This study unleashes the soliton solutions of the (3 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE), which is pertinent in areas such as optics, quantum mechanics, plasma physics, electromagnetic wave propagation, and optics. The domain of chiral soliton propagation in nuclear physics is very appealing because of its copious uses in communications and ultra-fast signal routing systems. The goals of this investigation are to achieve three principal objectives. Firstly, we intend to construct the soliton solutions of the (3 + 1)-dimensional CNLSE comprising hyperbolic, trigonometric, and exponential forms in their single and combined representations. These outcomes are extracted with the aid of two reliable schemes: the modified rational sine-cosine/sinh-cosh method and generalized exponential rational function method (GERFM). Secondly, we aim to conduct the modulation instability analysis of the aforementioned model with the assistance of the linear stability theory. Our third and final objective is to study the dynamics of the system through chaos, bifurcation, and sensitivity analyses, along with the determination of the associated Hamiltonian function. For the sake of understanding the dynamics of complicated physical phenomena and processes of the studied model, we sketch different graphs under the selection of appropriate parametric values. The current study makes an important contribution to our knowledge about wave dynamics in nonlinear optics and plasma physics, and the unique results obtained in the present study will be of utmost significance for future studies of nonlinear problems. The innovation of this paper lies in the fact that the presented model has not been previously investigated in this way.

In this study, we study the travelling wave solutions of the Modified Benjamin-Bona-Mahony Equation (MBBME) using the (G' /G,1/G)-expansion method (EM). The expansion method, along with an appropriate transformation, allows us to derive a set of explicit solutions for the MBBME in terms of rational, hyperbolic, and trigonometric functions. The travelling wave profiles that these solutions depict are observable in a variety of physical systems that the MBBME describes. The results obtained by this solution play important role in the mathematical physics. Moreover, the graphical analysis of (G' /G,1/G)- EM are visualized in 2D and as well as 3D given in this work.

This article presents the exact solution of a bipolar fuzzy heat equation based on bipolar fuzzy Fourier transform under generalized Hukuhara partial (gH-p) differentiability. A bipolar fuzzy Fourier transform is defined, and the related key propositions and fundamental characteristics are discussed. Further, a bipolar fuzzy heat equation model is constructed using gH-differentiability, and the analytical solution of a bipolar fuzzy heat equation with bipolar fuzzy Fourier transform approach is examined. Some illustrative examples are provided to check the suggested methodology's liability and efficiency. The type of differentiability and the solution of the bipolar fuzzy heat equation are shown graphically, demonstrating the versatility of the proposed methodology and elucidating the impact of differentiability types on the solution behavior of the bipolar fuzzy heat equation. Additionally, the impact of different parameters on the solution behavior is analyzed, revealing insights into the underlying dynamics.

The homotopy perturbation method is a semi-analytical method for solving linear and nonlinear ordinary/partial differential equations. Since it is extremely difficult to find exact solutions to bipolar fuzzy fractional partial differential equations, any perturbation strategy that satisfies the conditions is acceptable. It is important to emphasize that only a small number of terms are required to find the most accurate solution. In this paper, we present the concepts of Riemann–Liouville integrals, Riemann–Liouville derivatives, and Caputo derivatives in bipolar fuzzy environments. We introduce and investigate the analytical technique to solve bipolar fuzzy fractional heat equation using homotopy perturbation method. The series solution of bipolar fuzzy fractional heat equation is extracted under Hukuhar differentiability. Some examples are solved to demonstrate the effectiveness and efficiency of the proposed method. Results are presented graphically to illustrate the physical situation and the accuracy of the solution. Furthermore, our proposed method involves the practical application of heat transfer in biological tissues during hyperthermia treatment.

Fuzzy differential equations (FDEs) are the general concept of ordinary differential equations. FDE seems to be a natural way to model the propagation of cognitive uncertainty in dynamic environments. This article establishes the characteristics of the strongly generalized Hukuhara differentiability (SGHD)-based fifth-order derivative of the fuzzy-valued function (FVF). The Laplace operator is used in SGHD to create a strategy for solving the fifth-order fuzzy initial value problem (FIVP). Furthermore, some examples of FIVP are addressed to exploit liability and the efficiency of our proposed method. Furthermore, the switching points and solutions of FIVP are presented graphically to demonstrate and corroborate the theoretical findings. Additionally, an application of a mass-spring-damper system is solved by our proposed method.

In this work, we successfully implemented the unified method and modified direct algebraic method (MDAM) to analyse the dynamical nonlinear wave structures of the predator–prey (PP) model using the conformable derivative that plays a significant role in biology. We secured some novel hyperbolic, trigonometric and rational function solutions. Modulation instability (MI) analysis of the PP model was also observed. Furthermore, the physical behaviours of some of the reported results are shown as three-dimensional, two-dimensional and contour profiles choosing suitable parameters. The computation software Mathematica was used to verify all the reported outcomes by substituting them back into the PP model. The results show that our mentioned methods are simple, efficient and precise to follow and can be used for a variety of further complex problems. The resulting solutions are novel, intriguing and potentially useful in understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including biological sciences. The findings are exceptional and unique in comparison to previous findings in the literature. Furthermore, our findings are the first step towards understanding the structure and physical behaviour of complicated structures. These solutions define the wave performance of the governing model, actually. We feel that this work is timely and will be of interest to a wide spectrum of experts working on physics, engineering and biological models.

This article negotiates the investigation of optical pulses with the Biswas-Arshed equation in birefringent fibers that involves a two-component form for vector solitons in an optical fiber communication system. Various optical pulses are recovered in single and combo shapes like bright, dark, singular, bright-dark, and dark-singular solitons by the uses of the extended Fan-sub equation method (EFSEM). In addition, the hyperbolic, singular periodic waves, Jacobi’s elliptic function solutions are also derived. This method is beneficial for solving the nonlinear partial differential equations (NLPDEs), because this method is not only useful for finding the new solitary wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. The constraints conditions to ensure the existence criteria of reported optical soliton solutions are also listed. The stability analysis of the studied model is also examined by utilizing linear stability theory. In addition, by selecting different parametric values, the physical representation of some achieved solutions is plotted in 3D and contour graphs with the help of Mathematica. The reported results show that the proposed method is effective, concise, straightforward, powerful, and it can be used to tackle some more complex nonlinear systems.

The solitary wave solutions gained well-reputed significance because of their peculiar characteristics. Solitary waves are spatially localized waves and are found in a variety of natural systems from mathematical physics and engineering phenomena. This manuscript deals the different solitary wave solutions that have a great significance in mathematical physics. Various solutions are recovered in single and combo shapes like bright, dark, singular, bright-dark, and dark-singular solitons by the virtue of the generalized exponential rational function method (GERFM), ( G^'/G^2 )-expansion function method and the generalized Kudryashov method. Besides, the singular periodic wave and rational function solutions are also derived. The physical behavior of the reported results is sketched through several 3 dimensional, and 2 dimensional profiles with the assistance of suitable parameters. The acquired results are valuable in grasping the elementary scenarios of nonlinear sciences as well as in the related nonlinear higher dimensional wave fields. The achieved outcomes have been verified by putting them into the governing equation with the aid of Mathematica. Thus our strategies through the fortress of representative calculations give a functioning and intense mathematical execute for tackling complicated nonlinear wave problems. We anticipate, it will contribute us to observe the waves that occur in nonlinear complicated phenomenas. We believe that this work is timely and will be of interest to a broad range of experts involved in modeling.

In this manuscript, we secure solitary wave solutions of the generalized (2+1)-dimensional nonlinear conformable Schrödinger (NLCS) system by employing a powerful computational technique. The reported optical solitons can be segregated as bright, dark, singular, combo bright-dark, periodic, and mixed periodic solutions. In addition, by selecting different parametric values, the physical representation of some achieved solutions is plotted in 3D, 2D, and contour graphs and verified the correctness of these solutions using Mathematica. The findings demonstrate that the examined system theoretically contains a large number of solitary wave structures. The reported results show that the proposed method is effective, concise, straightforward, powerful, and it can be used to tackle some more complex nonlinear partial differential equations in the applied sciences.

•Modified nonlinear Schrodinger’s equation in case of ocean engineering is studied.•Extended sinh-Gordon equation expansion and (G′G2)-expansion function methods are employed.•Bright, dark, singular, periodic and combined soliton solutions are derived.•Moreover, three dimensional profiles with some suitable choice of parameter values have been depicted.

In this paper, a specific consideration is paid to the nonlinear dynamics of solitary waves to kinetics of phase separation in iron [Formula: see text] based on ternary alloys. The convective–diffusive Cahn–Hilliard (CH) equation is used as a mathematical model to describe the dynamics of the separation phase for the ternary alloys of iron. A variety of solitary wave solutions with unknown parameters are extracted in different shapes like kink-type, bell-shape, shock-type, combine soliton, trigonometric, hyperbolic and Jacobi’s elliptic function solutions with the assistant of recently computational tools, namely, extended Fan-sub equation method (EFSEM) and extended auxiliary equation method (EAEM). In addition, 3D, 2D, and their corresponding contour profiles of earned results are sketched in order to observe their dynamics with the choices of involved parameters. On the bases of achieved results, we may claim that the proposed computational methods are direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations.

The solitary wave solutions gained well-reputed significance because of their peculiar characteristics. Solitary waves are spatially localized waves and are found in a variety of natural systems from mathematical physics and engineering phenomena. This manuscript deals the investigation of optical pulses to the Biswas–Arshed equation with third order dispersion and self-steepening coefficients in nonlinear optics. Various optical pulses are recovered in single and combo shapes like bright, dark, singular, bright-dark, and dark-singular solitons by the virtue of extended sinh-Gordon equation expansion method and ( $$\frac{G^{\prime }}{G^2}$$ )-expansion function method. Besides, the singular periodic wave solutions are also derived. The constraints conditions to ensure the existence criteria of reported optical solutions are also listed. In addition, by selecting different parametric values, the physical representation of some achieved solutions is plotted in 3D graphs with the help of Mathematica. The reported results show that the proposed methods are effective, concise, straightforward, powerful, and they can be used to tackle some more complex nonlinear systems.

•The (2+1)-dimensional time-space fractional nonlinear Schrdinger equation is studied.•Extended rational sine-cosine/sinh-cosh and novel Φ6- expansion model approaches are employed.•Hyperbolic, trigonometric, bright, dark, singular, and combined optical soliton solutions are extracted.•Moreover, 3D, and contour profiles of some reported results with some suitable choice of parametric values have been depicted.

This paper reveals optical soliton solutions to fiber Bragg gratings (FBGs) with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are no doubt a technological spectacle that sustained balance between dispersion and the nonlinear effects that leads to a stable transmission of solitons across intercontinental distances. FBGs as sensor elements are used for measuring numerous engineering parameters such as temperature, strain, pressure, tilt, displacement, acceleration, load. Two recently developed mechanisms such as the unified method and extended sinh-Gordon equation expansion method (ShGEEM) are successfully employed to secure optical pulses in the shapes of the bright, dark, singular, complex combo, periodic, and plane wave solutions. The achieved solutions contain key applications in engineering and physics. These solutions define the wave performance of the governing models. By selecting suitable parametric values, the dynamics of the evaluated results are exemplified by sketching their 2-dimensional, 3-dimensional, and contour profiles to understand the real phenomena for such sort of nonlinear models. The novelty of the gained outcomes is manifested by a detailed comparison with the results that already exist.

•The (2+1)-dimensional Coupled Maccari’s system is considered.•New extended direct algebraic method, the unified method, and the extended Sinh-Gordon equation expansion method are employed.•Hyperbolic, trigonometric, bright, dark, singular, and combined exact soliton solutions are extracted.•Moreover, 3D, and contour profiles of some reported results with some suitable choice of parametric values have been depicted.

In this work, our main idea is to retrieve a variety of exact optical solitons of the generalized (2+1)-dimensional nonlinear conformable fractional Schrödinger system by employing three novel integration norms. The constructed optical solitons can be divided as dark, singular, combo dark-singular, periodic, mixed periodic, and plane wave solutions. In addition, by selecting different parametric values, the physical representation of some achieved solutions is plotted in 3-dimensional, 2-dimensional, and contour graphs. Computation packages are used to verify all of the secured results by putting them back into the original model. The findings demonstrate that the examined system theoretically contains a large number of optical solitons. The constructed results rendering that the consider methods in this article are effective, concise, straightforward, efficient, and they can be used to handle some more nonlinear partial differential equations in the applied sciences. The achieved outcomes in this article may be valuable in clarifying the true meaning of numerous nonlinear advancement circumstances that develop in various domains of nonlinear sciences. Also, these fresh solutions have many applications in physics and other branches of physical sciences.

This work devotes to explore abundant optical and other soliton solutions to fiber Bragg gratings with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are no doubt a technological spectacle that sustained balance between dispersion and nonlinear effects that leads to a stable transmission of solitons across intercontinental distances. Three reliable integration norms namely ([Formula: see text])-expansion function method, generalized tanh method and generalized Kudryashov method are engaging to devise optical dark, singular, combo, complex and periodic solutions. The reported solutions contain key applications in engineering and physics. These solutions define the wave performance of the governing models, actually. By the choice of suitable parametric values, the dynamics of the evaluated results are exemplified by sketching their 2D, 3D and contour profiles to understand the real phenomena for such sort of nonlinear models. The novelty of the gained outcomes is manifested by a detailed comparison with the results that already exist. The outcomes disclose that our schemes are very active reliable, concise, outspoken, appropriate, useful and skilled for attaining the exact solutions of such kind of problems which minimize the computational work and give it a wide range of applications.

The current study utilizes the extended sinh-Gordon equation expansion and ($frac{G^{prime}}{G^2}$)-expansion function methods in constructing various optical soliton and other solutions to the (2+1)-dimensional hyperbolic nonlinear Schr${ddot o}$dinger's equation which describes the elevation of water wave surface for slowly modulated wave trains in deep water in hydrodynamics. We secure different kinds of solutions like optical dark, bright, singular, combo solitons as well as hyperbolic and trigonometric functions solutions. Moreover, singular periodic wave solutions are recovered and the constraint conditions which provide the guarantee to the soliton solutions are also reported. In order to shed more light on these novel solutions, graphical features 3D, 2D and contour with some suitable choice of parameter values have been depicted. We also discuss the stability analysis of the studied nonlinear model with aid of modulation instability analysis.

The primary focus of this article is on cubic optical solitons (os) in a polarization-preserving fiber modeled by the nonlinear Schrodinger equation (NLSE). In this work, our main idea is to secure a variety of optical solitons in the forms of hyperbolic and trigonometric solutions, as well as a class of solitary wave solutions such as dark, bright-dark, singular, singular periodic, multiple-os, and mixed complex soliton solutions. To examine the governing model, a recently created integration technique called as the new extended direct algebraic method (NEDAM) is used. Furthermore, the investigated equation is explained in terms of two forms of nonlinearity. The constraints conditions for the resulting solutions are explicitly presented. The obtained findings demonstrate that the applicable computing system is direct, productive, and trustworthy, and that it may be applied to more complex phenomena.

In this study, we successfully apply Hirota’s bilinear method (HBM) to retrieve the different wave structures of the general [Formula: see text]th dispersionless Dym equation by considering the test function approaches. The studied model is used to describe the dynamics of deep water waves. We formally retrieve some novel lump periodic, some other new interaction, and breather wave solutions. Moreover, the physical behavior of the reported results is sketched through several three-dimensional, two-dimensional and contour profiles with the assistance of suitable parameters. The acquired results are valuable in grasping the elementary scenarios of nonlinear fluid dynamics as well as the dynamics of engineering sciences in the related nonlinear higher-dimensional wave fields. The gained results are checked and found correct by putting them into the governing equation with the aid of Mathematica. Thus, our strategies through the fortress of representative calculations give a functioning and intense mathematical execution for tackling complicated nonlinear wave issues.