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In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this nonlinear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation flows. To obtain the new exact solutions of the considered model, we have applied a novel analytical technique namely ( G ′ G ′ + G + A) -expansion method. Using the aforementioned method and computational software, we have obtained different kinds of periodic and singular solitary wave solutions of the generalized shallow water wave equation. The obtained solutions are exponential function and trigonometric function solutions. Using 2D and 3D plots of the wave solutions, the dynamic behaviors of the developed solutions are displayed. The retrieved solutions validated the effectiveness and robustness of the proposed technique. [ABSTRACT FROM AUTHOR]

This paper explores in detail the integrable Akbota equation, a Heisenberg ferromagnet-type problem that is essential to the study of surface and curve geometry. A variety of soliton families are represented by the generalized solitonic wave profiles that are produced using the improved modified Sardar sub-equation technique, which is renowned for its accuracy and dependability. There has never been a study that used this technique before the current one. As a result, the solitonic wave structures have kink, dark, brilliant, king-singular, dark-singular, dark-bright, exponential, trigonometric, and rational solitonic structures, among other characteristics. In order to check the energy conservation, the Hamiltonian function is created and energy level demonstrated. The sensitivity analysis is also presented at various initial conditions. The graphical representation is also depicted along with the appropriate parametric values. [ABSTRACT FROM AUTHOR]

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry. [ABSTRACT FROM AUTHOR]

The extended Gerdjikov–Ivanov equation in dense wavelength division multiplexed system for both kerr and parabolic law nonlinearities have been studied using the exp (- ϕ (ζ)) -expansion method. Completely new parameterized travel wave solutions are extracted. When special values are assigned to the parameters, Kink soliton solutions, periodic soliton solutions, and rational function solutions are derived. Figures have been added to indicate the behavior of solutions in some selected cases. [ABSTRACT FROM AUTHOR]

The inverse scattering transform is considered for the complex Sharma-Tasso-Olver equation with zero boundary condition by Riemann-Hilbert method. Under the reflection-less situation, we investigate the Riemann-Hilbert problem with one high-order pole and multiple high-order poles, respectively. By Laurent expansion of the Riemann-Hilbert problem and elimination of the integral factor involved in the solution, the explicit N-soliton solutions of the equation are derived. The interactions of several various solitons are displayed and their dynamics are analyzed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discover some new exact solutions for two kinds of nonlinear Schrödinger equation (NLSE), including the (2+1)-dimensional NLSE and the derivative NLSE, through the use of the variational principle method and amplitude ansatz method. We will find the solutions to these equations by selecting a trial function with a single nontrivial variational parameter, and it is continuous at all intervals in three cases from a region of a rectangular box, then using 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 quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Similarly, this trial function will be approximated by the tanh function. Additionally, we accept several new types of soliton solutions, including bright solitons, dark solitons, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. Also, conditions for the stability of the solutions will be proposed. These explanations are of great importance in the fields of applied science and engineering. The results will be shown through different types of graphs, including 2D, 3D, and contour plots. [ABSTRACT FROM AUTHOR]

Based on a modified Snyder–Mitchell model in nonlocal photorefractive media, spatiotemporal optical solitonary solutions are obtained and the formation and transmission characteristics of spatial solitons under the control of 2D optical lattice are studied by using self-similarity algorithm. The simulations show that the lattice order, modulation intensity and beam width significantly affect the shape and stability of spatial soliton in photonic lattice. The results have important guiding significance for the experimental study of spatial solitons. [ABSTRACT FROM AUTHOR]

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]

We investigate the standard binary Darboux transformation (SBDT) for an -component sdC integrable system. For this, we construct the Darboux matrix using specific eigenvector solutions associated to the Lax pair, not only in the direct space but also in the adjoint space, resulting in the binary Darboux matrix. By the iterative application of the SBDT, we derive quasi-Grammian soliton solutions of the -component sdC integrable system. We also examine the Darboux transformation (DT) applied to matrix solutions of the sdC integrable system, expressing solutions using quasideterminants. Additionally, we thoroughly discuss the DT applied to scalar solutions of the system, expressing solutions as ratios of determinants. Furthermore, we investigate the SBDT and its application to obtaining quasi-Grammian multikink and multisoliton solutions for the -component sdC integrable system. Additionally, we demonstrate that quasi-Grammian solutions can be simplified to elementary solutions by reducing spectral parameters. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is an important theoretical model for ultra-long and ultra-fast transportation. Therefore, the situation of its normal propagation and collision has significant implications for practical applications. For example, the nonlinear Schrödinger models can be utilized in the optical, plasma and fluid dynamics fields. However, current researches mainly focus on solutions of one dimensional or two dimensional Schrödinger equations with constant coefficients. Although these solutions possess good properties, they are often too unilateral. In this paper, our research mainly focuses on solutions of a (2+1)-dimensional coupled variable-coefficient Schrödinger equation. Through an explicit transformation, the solutions of this equation in the form of one-soliton, two-soliton, three-soliton, and breather-like bright-dark solitons can be acquired by utilizing the developed Hirota bilinear method. Furthermore, we will explicate that the collisions among two-soliton, two-breather-like bright-dark soliton are elastic with the help of the asymptotic analysis method. At last, we will exhibit their collisions via illustrations. [ABSTRACT FROM AUTHOR]

The generalized mK(m,n) model is used to portray the movement of nonlinear waves by applying four mathematical methods with assistance of Mathematica software. These methods are called enhanced simple equation (SE) method, Exp(−Ψ(ξ))-expansion method, (G′∕G)-expansion method and modified F-expansion method. For physical behavior, few solutions are plotted graphically in the form of two-dimensional (2D) and three-dimensional (3D). The techniques and soliton solutions that are developed provide computational tools for additional research in this area. With this work, we have gained a better knowledge of soliton motion and how it is used in dynamical systems and other related technologies. Hence, this work will help physicists envision some new ideas and hypotheses in nonlinear science. [ABSTRACT FROM AUTHOR]

In this paper, we use the Lie symmetry analysis to obtain the closed-form solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili Benjamin–Bona–Mahony (gKP-BBM) equation. We obtain the infinitesimal generators, commutator table of Lie algebra, adjoint representation of subalgebras, symmetry group, and similarity reduction for the gKP-BBM equation. Also, we evaluated the interactions of soliton solutions of the gKP-BBM equation. It is observed that multi-soliton complexes are self-localized in state wherein, several fundamental solitons including bright solitons and dark solitons are nonlinearly superimposed. A notable observation is the existence of topological defects in soliton interactions, where two solitons appear to be separated, with the adjoining structure being “out of phase” with each other. The kink, acting as a twist in the soliton’s value, overcomes topological defects, leading to a transition from one phase to another. This comprehensive analysis contributes to a deeper understanding of the dynamics and interactions of solitons in the (3+1)-dimensional gKP-BBM equation. [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]

This paper thoroughly investigates the integrable Akbota equation, a Heisenberg ferromagnet-type equation that plays a crucial role in exploring curve and surface geometry. The ϕ6-model expansion method, known for its proficiency and reliability, is applied to generate generalized solitonic wave profiles, spanning a diverse range of soliton families. Prior to this study, there is no existing study in which this technique is utilized that ensured the existence of solution via development of conditions. On the other hand, this method is provided with the Jacobi elliptic function-based solutions with the limiting cases. The analytical strategy presented a notable advantage by specifying a constraint for each solution, ensuring its existence. Consequently, the solitonic wave structures exhibit various attributes, including the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures. These characteristics emerge under previously unexplored existence conditions. The results are visually depicted through 2D, 3D, and contour plots, providing a clear illustration of the behavioral responses to pulse propagation and allowing for the inference of fitting values for system parameters. This visualization offers valuable insights into the characteristics and dynamics of soliton solutions derived from the integrable Akbota equation. [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]

The Riemann–Hilbert approach is applied to the mixed Chen–Lee–Liu equation. The corresponding Jost solutions are found, the analytic, asymptotic and symmetric properties of Jost solutions are studied, and a modified Riemann–Hilbert problem is constructed that satisfies the normalization condition. The formulas for multiple solitons related to the simple poles of the Riemann–Hilbert problem are given in determinant form. According to the Cauchy–Binet formula, the formulas for multiple solitons are given explicitly. Based on these explicit formulas, the first- and second-order solitons are obtained, and the multiple-soliton collisions are verified to be elastic. [ABSTRACT FROM AUTHOR]

In this paper, we employ some ansatz transformations to investigate various nonlinear waves for a well-known model, the generalized reaction Duffing model, including lump soliton, rogue waves, breather waves, Ma-breather, and Kuznetsov-Ma-breather. The standard Duffing equation is expanded upon in the generalized Duffing model, which adds more terms to take into consideration for more complex behaviors. The generalized reaction Duffing model is useful in many domains, such as electrical engineering, biomechanics, climate research, seismic research, chaos theory, and many more, due to its rich behavior and nonlinear dynamic. Lump soliton is a robust, confined, self-reinforcing wave solution to non linear partial differential equations. Breather waves are periodic, specific solutions in nonlinear wave systems that preserve their amplitude and structure. Rogue waves, which pose a hazard to marine safety, are unexpectedly strong and sharp ocean surface waves that diverge greatly from the surrounding wave pattern. They frequently appear in solitary and apparently random situations. The solutions are graphically displayed using contour, 3D, and 2D graphs. [ABSTRACT FROM AUTHOR]

In this study, we present an amalgamation of solitary wave solutions for the fractional coupled Drinfeld-Sokolov-Wilson (FCDSW) equations, a versatile mathematical model with significant applications in fluid dynamics, plasma physics, and nonlinear dynamics. By leveraging the two recently developed computational approaches, namely the modified Sardar sub-equation (MSSE) method and the improved F - expansion method, we manifested the novel soliton solutions in the form of dark, dark-bright, bright-dark, singular, periodic, exponential, and rational forms. Furthermore, we also extracted W-shape, V-shape, mixed trigonometric, and hyperbolic soliton wave solutions, which are not reported in previously. Also, the modulation instability (MI) of the proposed model is also examined. To further understand the dynamics of the FCDSW equations, we conducted a detailed bifurcation analysis to investigate the bifurcation events exhibited by these equations. A sensitivity analysis was also performed to assess the model’s robustness against variations in initial conditions and parameters, offering valuable insights into the system’s susceptibility to perturbations. We proved the effectiveness of our suggested techniques in the analysis of the FCDSW equations using analytical tools and numerical simulations. Our results provide new insights into the behavior and solutions of the FCDSW equations and enhance the mathematical tools for investigating nonlinear partial differential equations (NLPDEs). These findings have potential applications in fields like fluid flow modeling, wave propagation in plasmas, and the study of nonlinear optical phenomena in physics and applied mathematics. [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]

The presented study deals with the investigation of nonlinear Bogoyavlenskii equations with conformable time-derivative which has great importance in plasma physics and non-inspectoral scattering problems. Travelling wave solutions of this nonlinear conformable model are constructed by utilizing two powerful analytical approaches, namely, the modified auxiliary equation method and the Sardar sub-equation method. Many novel soliton solutions are extracted using these methods. Furthermore, 3D surface graphs, contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica. The constructed solutions will help to understand the dynamical framework of nonlinear Bogoyavlenskii equations in the related physical phenomena. [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.

In this study, the modified exponential approach is used to investigate the fractional order longitudinal wave equation in a magneto-elastic circular rod, which represents the nonlinear interplay between dispersion and longitudinal wave velocity depending on the rod's material and geometry. The time-fractional order is used in the standpoint of conformable derivative. Comprehensive and descriptive waveform solutions, including kink-shaped, bell-shaped, and bright bell-shaped solitons, are obtained. The produced soliton solutions are advanced, unique, developed, and crucial. To demonstrate the physical appearance of the obtained solutions, 2D, 3D, and contour graphs of some of the obtained solutions are plotted. The extracted solutions demonstrate the physical significance of the model and express the pressure and electrostatic potential for the longitudinal wave equation and can be used to elucidate more complicated nonlinear models of classical and fractional order. The obtained results are helpful in finding the amplitude of tsunami waves which is of great interest to researchers in the field of ocean engineering. If a tsunami extends across a large area, it becomes a manifestation of solitons. The comparison of the obtained solutions is also given in the form of remarks 1 and 2 which shows the novelty of our work. [ABSTRACT FROM AUTHOR]

In this study, we particularly address the generalized (3+1)-dimensional Kortewegde Vries (KdV) problem as one variation of the KdV equation. This equation can be utilized to simulate a wide range of physical events in a variety of domains, such as nonlinear optics, fluid dynamics, plasma physics, and other fields where coupled wave dynamics are significant. We first construct a Hirota bilinear form for the generalized KdV equation, and then we derive two different Bäcklund transformations (BT). The first Bäcklund transformation includes eleven arbitrary parameters, while the second form contains eight parameters. Rational and exponential traveling wave solutions with random wave numbers are found based on the suggested bilinear Bäcklund transformation. These solutions of the rational and exponential functions lead to the formation of dark and bright solitons. Moreover, we utilize the bilinear form of the equation to fully comprehend the behavior of lump-kink, breather, rogue, two-wave, three-wave, and multi-wave solutions. In-depth numerical simulations using 3-D profiles and contour plots are carried out while carefully taking into account relevant parameter values, offering more insights into the unique characteristics of the solutions that are obtained. Our results demonstrate the effectiveness and efficiency of the method used to obtain analytical solutions for nonlinear partial differential equations. [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 $ \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.

In this paper, the Riemann‐Hilbert (RH) approach for the integrable discrete Hirota equation with bounded boundary conditions is presented. In the direct scattering problem, we study the analyticity, asymptotics, symmetries of the eigenfunctions, and scattering coefficients and analyze the distribution of discrete eigenvalues. In the inverse scattering problem, the RH problem is constructed and solved as well as the reconstruction formula of potential is derived based on asymptotics. Finally, combining the time evolution, we solve the first‐ and second‐order dark soliton solutions on the nonzero background under the reflectionless condition. [ABSTRACT FROM AUTHOR]

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]

A generalized variable-coefficient Gardner equation with an external-force term is investigated which can model the propagation and interaction of some nonlinear waves in a fluid or plasma. Via the Hirota bilinear method, we have obtained some bilinear forms using a dependent variable transformation under certain constraints. The bilinear Bäcklund transformations, Lax pair and Lax-type Bäcklund transformations have also been constructed via the bilinear forms. By virtue of the extended variable-coefficient homogeneous balance method and truncated Painlevé expansion, we have obtained the same Bäcklund transformations and three different kinds of the analytical solutions. Additionally, profiles of some types of the obtained solutions are illustrated graphically. Graphic analysis on those waves shows that: (i) For the soliton-like and rational waves, the characteristic lines and velocities are related to the dispersive, dissipative, perturbed and external-force coefficients; the backgrounds are related to the perturbed and external-force coefficients; (ii) For the periodic waves, the velocities are related to the dispersive, dissipative, perturbed and external-force coefficients; the amplitudes are related to the dispersive, perturbed and external-force coefficients, while the backgrounds are related to the perturbed and external-force coefficients. [ABSTRACT FROM AUTHOR]

The dualism of the theories of soliton solutions and solutions to functional differential equations of pointwise type is discussed. We describe the foundations underlying the formalism of this dualism, the central element of which is the concept of a soliton bouquet, as well as a dual pair "function–operator." Within the framework of this approach, it is possible to describe the entire space of soliton solutions with a given characteristic and their asymptotics in both space and time. As an example, the model of traffic flow on the Manhattan lattice is used to describe the whole family of bounded soliton solutions. [ABSTRACT FROM AUTHOR]

Variable-coefficient equations can be used to describe certain phenomena when the inhomogeneous media and nonuniform boundaries are taken into consideration. It is meaningful to solve the exact solution of variable-coefficient equations. In this paper, a (2+1)-dimensional variable-coefficient Boussinesq equation is investigated. The integrability is firstly examined by the Painlevé analysis method. Secondly, the Bäcklund transformations, one- and two-soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq equation are studied by virtue of the Hirota bilinear method. Propagation characteristics and interaction behaviors of the solitons are discussed: (i) soliton shapes and interaction behaviors are affected by the variable coefficients, and (ii) the two-soliton interaction is elastic, and the shape and velocity does not change after the collision, and only the shift changes. [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]

In this paper, new exact solutions of the perturbed Chen-Lee-Liu equation (CLL), which represents the propagation of an optical pulse in plasma and optical fiber, are obtained. Using a traveling wave ansatz, the nonlinear CLL is transformed into two nonlinear ordinary differential equations. Consequently, these equations are reduced to a first-order ordinary differential equation that has a well-known exact solution. The obtained exact solutions are in the form of kink wave, peakon, dark, and bright soliton solutions. [ABSTRACT FROM AUTHOR]

This investigation discusses the modified M-truncated form of the perturbed Chen–Lee–Liu (pCLL) dynamical equation. The pCLL equation is a generalization of the original CLL equation, which describes the propagation of optical solitons in optical fibers. The pCLL equation includes additional terms that account for various influences such as chromatic dispersion, nonlinear dispersion, inter-modal dispersion, and self-steepening. A new version of the generalized exponential rational function method is utilized to obtain multifarious types of soliton solutions. Moreover, the planar dynamical system of the concerned equation is created using a Hamiltonian transformation, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are added to the analytical results to better understand the dynamic behavior of these solutions. The study's findings could be extremely useful in solving additional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Highly important is a three-dimensional nonlinear partial differential equation because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that describes how the system changes in time. Thus, this article comprehensively reveals the investigation carried out on a (3+1)-dimensional generalized fifth-order Zakharov–Kuznetsov equation with power-law as well as dual power-law nonlinearities analytically, where the fifth-order term involved is regarded as a dispersion perturbation term. We utilize the well-celebrated Noether's theorem to comprehensively construct conserved currents of the underlying equation. A detailed Lie group analysis of the understudied model consisting of power-law nonlinearities is further performed. This involves performing reductions of the underlying models using their Lie point symmetries. In consequence, various invariants are found. In addition, the equation reduces to diverse ordinary differential equations using its point symmetries and consequently diverse solutions of interest were achieved. Moreover, we derive some solitary wave solutions by invoking the newly introduced logistic function technique for some particular cases of the equation under consideration. In consequence, we achieve some exponential function solutions. In addition, the physical meaning of the results is put on the front burner by revealing the wave dynamics of these solutions via graphical depictions. Finally, the significance of the robust and detailed findings in the work are further corroborated with various real-world applications. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional hyperbolic nonlinear Gardner–Kadomtsov–Petviashvili (GKP) equation is a mathematical model that describes the propagation of nonlinear waves in a four-dimensional space-time. The GKP equation is used as a mathematical tool for studying the dynamics of nonlinear waves in a variety of physical systems including fluid dynamics, nonlinear optics, plasma physics. The GKP equation is a nonlinear partial differential equation (PDE) with mixed spatial and temporal derivatives. It exhibits various interesting phenomena, such as soliton solutions, wave interactions and wave turbulence depending on the values of the conditions. The improved F-expansion technique and generalized Kudryashov technique are successfully employed to evaluate the soliton solutions of governing model. The GKP equation supports various types of soliton solutions, such as bright solitons and dark solitons, depending on the specific parameters. The study of solitons in the GKP equation provides insights into the stability, propagation, and interaction of nonlinear waves in higher-dimensional spaces. The nonlinear terms in the GKP equation are responsible for wave–wave interactions. When multiple waves co-exist in a medium, they can interact with each other, leading to complex phenomena. The nature of wave interactions in the GKP equation depends on the specific values of the coefficients. The GKP equation has many applications in various areas of physics and mathematics. It has been used to model nonlinear waves in several fields and Bose–Einstein condensates. [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 enduring stability exhibited by solitons is strikingly demonstrated as a soliton pulse traverses an ideal lossless optical fibre, thereby highlighting a compelling attribute for their integration into optical communication systems. In this study, we employ the improved Bernoulli sub-equation function method to systematically derive stable and functionally robust soliton solutions for the Riemann wave equation. The stability of the soliton solutions is demonstrated through their composition involving hyperbolic and exponential functions, among others. The physical significance of these solutions is meticulously analyzed by presenting 2D and 3D graphs that illustrate the behaviour of the solutions for specific parameter values. Additionally, a comprehensive investigation into the influence of the nonlinear parameter on the wave velocity and solution curve is conducted. The study further explores local stability through bifurcation and phase plane analysis. Our findings affirm the reliability of the improved Bernoulli sub-equation function method and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering. [ABSTRACT FROM AUTHOR]