Bernoulli Sub-equation Function Method Research Articles

ON THE SOLITARY WAVE SOLUTIONS OF DIFFERENT VERSIONS OF FRACTIONAL 3D- WAZWAZ -BENJAMIN-BONA-MAHONY EQUATIONS

Nonlinear fractional Wazwaz -Benjamin-Bona-Mahony (WBBM) equations play an important role in physics. The equations form an important model for studying the approximately unidirectional propagation of small amplitude long waves in certain nonlinear distribution systems as an alternative to Kortweg and de Vries (KDV). In this study, the fractional 3D-WBBM equations are solved by using the Improved Bernoulli Sub-Equation Function (IBSEF) method. 3D, 2D and contour plots are given to show the physical properties of the solutions. The main aim of this method is to clarify obvious the exact solutions to the equations. Moreover, the effectiveness of the method is demonstrated by the findings presented in this paper.

Exact solutions, symmetry groups and conservation laws for some (2+1)-dimensional nonlinear physical models

In this paper, the Bernoulli sub-equation function method is used to construct new exact travelling wave solutions for two important physical models: (2+1)-dimensional hyperbolic nonlinear Schrödinger (HNLS) equation and (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation. These solutions provide valuable insights into the behavior of these models, described in terms of exponential and hyperbolic tangent (tanh) functions. The study also involves an exploration of the infinitesimal generators and symmetry groups through the Lie symmetry method. In addition, by using multiplier approach, the conservation laws are established for these models. Graphical simulation of some solutions in the form of two-dimensional and three-dimensional are plotted to understanding of the underlying physical phenomena and mathematical properties of the (2+1)-dimensional HNLS and HFSC equations. The solutions and graphing are performed using Maple software.

Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

In the present work, we aim to explore the new (3+1)-dimensional integrable fourth-order nonlinear equation(IFNE) for describing the shallow water waves. First, we study its N-soliton solutions via the bilinear form which is constructed by applying the Cole-Hopf transform. The resonance conditions of the soliton molecular are extracted and the soliton molecules are obtained. Second, the ansatz function method together with the symbolic computation, is implemented to develop the interaction wave solutions(IWSs). Finally, we take advantage of the Bernoulli sub-equation function method(BSFM) to look into the travelling wave solutions(TWSs). Different kinds of the TWSs like the singular-kink and kink solitary wave solutions are found. Correspondingly, the dynamic performances of the solutions are depicted graphically to present the physical interpretations. And for all we know, the solutions got in this work are all new and can be regarded as an extension of the solutions for the new (3+1) dimensional IFNE, which are expected to have practical significance for the application of these equations in physics.

A study of the wave dynamics of the space–time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach

The space–time fractional nonlinear Klein-Gordon and modified regularized long-wave equations explain the dynamics of spinless ions and relativistic electrons in atom theory, long-wave dynamics in the ocean, like tsunamis and tidal waves, shallow water waves in coastal sea areas, and also modeling several nonlinear optical phenomena. In this study, the improved Bernoulli sub-equation function method has been used to generate some new and more universal closed-form traveling wave solutions of those equations in the sense of beta-derivative. Using the fractional complex wave transformation, the equations are converted into nonlinear differential equations. The achieved outcomes are further inclusive of successfully dealing with the aforementioned models. Some projecting solitons waveforms, including, kink, singular soliton, bell shape, anti-bell shape, and other types of solutions are displayed through a three-dimensional plotline, a plot of contour, and a 2D plot for definite parametric values. It is significant to note that all obtained solutions are verified as accurate by substituting the original equation in each case using the computational software, Maple. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed technique is effective, computationally attractive, and trustworthy to establish more generalized wave solutions.

Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics

Abstract In this article, we investigated the Landau–Ginzburg–Higgs (LGH) equation, focusing on the analysis of isolated soliton solutions and their stability. To compute the isolated soliton solutions, we used the advanced auxiliary equation (AAE) approach, which has proven to be a powerful and efficient method for extracting soliton solutions in various nonlinear partial differential equations (NLPDEs). We provided a detailed explanation, both graphically and physically, of the obtained soliton solutions in this article. Furthermore, we used the linear stability technique to conduct a stability analysis of the LGH equation. Additionally, we studied the bifurcation and stability of the equilibria and performed phase plane analysis of the model. We also provided a discussion on the comparisons between the AAE method and two other well-known approaches: the generalized Kudryashov method and the improved Bernoulli sub-equation function method. The application of the AAE approach in this study demonstrates its effectiveness and capability in analysing and extracting soliton solutions in NLPDEs.

Optical waves solutions for the perturbed Fokas–Lenells equation through two different methods

This study examines the perturbed Fokas–Lenells equation using two methods: the Bernoulli sub-equation function method and the 1/G′-expansion method. A traveling wave transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The perturbed Fokas–Lenells equation arises in optical fibers, and the article constructs various optical soliton solutions, including periodic dark-bright, singular, and periodic singular soliton solutions. The article also presents the necessary constraint conditions for the existence of valid solitons and graphs 2D, 3D, and contour surfaces based on different parameter values.

Diverse wave structures to the modified Benjamin–Bona–Mahony equation in the optical illusions field

In this work, the modified Benjamin–Bona–Mahony equation (MBBME), which has an important role in the optical illusions field, is investigated by three analytical methods, namely, the Bernoulli sub-equation function method, variational method and the Hamiltonian method. Abundant wave solutions expressed in the form of the exponential function, hyperbolic function and trigonometric function are obtained. Finally, the dynamic behaviors of the solutions are illustrated via the 3D plots, 2D contours and 2D curves with the help of Mathematica software, and their corresponding physical interpretations are also given. The ideas in this work are expected to provide some new enlightenment for the study of the exact solutions of the PDEs in optical physics.

Exact Solutions of Nonlinear Time Fractional Schrödinger Equation with Beta-Derivative

This article consists of Improved Bernoulli Sub-Equation Function Method (IBSEFM) to get the new solutions of nonlinear fractional Schrödinger equation described by beta-derivative. Foremost, it is dealt with derivative of Atangana. Secondly, basic properties of the IBSEFM are given. Finally, the proposed method has been applicated to the considered equation to get its new solutions. Moreover, the graphs of the obtained solutions are plotted via Mathematica. It is inferred from the results that IBSEFM is effectual technique for new solutions of nonlinear equations containing conformable derivatives.

Propagation of shock wave of the time‐fractional Gardner Burger equation in a multicomponent plasma using novel analytical method

The time‐fractional Gardner Burger (TFGB) equation is an efficient tool for studying nonlinear fluctuations of different types of the wave profiles, such as the gravity solitary waves in the ocean and the dust ion acoustic wave (DIAW) in a plasma environment. Here, to create an example for the existence of the classical Gardner Burger (GB) equation, a multicomponent plasma environment is considered and a classical GB equation has been derived using the basic governing equation by employing reductive perturbation technique (RPT). In addition, employing Agrawal's technique, the classical GB equation is transformed to the TFGB equation by applying the Riesz fractional derivative to the time‐fractional term. To solve TFGB equations, a new approach, utilizing the improved Bernoulli subequation function method (IBSEFM), is adopted. The physical structures of the solution are explored through some two‐ and three‐dimensional figures, and the significant impact of Burgers term and fractional order of the TFGB equation are demonstrated numerically. Finally, the wave dynamics of the DIAW in the parametric space for the present plasma environment are described from the numerical understanding.

Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations

Abstract This paper applies a powerful scheme, namely Bernoulli sub-equation function method, to some partial differential equations with high non-linearity. Many new travelling wave solutions, such as mixed dark-bright soliton, exponential and complex domain, are reported. Under a suitable choice of the values of parameters, wave behaviours of the results obtained in the paper – in terms of 2D, 3D and contour surfaces – are observed.

New exact solutions of space-time fractional Schr¨odinger-Hirota equation

In this study, improved Bernoulli sub-equation function method (IBSEFM) is presented to construct the exact solutions of the nonlinear conformable fractional Schr¨odinger-Hirota equation (FSHE). By using the traveling wave transformation FSHE turns into the ordinary differential equation (ODE) and by the aid of symbolic calculation software, new exact solutions are obtained. 2D, 3D figures and contour surfaces acquired from the values of the solutions are plotted. The results show that the proposed method is powerful, effective and straightforward for formulating new solutions to various types of nonlinear fractional partial differential equations in applied sciences.

Analysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches

Abstract The variant Boussinesq equation has significant application in propagating long waves on the surface of the liquid layer under gravity action. In this article, the improved Bernoulli subequation function (IBSEF) method and the new auxiliary equation (NAE) technique are introduced to establish general solutions, some fundamental soliton solutions accessible in the literature, and some archetypal solitary wave solutions that are extracted from the broad-ranging solution to the variant Boussinesq wave equation. The established soliton solutions are knowledgeable and obtained as a combination of hyperbolic, exponential, rational, and trigonometric functions, and the physical significance of the attained solutions is speculated for the definite values of the included parameters by depicting the 3D profiles and interpreting the physical incidents. The wave profile represents different types of waves associated with the free parameters that are related to the wave number and velocity of the solutions. The obtained solutions and graphical representations visualize the dynamics of the phenomena and build up the mathematical foundation of the wave process in dissipative and dispersive media. It turns out that the IBSEF method and the NAE are powerful and might be used in further works to find novel solutions for other types of nonlinear evolution equations ascending in physical sciences and engineering.

Computational simulations; propagation behavior of the Riemann wave interacting with the long wave

The analytical solutions to the (2+1)-dimensional integrable Schwarz-Korteweg-de Vries (SKdV) problem are investigated in this article. To get solitary wave formulae such as complex, exponential, hyperbolic, and trigonometric function solutions, the modified Khater technique and the improved Bernoulli sub-equation function method (IBSEFM) are used. Furthermore, their stability characteristics are investigated. The solutions produced using the two methods are compared in order to uncover relationships between them and their features as indicated by the appropriate parameter values.

Two-component plasma and electron trapping’s influence on the potential of a solitary electrostatic wave with the dust-ion-acoustic speed

In this paper, the ion-acoustic waves’ dynamic behavior in plasma and dusty plasma is investigated to explain the electron trapping’s effect on the potential of a solitary electrostatic wave with the dust-ion-acoustic speed in two-components plasma. This study investigates the soliton wave solutions of the Schamel–Korteweg–de Vries (S–KdV) equation that Hans Schamel derived in 1973. In a nonlinear dispersive medium, the S–KdV equation is used to describe a coherent localized wave structure propagates are demonstrated. One-dimensional, unmagnetized plasma with positive ions, negative ions, trapped electrons, and stationary dust grains with both positive and negative charges is studied for the nonlinear propagation of dust-ion-acoustic (DIA) waves. The Bernoulli sub-equation function (BSEF) method is applied to the S–KdV equation for constructing some novel soliton wave solutions that explain the effect of the plasma parameters on the DIA solitary waves. The obtained solitary wave solutions are represented through some different graphs by Mathematica 12 software. Additionally, the axisymmetric pulse propagation is demonstrated in some stream plots. All the obtained solutions are checked by putting them back into the original model.

Studying on Kudryashov-Sinelshchikov dynamical equation arising in mixtures liquid and gas bubbles

In this paper, some new exact traveling and oscillatory wave solutions to the Kudryashov-Sinelshchikov non-linear PDE are investigated by using Bernoulli sub-equation function method. Profiles of obtained solutions are plotted.

Study of the parametric effects on soliton propagation in optical fibers through two analytical methods

The dual-core optical fiber has significant applications in optical electronics for long-wave propagation, especially in telecommunication fibers. The aim of this article is to study the parametric effects on solitary wave propagation and characteristic aspects of long-wave traveling through optical fibers by establishing some standard and wide-spectrum solutions via the improved Bernoulli sub-equation function (IBSEF) method and the new auxiliary equation (NAE) approach. The investigated solitary wave solutions are ascertained as an integration of hyperbolic, exponential, rational and trigonometric functions and can be extensively applicable in optics. The physical significance of the solutions attained is illustrated for the definite values of the included parameters through depicting the 3D profiles. The solitons profile represents different types of waves associated with the free parameters which are related to wave number and velocity. It turns out that the solutions obtained through both the methods are potential and can be used to interpret signals in telecommunication fibers and other works which can reduce casualties that ensue in essence.

An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional equation

The nonlinear conformable time-fractional modified Camassa-Holm (MCH) equation plays an important role in physics. It is an interesting model to define change waves with weak nonlinearity. The aim of this study is to present the new exact solutions of conformable time-fractional MCH equation. For this purpose, an effective method which is the Improved Bernoulli Sub-Equation Function Method (IBSEFM) has been used. The 2D and 3D graphs and contour surfaces acquired from the values of the solutions are plotted by the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable time-fractional partial differential equations arising in mathematical physics.

Stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation

The low-pass nonlinear electrical transmission lines and the Cahn-Allen equation are important nonlinear model equations to figure out different tangible systems, namely, electrical engineering, fluid dynamics etc. The contrivance of this study is to introduce advanced Bernoulli sub-equation function method to search for stable and effective solitary solutions of the described wave equations. Stable solitary solutions are reported as an integration of exponential functions, hyperbolic functions, etc., and the graphical implications for specific values of the corresponding parameters are explained in the solutions obtained in order to uncover the inmost structure of the tangible phenomena. It is establish that the IBSEF method is reliable, contented and might be used in further works to found ample novel soliton solutions for other types of NLEEs arising in physical science and engineering.

Study of the parametric effect of self-control waves of the Nizhnik-Novikov-Veselov equation by the analytical solutions

A significant model of the preserved scalar nucleons associated with neutral scalar masons is the (2 + 1)-dimensional Nizhnik-Novikov-Veselov equation. Wave solutions are computed concerning exponential, hyperbolic and trigonometric structures balancing the powers in linear and nonlinear terms of the highest order from which scores of typical wave profiles including kink, bell-shape soliton, singular soliton, bright soliton, periodic soliton, dark soliton and dark-bright waves have been extracted. In this article, we have demonstrated that the nature of wave profiles fluctuates regarding the change of the free parameters associated with it and are principally controlled by the linear and nonlinear effect. The effects of other free parameters and wave velocity in the wave profile have also been discussed. The wave solutions are arranged through the improve Bernoulli sub-equation function (IBSEF) method and extending higher dimensional sine-Gordon expansion (SGE) method. Comparing the results obtained from both methods and analyzed the solutions by outlining figures for different values of corresponding variables, and it is detected that the attributes of those solutions are pivotal on the selection of parameters.

Extraction Complex Properties of the Nonlinear Modified Alpha Equation

This paper applies one of the special cases of auxiliary method, which is named as the Bernoulli sub-equation function method, to the nonlinear modified alpha equation. The characteristic properties of these solutions, such as complex and soliton solutions, are extracted. Moreover, the strain conditions of solutions are also reported in detail. Observing the figures plotted by considering various values of parameters of these solutions confirms the effectiveness of the approximation method used for the governing model.