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In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

In this study, the optimization of the fixed point in function spaces and exact solutions to the biofilm model are studied to describe growth, and bacteria working together, including quorum sensing. Engineering applications that involve biotechnology, environmental engineering, and the creation of medical devices are particularly pertinent to biofilm models that include quorum sensing processes. This model is applied to cellular processes such as cell growth in mathematical biology. The optimization of the solution is also discussed by using the Banach space in an optimal ball. The existence of a solution is shown by using the Schauder fixed point theorem. Diverse exact solutions are extracted in hyperbolic, trigonometric, and plane waveforms are archive by using two techniques such as modified extended direct algebraic (MEDA) and G′/G-expansion technique. Additionally, the 3D plots and their corresponding contour graphs are also drawn to show the physical behavior of the solutions.

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 In this manuscript, we implement the travelling wave solutions of the fractional (3+1) generalized computational nonlinear wave equation with gas bubbles via application of five mathematical methods. Liquids with gas bubbles primarily arise in various applications like science, engineering, and mathematical physics. The obtained solitary waves solutions have fruitful applications in engineering, science, life, nature and physics. Several novel soliton solutions of concerned model are established in the form of hyperbolic, trigonometric, exponential and rational functions. To handle all calculations and verification of obtained results, computational software Mathematica 12.1 is used. For the demonstration of the physical behaviour of concern model, some solutions are plotted graphical in 2-dimensional and 3-dimensional by imparting specific values to the parameters under constrain conditions. Finally, we intrigue both two and three dimensional to explain the physical behavior of the model.

This research paper focused on the solution of systems of fractional integro-differential equations (FIDEs) of the Volterra type with variable coefficients. The proposed approach combined the tau method and shifted Gegenbauer polynomials in a matrix form. The investigation of the existence and uniqueness of solutions for these systems was carried out using Krasnoselskii's fixed point theorem. The equations employed Caputo-style derivative operators, and to minimize computational operations involving derivatives and multiplications, integral and product operational matrices were derived. By introducing suitable polynomial approximations and employing the tau spectral method, the original system of FIDE was transformed into an algebraic system. Solving this algebraic system provided approximate solutions to the main system. Error bounds were computed in the Gegenbauer-weighted Sobolev space. The proposed algorithm was implemented and tested on two systems of integro-fractional differential equations to demonstrate its efficiency and simplicity. By varying the parameter $ \sigma $ in the Gegenbauer polynomials, the impact of this variation on the approximate solutions can be observed. A comparison with another method utilizing the block-by-block approach was also presented.

Learned from wrinkle wave motions, we concentrated on bifurcation phenomena in substrate-supported graphene sheets by obtaining the bifurcation solitons of thermophoretic motion equation. We find new soliton solutions by applying the bilinear technique and correctly choosing the auxiliary function involved in the bilinear form. Use of the Hirota bilinear technique combined with symbolic computation yields lump solutions as well as lump-type solutions. The governing model is subject to computation of the lump one strip, lump two strip, Y-type, lump periodic solution, generalised breather, and lump periodic solution. When simulating waves in different physical systems, such as water waves, fibre optics, and plasma physics, lump and lump type solutions are crucial. Rogue waves are studied in relation to large and unforeseen ocean wave phenomena that might affect coastal buildings and ship safety. In communication systems, including optical fibres, generalised breathers are used to comprehend and regulate pulse propagation.

In this article, the nonlinear plasma model known the damped modified Korteweg–de Vries equation under consideration on the base of analytical technique to know the novel behavior of this nonlinear model. We formulated the nonlinear plasma model by utilizing the RP (reductive perturbation) technique with basic set of fluid system which consisted on cold negative charge dust fluid, immobile background neutral charges, positive charge ion fluid, fast (ionized) electrons and thermally electrons. We also constructed various kinds of novel soliton solutions including kink wave solitons, periodic solitary waves, anti-kink wave solitons for the damped modified KdV equation. The determined soliton solutions are novel, interested and did not determined in the previous research. The physical interpretation of demonstrated solutions represent in contour, two-dim and three-dim with computational software. The study prove that our proposed approach is powerful, efficient and can also be fruitful for study of other nonlinear models.

We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves traveling in both directions, the zero-mass assumption, which is required for many scientific applications, is not required by the KP-B equation. In several applications in engineering and physics, taking these features into account allows researchers to acquire more precise conclusions, particularly in studies pertaining to the dynamics of water waves. The foremost purpose of this manuscript is to establish diverse solutions in the form of exponential, trigonometric, hyperbolic, and rational functions of the (3+1)-dimensional fractional (KP-B) via the application of four analytical methods. This KP-B model has fruitful applications in fluid dynamics and plasma physics. Additionally, in order to better explain the potential and physical behavior of the equation, the relevant models of the findings are visually indicated, and 2-dimensional (2D) and 3-dimensional (3D) graphics are drawn.

In this paper, we explore optical nonlinear waves in a birefringent fiber by investigating the dynamics of a coupled nonlinear Schrödinger system with four-wave mixing factors. Utilizing the Jacobi elliptic function approach, we achieve this goal by building cusp-like singular soliton solutions in this optical system. Depending on the characteristics of the focusing and defocusing optical medium, we investigate four different physical possibilities. We identify numerous cusp-type singular soliton profiles in these settings, namely cusp-like bright, dark, kink, and antikink solitons. Group velocity dispersion, spatio-temporal dispersion, and four-wave mixing effects are further evaluated in terms of their influence on the constructed singular soliton solutions. We show that the intensity of the singular soliton profiles can be suppressed or enhanced by altering the dispersion values. We also witness interesting transitions: when four-wave mixing parameters are tuned, cusp-like kink-type solitons can become cusp-like antikink solitons and vice versa. These novel findings about singular solitons in birefringent fibers have great promise for experimental uses in the realm of optics, as well as for advancing our knowledge of four wave mixing and nonlinear propagation.

In this research, the complete discriminant system (CDS) of polynomial method (CDSPM) will be applied to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) with a particular emphasis on the introduction of various optical solitons and wave structures. Our analysis of the CQNLSE-AC illustrates the importance of CDSPM, and we give dynamic results, such as critical conditions and bifurcation points for solutions. Additionally, we determine several types of optical soliton solutions, including Jacobian elliptic function (JEF), hyperbolic function, and trigonometric function solutions, and also convert JEF into solitary wave (SW) solutions.

In this paper, our aim is to further expand the use of the two-variable [Formula: see text]-expansion approach to a new coupled KdV and Z-K system, which has various significant applications in different fields of applied sciences. The KdV equation, along with shallow-water waves and long internal waves in oceans, basically explains how long, one-dimensional waves propagate in a variety of physical conditions. The study of coastal waves on the basis of the ocean is done using the Zakharov–Kuznetsov (Z-K) equation and this model is utilized to illustrate ion-acoustic wave propagation. By using this method, different forms of analytical solutions of the new coupled KdV (NCKdV) system and the new coupled Z-K (NCZ-K) system, such as solitons, multi-peak solitons, solitary waves, trigonometric, hyperbolic and rational functions and other wave solutions are constructed. The significant features of multi-peak solitons induced by the higher-order effects, including velocity variations, localization or periodicity attenuation and state transitions, are revealed. When the localization disappears then the multi-peak soliton becomes a periodic wave. The constructed solutions are also presented graphically having their applications in engineering, etc. The stability of the solution is examined by utilizing modulation instability. The results obtained show that the proposed technique is universal and efficient. In addition, this technique can also be applied to lots of other new coupled systems arising in other areas of applied sciences.

In this article, the coupled Drinfeld–Sokolov–Wilson equation under investigation base on computational approach named extension of modified rational expansion approach. By this computational approach secured various kinds of solitons named kink solitons, dark solitons, anti kink solitons, periodic solitons, bright solitons, kink dark solitons, kink bright solitons, anti kink dark solitons, anti kink bright solitons for coupled DSW system. The calculated results are interesting, precise, novel, and different which have not been secured in the previous research. We represent the graphical behaviour of determined solutions by 2-D, 3-D and contour through computational software. The determined results may helpful and play significant role in the study of nonlinear behaviour in various branches of physical sciences such as laser optics, communication system, fluid dynamics, transfer of heat, shallow water waves, acoustics, optics and many others. The constructed results prove that proposed computational approach is concise, powerful, and straightforward for securing the solitons results of other nonlinear partial differential equations.

We investigate the rational solitons of the nonlinear (3+1)-dimensional Geng (NG)-equation via symbolic computation with ansatz functions technique and the logarithmic transformation. The (3+1) NG-equation uses in shallow water wave dynamics. Multiwave solitons are reported by using three-waves technique. Homocentric breathers (HBs) approach is also applied to built new forms of exact solutions. Furthermore periodic cross rational solitons (PCRS) and kink cross rational solitons (KCRS) are examined through suitable transformations. We construct M-shape solitons and their dynamics are shown via the appropriate values of parameters. Additionally, two types of interactions for rational soliton with kink wave are investigated.

Some natural phenomena that arise in nonlinear sciences can be sometimes discussed using discrete-time systems. In this work, we investigate the periodicity, boundedness, oscillation, stability, and some exact solutions of nonlinear difference equations with generalized order. Exact solutions are extracted using the traditional iteration and the modulus operator. The stability of the equilibrium points is examined using some well-known theorems. Some numerical examples are also introduced to confirm the validity of the theoretical work. We use MATLAB to implement the numerical part. The used technique can be easily utilized to deal with other rational recursive problems.

This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion. The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary, kink, anti-kink, combo, and singular-periodic wave solutions. The attained solutions are expressed by the trigonometric and hyperbolic functions including tan, sec, cot, csc, tanh, sech, coth, csch, and of their combination. In addition, the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions. Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family. For the bright wave solutions in terms of Atangana’s conformable derivative, the amplitudes of the bright wave gradually decrease, but the smoothness increases when the fractional parameters α and β increase. On the other hand, the periodicities of periodic waves increase. The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases. The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.

This paper examines the propagation of M-shape solitons and their interactions with kink waves to the (2 + 1)-dimensional integrable Schwarz-Korteweg-de Vries (ISKdV) problem by applying the symbolic computation with ansatz functions technique and logarithmic transformation. The governing model usually appears in the nonlinear shallow water waves and fluid mechanics. We discuss various nonlinear waves like multiwave solutions (MSs), homoclinic breather (HB), M-shape solitons, single exponential form (one-kink), and double exponential form (two-kink). These waves have lot of applications in fluid dynamics, nonlinear optics, chemical reaction networks, biological systems, climate science, and material science. We also study interaction among M-shape solitons with kink wave. At the end, we discuss the stability characteristics of all solutions.

This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable (G′/G,1/G)-expansion and generalized exp(−ϕ(ξ))-expansion techniques are presented to construct the ion-acoustic wave results of a (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) model. As a result, the novel soliton and other wave solutions in a variety of forms, including kink- and anti-kink-type breather waves, dark and bright solitons, kink solitons, and multi-peak solitons, etc., are attained. With the help of software, the solitary wave results (that signify the electrostatic potential field), electric and magnetic fields, and quantum statistical pressures are also constructed. These solutions have numerous applications in various areas of physics and other areas of applied sciences. Graphical representations of some of the obtained results, and the electric and magnetic fields as well as the electrostatic field potential are also presented. These results demonstrate the effectiveness of the presented techniques, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields.

In this article, lump solutions, lump with I-kink, lump with II- kink, periodic, multiwaves, rogue waves and several other interactions such as lump interaction with II-kink, interaction between lump, lump with I-kink and periodic, interaction between lump, lump with II-kink and periodic are derived for Pavlov equation by using appropriate transformations. Additionally, we also present 3-dimensional, 2-dimensional and contour graphs for our solutions.

This article studies diverse forms of lump-type solutions for coupled nonlinear generalized Zakharov equations (CNL-GZEs) in plasma physics through an appropriate transformation approach and bilinear equations. By utilizing the positive quadratic assumption in the bilinear equation, the lump-type solutions are derived. Similarly, by employing a single exponential transformation in the bilinear equation, the lump one-soliton solutions are derived. Furthermore, by choosing the double exponential ansatz in the bilinear equation, the lump two-soliton solutions are found. Interaction behaviors are observed and we also establish a few new solutions in various dimensions (3D and contour). Furthermore, we compute rogue-wave solutions and lump periodic solutions by employing proper hyperbolic and trigonometric functions.

This paper provides the effects of nutrients concentrations on the algae growth via solitary wave analysis and classical existence theory. There is a well known Diffusive Nutrient-Algae model based on the Sanyang wetland, is under consideration. The new families of exact solutions which represent nutrient concentration and algae density in different form are observed using the ϕ6-model expansion. Furthermore, the existence of these solutions are also discussed under different restrictive conditions and variables of nutrient concentration and algae density are represented in hyperbolic, trigonometric and rational forms. The graphics of these solutions are also sketched in 3D and 2D representations.

We have utilized three novel methods, called generalized direct algebraic, modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations and modified Benjamin-Bona-Mahony equation. After substituting particular values of the parameters, different solitary wave solutions are derived from the exact traveling wave solutions. It is shown that these employed methods are more powerful tools for nonlinear wave equations.

In this article, we consider the improved perturbed nonlinear Schrödinger Equation (IP-NLSE) with dual power law nonlinearity, which arises in optical fibers and photovoltaic-photo-refractive materials. We found grey and black optical solitons of the governing equation by means of a suitable complex envelope ansatz solution. By using the Chupin Liu’s theorem (CLT) for the grey and black solitons, we evaluated new categories of combined optical soliton (COS) solutions to the IP-NLSE. The propagation behaviors for homoclinic breathers (HB), multiwaves and M-shape solitons will be analytically examined. All new analytical solutions will be found by an ansatz function scheme and suitable transformations. Multiwave solitons have been reported by using a three-waves technique. Furthermore, two kinds of interactions for M-shape soliton through exponential functions will be examined.

We explore stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) equations for some wave solutions such as the cross-kink rational wave solution, periodic cross-rational wave solution and homoclinic breather wave solution. We also examine some M-shaped solutions such as the M-shaped rational solution, M-shaped rational solution with one and two kink waves. We also derive the M-shaped interaction with rogue and kink waves and the M-shaped interaction with periodic and kink waves. This model is used in mathematical physics, surface physics, plasma physics, population dynamics and applied sciences. Moreover, we also show our results graphically in different dimensions. We obtain these solutions under some constraint conditions.

In this study, based on the Hirota bilinear form, the exact analytic solutions of the (3 + 1) dimensional Vakhnenko–Parkes equation with various physical properties were constructed with the help of the Maple package program and symbolic computation. These solutions are the type of multi-waves, breather wave, lump–kink, lump–periodic solutions and interaction solutions (between lump and hyperbolic wave solutions). The constructed solutions have expanded and enriched the solution forms of this new model existing in the literature. By means of Maple package program, 3D and 2D graphs were drawn for the special choices of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. The solutions obtained can be used in the explanation of physical phenomena occurring in the propagation of high-frequency waves in a relaxing medium.

In this study the generalized exponential rational function approach is used to extract some new exact solitary wave solutions of the generalized coupled nonlinear Schrödinger–Korteweg–de Vries equations. These coupled equations appear in interactions between short and long dispersive waves which are important in a variety of applied sciences and engineering domains. With the help of the Mathematica package software, the solutions found in this study were verified. We also show graphical representations of the solutions for a better understanding of the coupled equations’ behavior and physical structures.

Atangana conformable Boussinesq-like equations (NLACBEs) plays an important role in shape-memory alloys, nonlinear string and coupled electrical circuits. In this paper, we study the different analytical solutions of various types of NLACBEs like multi-wave, homoclinic breather approach, rogue waves, M-shaped rational solutions, lump with one kink, lump interaction with kink and rogue wave, periodic-cross lump, lump with two kink, periodic cross-kink and periodic solutions. We compute the essential relationship between lump interaction with kink and rogue wave, periodic cross-kink and other types of exact solutions by studying the unknown nonlinear confirmable differential equations for the corresponding parameters. We show the graphical representation for our retrieve analytical solutions to observe the behavior of effective waves.

This manuscript is related to investigate analytical solutions to some linear fractional partial fuzzy differential equations under certain conditions. For the concerned investigation, we utilize Laplace transform along with some decomposition method to compute series type solution under fuzzy concept. Some examples are solved to demonstrate the proposed method.

There are a lot of physical phenomena which their mathematical models are decided by nonlinear evolution (NLE) equations. Our concern in the present work is to study a special type of NLE equations called the (3 + 1)-dimensional generalized breaking soliton (3D-GBS) equation. To this end, the linear superposition (LS) method along with a series of specific techniques are utilized and as an achievement, multiwave, multicomplexiton, and positive multicomplexiton solutions to the 3D-GBS equation are formally constructed. The study confirms the efficiency of the methods in handling a wide variety of nonlinear evolution equations.

The propagation of waves in dispersive media, liquid flow containing gas bubbles, fluid flow in elastic tubes, oceans and gravity waves in a smaller domain, spatio-temporal rescaling of the nonlinear wave motion are delineated by the compound Korteweg-de Vries (KdV)-Burgers equation, the (2+1)-dimensional Maccari system and the generalized shallow water wave equation. In this work, we effectively derive abundant closed form wave solutions of these equations by using the double (G′/G, 1/G)-expansion method. The obtained solutions include singular kink shaped soliton solutions, periodic solution, singular periodic solution, single soliton and other solutions as well. We show that the double (G′/G, 1/G)-expansion method is an efficient and powerful method to examine nonlinear evolution equations (NLEEs) in mathematical physics and scientific application.

The research of unmagnetized dusty plasmas is extremely amiable as long as theoretical aspects and their applicability. They are an outstanding mechanism for generating exact solitary waves and solitons. The present article examines the KdV-Burgers type equation in an unmagnetized dusty plasma and the Kadomtsev-Petviashvili dynamical equation in unmagnetized dust plasma. We present the modified (G′G)-expansion process to secure few exact solitary wave answers. The acquired outcomes confirm that the studied method is an outspoken and useful analytical device for NLEEs in mathematical physics.

In this paper, modified variational iteration algorithm-II is investigated for finding approximate solutions of nonlinear Parabolic equations. Comparisons of the MVIA-II with trigonometric B-spline collocation method, variational iteration method, homotopy perturbation transform method, Adomian decomposition method, and modified variational iteration method are carried out, which show that the proposed algorithm (MVIA-II) is robust one. Some nonlinear Parabolic equations are given to demonstrate the implementation and accuracy of the MVIA-II.

In this study, we consider the third order nonlinear Schrödinger equation (TONSE) that models the wave pulse transmission in a time period less than one-trillionth of a second. With the help of the extended modified method, we obtain numerous exact travelling wave solutions containing sets of generalized hyperbolic, trigonometric and rational solutions that are more general than classical ones. Secondly, we construct the transformation groups which left the equations invariant and vector fields with the Lie symmetry groups approach. With the help of these vector fields, we obtain the symmetry reductions and exact solutions of the equation. The obtained group-invariant solutions are Jacobi elliptic function and exponential type. We discuss the dynamic behaviour and structure of the exact solutions for distinct solutions of arbitrary constants. Lastly, we obtain conservation laws of the considered equation by construing the complex equation as a system of two real partial differential equations (PDEs).

The (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is studied, which describes the incompressible fluid. By virtue of an ansätz functions and the bilinear form, many entirely new complexiton solutions and periodic-soliton solutions are derived. With the aid of symbolic computation, their dynamical behaviors are demonstrated in some three-dimensional plots by choosing different values of the parameters.

In this article, the analytical solutions for modified Benjamin-Bona-Mahony and coupled Drinfel’d-Sokolov-Wilson equations have been extracted with the help of very simple transformation. These results hold numerous traveling wave solutions that are of key importance in elucidating some physical circumstance. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. Keyword: Modified Benjamin-Bona-Mahony equation, Drinfel’d-Sokolov-Wilson equation, Auxiliary equation method

An extended fifth order Korteweg-de-Vries (efKdV) equation is an important equation in fluids dynamics for the description of nonlinear wave processes, and contains quite a number of KdV-type equations including the Sawada-Kotera equation, the Caudrey-Dodd-Gibbon equation, the Lax equation, the Kaup-Kuperschmidt equation and the Ito equation among others. However, in this paper, we examine the efKdV by extracting three different fifth order Korteweg-de-Vries (fKdV) equations. Solitary wave solutions to the extracts are found by means of an exponential function ansatz specifically constructed for the study. The obtained solutions can be used in description of shallow-water waves. Keywords: Fifth order KdV equation, KdV equations, Exponential function ansatz, Solitary wave

In present work, we discussed the (3+1)-dimensional generalized Korteweg–de-Vries–Zakharov–Kuznetsov equation (gKdV-ZKe) which describes the effect of magnetic field on weak nonlinear ion-acoustic waves studied in the field of plasma comprised of cool and hot electrons. The solutions of gKdV-ZKe are acquired by new modified extended direct algebraic method (MEDAM) and are observed in the form of solitary, shock, singular, shock-singular, solitary-shock and double singular soliton solutions. Moreover, the families of rational solutions are also emerged during the derivation. The results we derive, in this article, are useful to study and verify the analytical solutions with numerical and experimental solutions in plasma physics. The obtained solutions are also illustrated graphically.

The famous Hamiltonian amplitude equation (HAE) and the well known cubic nonlinear Schrödinger equation (CNLSE) with repulsive delta potential have been taken into account for integrability analysis, establishment of conservation laws (CLs) and traveling wave (soliton) solutions in the polynomial forms. The integrability investigation has been done with the help of Painlevé test (P-test), conserved densities and associated fluxes required for the establishment of CLs are found with the help of scaling invariance approach whereas the traveling wave solutions in the form of polynomials are derived by unified method (UM).

In this research, we study the dispersive phenomena for the (2+1)-dimensional generalized breaking soliton system (GBSS), which describe the interaction phenomena between Riemann wave and long wave via two space variable in nonlinear media. The constructed solitary wave results for the (2+1)-dimensional GBSS via extended modified rational expansion (EMRE) method in the form of rational, elliptic, and trigonometric functions including mixed solitons, single bright and dark solitions, combined bright and dark solitons, kink and anti-kink wave solitions, traveling waves, and periodic waves. The physical phenomena of demonstrated results represented by higher dimensional and contour plots graphically with the help of symbolic computation. The obtained results are very helpful in the study of interaction phenomena in fiber optics, mathematical physics, fluid dynamics, geophysics, engineering and many other various areas of scientific fields.

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

We have investigated wave solutions of the Predator–Prey (PP) model with fractional derivative order by novel three modified mathematical methods with the help of the Mathematica platform. The derived solutions are in the form of distinct functions such as trigonometric, hyperbolic, exponential and rational functional. For the physical phenomena of fractional model, some solutions are plotted in 2-dimensional and 3-dimensional by inserting specific values to attached parameters under sufficient condition on each solution. Hence, proposed schemes are enormously superbly mathematical tools to review wave solutions of several fractional models in nonlinear science.

Several types of solitary wave solutions of (3 + 1)-dimensional nonlinear extended and modified quantum Zakharov–Kuznetsov equations are established successfully via the implantation of three mathematical methods. The concerned models have many fruitful applications to describe the waves in quantum electron–positron–ion magnetoplasmas and weakly nonlinear ion-acoustic waves in plasma. The derived results via the MEAEM method, ESE method, and modified F-expansion have been retrieved and will be expedient in the future to illuminate the collaboration between lower nonlinear ion-acoustic waves. For the physical behavior of the models, some solutions are plotted graphically in 2D and 3D by imparting particular values to the parameters under the given condition at each solution. Hence explored solutions have profitable rewards in the field of mathematical physics.

In this study, we implement the generalized (G′/G)-expansion method established by Wang et al. to examine wave solutions to some nonlinear evolution equations. The method, known as the double (G′/G, 1/G)-expansion method is used to establish abundant new and further general exact wave solutions to the (3 + 1)-dimensional Jimbo–Miwa equation, the (3 + 1)-dimensional Kadomtsev–Petviashvili equation and symmetric regularized long wave equation. The solutions are extracted in terms of hyperbolic function, trigonometric function and rational function. The solitary wave solutions are constructed from the obtained traveling wave solutions if the parameters received some definite values. Graphs of the solutions are also depicted to describe the phenomena apparently and the shapes of the obtained solutions are singular periodic, anti-kink, singular soliton, singular anti-bell shape, compaction etc. This method is straightforward, compact and reliable and gives huge new closed form traveling wave solutions of nonlinear evolution equations in ocean engineering. Keywords: Exact traveling wave solutions, (G′/G, 1/G)-expansion method, (3+1)-dimensional Jimbo–Miwa equation, (3+1)-dimensional Kadomtsev–Petviashvili equation, Symmetric regularized long wave equation

In this research, we constructed the exact travelling and solitary wave solutions of the Kudryashov–Sinelshchikov (KS) equation by implementing the modified mathematical method. The KS equation describe the phenomena of pressure waves in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. Our new obtained solutions in the shape of hyperbolic, trigonometric, elliptic functions including dark, bright, singular, combined, kink wave solitons, travelling wave, solitary wave and periodic wave. We showed the physical interpretation of obtained solutions by three-dimensional graphically. These new constructed solutions play vital role in mathematical physics, optical fiber, plasma physics and other various branches of applied sciences.

In this article, we recover a variety of new families of shallow water wave and solitary wave solutions to the (1+1)-dimensional Dullin “Gottwald” Holm (DGH) system by employing new extended direct algebraic method (NEDAM). The derived results are obtained in diverse hyperbolic and periodic function forms. The attained solutions are new addition in the study of solitary wave and shallow water wave theory. In addition, 3-dimensional, 2-dimensional, and contour graphs of secured results are plotted in order to observe their dynamics with the choices of involved parameters. On the basis of achieved outcomes, we may claim that the proposed computational method is direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations.

In this study, the monochromatic beam propagation interprets non-scattering and non-dissipation spatiotemporally localized wave parcels proliferated in the announcement of optical Kerr. By using the mathematical techniques some elliptic, rational, and soliton solutions of dimension-less (time-dependent) paraxial wave structure are established. The Sardar subequation method (SSM) and mathematica 11.0 are used to find the exact solution of the paraxial wave equation. The solutions which we will be obtained explain some key implementations in engineering and physics. Some graphical representation has been explained in modulus, real and imaginary graph by considering relevant values of the framework. The solidity of this work explains that the soliton solutions are secure and perfect.

Our aim in this paper is to determine rogue-wave solutions for Maccari-system. We also construct multi-waves, homoclinic breathers, M-shaped rational and periodic cross kink solutions with the combination of exponential, rational, trigonometric functions and various bilinear forms. We will also draw graphical structures of our newly attained results and explain their physique.

In this article, we study the nonlinear higher order of extended KdV equation with free surface displacement. The modified exponential rational function method is used in order to find exact solutions of the extended KdV equation. As a result, various exact solutions for the equation under consideration are obtained. To illustrate the graphical behavior of the results, some of the obtained solutions are presented graphically.

The article studies the exact traveling wave solutions to the Schrödinger-Poisson system which has applications in gravity’s role of quantum state and approximate the coupling between quantum mechanics with gravitation. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtain using two norms of integration. For this sake modified extended direct algebraic (MEDA) and (G′/G)-expansion techniques are used. The 3D plots and their corresponding contour graphs are also depicted. The constraints conditions for the exact of solutions are also emerged during the derivation of solution.

The variant Boussinesq and the Lonngren wave equations are underlying to model waves in shallow water, such as beaches, lakes, and rivers, as well as electrical signals in telegraph lines based on tunnel diodes. The aim of this study is to accomplish the closed-form wave solutions by means of the generalized Kudryashov technique to the formerly stated models. The process provides further generic and inclusive wave solutions integrated with physical parameters and for definite values of these constraints reveal distinct special shapes of the waveform, namely kink soliton, bell-shape soliton, singular soliton, anti-bell shape soliton, flat soliton, and other different types of soliton. To estimate the topography of solitons’ form, we have outlined 3D and contour plots of some of the defined solutions. It has been demonstrated that the technique used to investigate nonlinear evolution equations (NLEEs) is consistent, compatible, and significant and yield typical and further generic wave solutions. Moreover, the method shows a comprehensive extension of application to function the other types of nonlinear wave phenomena.

This paper presented exact solutions and stability analysis to the nonlinear longitudinal wave equation (NLLWE) in a magneto electro-elastic (MEE) circular rod. The new generalized exponential rational function method (GERFM) has been applied for finding the exact solutions of NLLWE in MEE circular rod. It has the capacity to generate many types of solutions in different form. Therefore, it is effective in extracting new types of solutions in other nonlinear partial differential equations (NLPDEs) that commonly model applications in engineering, mathematical physics. Furthermore, some 2D, 3D graphs are obtained to understand the physical phenomena of this longitudinal wave equation and the stability of governing equation is investigated.