresses the significant issue of plasma waves interacting with non-linear dynamical systems in both perturbed and unperturbed states, as modeled by the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBK-BK) Equations. We investigate analytical solutions and the subsequent emergence of chaos within these systems. Initially, we apply advanced mathematical techniques, including the transform method and the G′G2 method. These methods allow us to derive new precise solutions and enhance our understanding of the non-linear processes dominating plasma wave dynamics. Through a systematic analysis, we identify the conditions under which the system transitions from orderly patterns to chaotic behavior. This investigation provides valuable insights into the fundamental mechanisms of non-linear wave propagation in plasmas. Our results highlight the dynamic interplay between non-linearity and variation, leading to chaos, which may be useful in predicting and potentially controlling similar phenomena in practical applications.

The fractional regularized long wave equation and the fractional nonlinear shallow-water wave equation are the noteworthy models in the domains of fluid dynamics, ocean engineering, plasma physics, and microtubules in living cells. In this study, a reliable and efficient improved F-expansion technique, along with the fractional beta derivative, has been utilized to explore novel soliton solutions to the stated wave equations. Consequently, the study establishes a variety of reliable and novel soliton solutions involving trigonometric, hyperbolic, rational, and algebraic functions. By setting appropriate values for the parameters, we obtained peakons, anti-peakon, kink, bell, anti-bell, singular periodic, and flat kink solitons. The physical behavior of these solitons is demonstrated in detail through three-dimensional, two-dimensional, and contour representations. The impact of the fractional-order derivative on the wave profile is notable and is illustrated through two-dimensional graphs. It can be stated that the newly established solutions might be further useful for the aforementioned domains.

The thin-film ferroelectric materials equation is a fundamental mathematical model to explain the behavior of polarized electric fields in ferroelectric materials possessing unique properties such as piezoelectricity, pyroelectricity, electrostriction, hysteresis, spontaneous polarization, etc. These significant properties of ferroelectric materials are important in the progress of capacitors, non-volatile memories, electrical sensors, as well as optical and biomedical devices, among other applications. The aim of the present study is to explore applicable soliton solutions to the fractional-order model to the aforementioned equation by addressing the improved F-expansion method. Through analysis and rigorous identification, various radical solutions, including trigonometric, hyperbolic, and rational forms of the model, are established, and significant solitons are restored by setting appropriate parameter values in the analytical solutions obtained, which are demonstrated through 3D, 2D, and contour plots. We also explore the impact of fractional derivative as well as the parameters, like pressure, temperature, and others included in the model to determine the polarization effect in ferroelectric materials. This study uncovers several novel solutions along with their states in response to external or internal field parameters, which might be helpful in understanding the nature of ferroelectric materials.

In this study, the Lie point symmetries and optimal system have been established. We discuss the biofilm model’s soliton solutions. To examine a nonlinear dynamical biofilm system, which is simply a bistable Allen–Cahn equation with quartic potential, to determine solitary wave profiles using the considered equation, a new auxiliary equation technique is used. A suitable variable transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The unified approach is utilized to evaluate periodic solutions, solitary and soliton solutions as well as several newly discovered exact solitary wave solutions, it can be accomplished via Mathematica (https://www.wolfram.com/mathematica/online/). The new auxiliary equation approach is an efficient method for creating unique wave profiles based on a variety of soliton families. Also, the results are graphically visualized, by using the appropriate parametric settings. The outcomes are shown graphically in two dimensions, three dimensions, and contour form. The Hamiltonian conditions are satisfied by the planer dynamical framework of equations to ensure as the system that was generated is a conservative Hamiltonian dynamical system which includes all traveling wave structures. A sensitivity evaluation is used to explore the governing model’s thoroughly dynamical properties. It is demonstrated that the model becomes more dependent on the beginning conditions than the variables. The strategy could be used to look for exact solutions to various nonlinear evolution equations. We believe that this study will have important applications in different areas of science.

As a result of oversimplified anti-cubic non-linearity, we are interested in form optical solitons to the nonlinear Schrödinger equation (NLSE). The modified G′G2 creates new solitons that include periodic type solitons in chirped forms. These findings are extremely beneficial to the study of optics. The acquired outcomes are confirmed and explained using various graphs. Computational results show that the preceding analytical expansion scheme is efficient. We present some sufficient conditions for the presence of random quasi-periodic measures for various partial differential equations and sufficient conditions for the presence of quasi-periodic measurements and satisfaction of the NLSE equation. It is discovered that a saddle point bifurcation causes the transition from periodic to quasi-periodic actions in a sensitive area. Further investigation reveals favorable conditions for the multidimensional bifurcation of increasingly complex solutions. Our results provide insight into the complex interplay between periodic and quasi-periodic behaviors and further our understanding of the complex dynamics displayed by these PDE‘s. The outcomes that are being given broaden the field's current understanding and provide opportunities for more investigation into the intricate mathematical structures that underlie partial differential equation solutions.

The precise solutions of the (2+1)-dimensional modified KdV-KP equation are examined in this study. Three reliable methods, namely, the modified auxiliary equation scheme, extended rational sine–cosine methodology, and extended rational sinh–cosh technique, are utilized for the first time to extract multiple soliton solutions for the governing model. These types of solutions have applications in plasma, solid state, neuronal, biological production, and diffusion processes. Graphical simulations of the obtained solutions are included in the form of 3 dimensional surface plots to reveal their physical structure and dynamical features.

This study examines the thin-film ferroelectric materials equation via two most efficient and reliable integrating schemes. These schemes include the (G′G,1G)-expansion method and the generalized Kudryashov method. The thin-film ferroelectric materials equation plays a crucial part in physics, optics, and other branches of science. The traveling wave transformation is applied to reduce the governing model into ordinary differential equation. The suggested methods extract hyperbolic, trigonometric and rational function solutions. Constraint conditions for the validity of obtained solutions are also discussed. Moreover, for appropriate parametric values, acquired findings are displayed via contour, 2D and 3D graphics that show the physical relevance and dynamical behavior of the proposed equation.