Calogero – Bogoyavlenskii – Schiff Research Articles

On the solutions of space-time fractional CBS and CBS-BK equations describing the dynamics of Riemann wave interaction

In this paper, Kudryashov and modified Kudryashov methods are implemented for the first time to compute new exact traveling wave solutions of the space-time fractional [Formula: see text]-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation and Calogero–Bogoyavlenskii–Schiff and Bogoyavlensky Konopelchenko (CBS-BK) equation. With the help of wave transformation, the aforementioned fractional differential equations are converted into nonlinear ordinary differential equations. The purpose of this paper is to devise novel exact solutions for the space-time-fractional [Formula: see text]-dimensional CBS and the space-time-fractional CBS-BK equations by utilizing the Kudryashov and modified Kudryashov techniques. The solutions, thus, acquired are demonstrated in figures by choosing appropriate values for the parameters. The solutions derived take the form of various wave patterns, including the kink type, the anti-kink type and the singular kink wave solutions. The obtained solutions are indeed beneficial to analyze the dynamic behavior of fractional CBS and CBS-BK equations in describing the interesting physical phenomena and mechanisms. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. The techniques presented here are very simple, efficacious and plausible and hence can be employed to attain new exact solutions for fractional PDEs.

Multiple soliton and traveling wave solutions for the negative-order-KdV-CBS model

The (3+1)-dimensional new negative-order-KdV-CBS model is investigated in this study. The suggested model combines the Korteweg-de Vries (KdV) and Calogero-Bogoyavlenskii-Schiff (CBS) equations. This research provides multiple soliton solutions and traveling wave solutions for the KdV-CBS model. Multiple exp-function methods have been used for extracting soliton solutions. For this aim, the extended sinh-Gordon equation expansion approach was selected to get traveling wave solutions. The findings are graphically examined by selecting appropriate values for arbitrary parameters.

Exact solutions of (1 + 1)-dimensional integro-differential Ito, KP hierarchy, CBS, MCBS and modified KdV-CBS equations.

The present study computes the Lie symmetries and exact solutions of some problems modeled by nonlinear partial differential equations. The (1 + 1)-dimensional integro-differential Ito, the first integro-differential KP hierarchy, the Calogero-Bogoyavlenskii-Schiff (CBS), the modified Calogero-Bogoyavlenskii-Schiff (CBS), and the modified KdV-CBS equations are some of the problems for which we want to find new exact solutions. We employ similarity variables to reduce the number of independent variables and inverse similarity transformations to obtain exact solutions to the equations under consideration. The sine-cosine method is then utilized to determine the exact solutions.

New modified tanh-function method for nonlinear evolution equations

The aim of this paper is to calculate the Traveling waves solutions by using a new technique which is called modified tanh-function method, which are successfully performed to get analytical solutions for Korteweg-deVries (KdV)–Burgers’ equation and (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. As a result, when the equation parameters are taken as special values, some new solitary wave solution are obtained. Moreover we find in this work that the modified tanh-function method give some new results which are easier and faster to compute by the help of a symbolic computation system. The results obtained were compared with standard tanh-function method.

M-lump solutions to the (2+1)-dimensional generalized Calogero–Bogoyavlenshii–Schiff equation

We study M-lump solutions of the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation. The amplitude of these solutions are rational in their independent variable, owing to they are constructed via taking “long wave” limit of the corresponding N-soliton solutions acquired by direct methods. The dynamic properties of these solutions are shown in figures. The results of this work may be advantageous to understand the propagation of localized wave.

Solitary Wave Solutions to the Modified Zakharov–Kuznetsov and the (2 + 1)-Dimensional Calogero–Bogoyavlenskii–Schiff Models in Mathematical Physics

The modified Zakharov–Kuznetsov (mZK) and the (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) models convey a significant role to instruct the internal structure of tangible composite phenomena in the domain of two-dimensional discrete electrical lattice, plasma physics, wave behaviors of deep oceans, nonlinear optics, etc. In this article, the dynamic, companionable, and further broad-spectrum exact solitary solitons are extracted to the formerly stated nonlinear models by the aid of the recently enhanced auxiliary equation method through the traveling wave transformation. The implication of the soliton solutions attained with arbitrary constants can be substantial to interpret the involuted phenomena. The established soliton solutions show that the approach is broad-spectrum, efficient, and algebraic computing friendly and it may be used to classify a variety of wave shapes. We analyze the achieved solitons by sketching figures for distinct values of the associated parameters by the aid of the Wolfram Mathematica program.

Conservation Laws and Travelling Wave Solutions for a Negative-Order KdV-CBS Equation in 3+1 Dimensions

In this paper, we study a new negative-order KdV-CBS equation in (3+1) dimensions which is a combination of the Korteweg-de Vries (KdV) equation and Calogero–Bogoyavlenskii–Schiff (CBS) equation. Firstly, we determine the Lie point symmetries of the equation and conservation laws by using the multiplier method. The conservation laws will be used to obtain a triple reduction to a second order ordinary differential equation (ODE), which lead to line travelling waves and soliton solutions. Such solitons are obtained via the modified form of simple equation method and are displayed through three-dimensional plots at specific parameter values to lend physical meaning to nonlinear phenomena. It illustrates that these solutions might be extremely beneficial in understanding physical phenomena in a variety of applied mathematics areas.

Analytical solutions of (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics/plasma physics using the New Kudryashov method

This study investigates various analytic soliton solutions of the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation in fluid dynamics and plasma physics using a recently introduced technique which is the New Kudryashov method. Moreover, it is examined how the wave propagation in both directions represented by the CBS equation occurs. The considered equation describes the interaction of the long propagating wave in the x axis with the Riemann propagating wave along the y axis. To get traveling wave solutions of the CBS equation, it is transformed into a nonlinear ordinary differential equation (NLODE) using a proper wave transformation. Supposing that the NLODE has some solutions in the form provided by the method, one can obtain a nonlinear system of algebraic equations. The unknowns in the system can be found by solving the system via computer algebraic systems such as Mathematica and Maple, etc. Substituting the unknowns into the trial solutions provided by the method, we get the solutions of the NLODE. Then, putting wave transformations back into the solutions of NLODE, we get the solutions of the considered CBS equation. We present the 2D, 3D and contour plots to illustrate the physical behavior of the obtained solutions using the appropriate parameters. Besides, the schematic representation of wave motion of the soliton along both spatial axes and its interpretation are given. The used novel technique can be used for a wide range of partial differential equations (PDEs) in the real world. It is expected that the derived soliton solutions might be helpful for better understanding the wave behavior and so, it might contribute to future studies in various disciplines.

The nonlocal potential transformation method for solitary wave packets of a shock-breaking dynamics system

In this work, we employ the potential similarity transformation method to derive some solitary wave packet solutions for the Calogero-Bogoyavlenskii-Schiff (CBS) equation. Exploiting nonsingular local multipliers, a set of local conservation laws is presented for the equation. The nonlocally related partial differential equation (PDE) systems were observed. Considering the nonlocal-related PDE systems, we present 46 analytic solutions of the CBS equation that involves several arbitrary functions, enabling the spread of the equation’s solution to understand Riemann waves dynamics extensively. These are helpful in severalapplications such as electromagnetic waves in an optical tsunami in fibers, magneto-sonic and ion in plasma, the acoustic waves in compressible fluids, surface, and interval waves in oceans, tidal and tsunami in rivers.

The closed-form soliton solutions of the time-fraction Phi-four and (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff model using the recent approach

The Phi-four (PF) model is a variation of the familiar Klein–Fock–Gordon​ (KFG) model. Quantum effects, ultra-short pulse propagation in optical fibers, character of de-Broglie waves, wave-particle duality, composite particles of spineless relativistic, etc. are investigated through the PF model. The interface of Riemann waves in two spatial dimensions is explained by the Calogero–Bogoyavlenskii–Schiff (CBS) model. Many physical phenomena such as, the magneto-sound waves in plasmas, tsunami and tidal in rivers, and the internal waves in oceans can be describe through the Riemann wave. This article investigates the unique and wide-ranging analytic soliton solutions to the above-stated models in the sense of the fractional conformable derivative by using the auxiliary equation technique in terms of exponential, trigonometric, hyperbolic, and rational functions. To analyze the underlying wave structures and to examine the active behavior of the obtained solutions, three-dimensional (3D) and contour graphs are plotted using the Wolfram Mathematica program. The attained solutions demonstrate that the method is compatible, effective, and scientifically efficient.

Exact solutions of the different dimensional CBS equations in mathematical physics

The focus of this article is to find the exact traveling wave solutions to Calogero–Bogoyavlenskii–Schiff (CBS) equation by employing the exp−Φζ expansion method. The traveling wave solutions are expressed by the hyperbolic function solutions, trigonometric function solutions and rational function solutions. 3D and 2D charts are drawn from the obtained solutions by Mathematica software and discussed the graphical behavior of solutions. Finally, an effective mathematical tools for solving nonlinear evolution equations in mathematical physics are provided by the proposed method. This method is more common and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations.

Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation

This paper studies the analytical and semi-analytic solutions of the generalized Calogero–Bogoyavlenskii–Schiff (CBS) equation. This model describes the (2 + 1)–dimensional interaction between Riemann-wave propagation along the y-axis and the x-axis wave. The extended simplest equation (ESE) method is applied to the model, and a variety of novel solitary-wave solutions is given. These solitary-wave solutions prove the dynamic behavior of soliton waves in plasma. The accuracy of the obtained solution is verified using a variational iteration (VI) semi-analytical scheme. The analysis and the match between the constructed analytical solution and the semi-analytical solution are sketched using various diagrams to show the accuracy of the solution we obtained. The adopted scheme’s performance shows the effectiveness of the method and its ability to be applied to various nonlinear evolution equations.

The generalized (2 + 1) and (3+1)-dimensional with advanced analytical wave solutions via computational applications

The analytical solutions for an important generalized Nonlinear evolution equations NLEEs dynamical partial differential equations (DPDEs) that involve independent variables represented by the (2 + 1)-dimensional breaking soliton equation, the (2 + 1)-dimensional Calogero--Bogoyavlenskii--Schiff (CBS) equation, and the (2 +1)-dimensional Bogoyavlenskii's breaking soliton equation (BE), and some new exact propagating solutions to a generalized (3+1)-dimensional KP equation with variable coefficients are constructed by using a new algorithm of the first integral method (NAFIM) and determined some analytical solutions by appointing special values of the parameters. In addition to that, we showed a new variety and unique travelling wave solutions by graphical illustration with symbolic computations.

A variety of completely integrable Calogero–Bogoyavlenskii–Schiff equations with time-dependent coefficients

PurposeThe purpose of this paper is to introduce a variety of new completely integrable Calogero–Bogoyavlenskii–Schiff (CBS) equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for each of the developed models.Design/methodology/approachThe newly developed models with time-dependent coefficients have been handled by using the simplified Hirota’s method. Moreover, multiple complex soliton solutions are derived by using complex Hirota’s criteria.FindingsThe developed models exhibit complete integrability, for specific determined functions, by investigating the compatibility conditions for each model.Research limitations/implicationsThe paper presents an efficient algorithm for handling integrable equations with analytic time-dependent coefficients.Practical implicationsThe work presents new integrable equations with a variety of time-dependent coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.Social implicationsThis study presents useful algorithms for finding and studying integrable equations with time-dependent coefficients.Originality/valueThe paper gives new integrable CBS equations which appear in propagation of waves and provide a variety of multiple real and complex soliton solutions.

Lump, multi-lump, cross kinky-lump and manifold periodic-soliton solutions for the (2+1)-D Calogero–Bogoyavlenskii–Schiff equation

A bilinear form of the (2+1)-dimensional nonlinear Calogero–Bogoyavlenskii–Schiff (CBS) model is derived using a transformation of dependent variable, which contain a controlling parameter. This parameter can control the direction, wave height and angle of the traveling wave. Based on the Hirota bilinear form and ansatz functions, we build many types of novel structures and manifold periodic-soliton solutions to the CBS model. In particular, we obtain entirely exciting periodic-soliton, cross-kinky-lump wave, double kinky-lump wave, periodic cross-kinky-lump wave, periodic two-solitary wave solutions as well as breather style of two-solitary wave solutions. We present their propagation features via changing the existence parametric values in graphically. In addition, we estimate a condition that the waves are propagated obliquely for η≠0 , and orthogonally for η=0.

Investigation of breaking dynamics for Riemann waves in shallow water

Abstract Breaking dynamics of Riemann waves, has a profound impact on studying the dynamics of shock-breaking systems. Calogero– Bogoyavlenskii– Schiff (CBS) equation describes the interaction between Riemann propagating wave along y-axis with long propagating wave along x-axis. Two extensions of the CBS (2+1)-dimensional equation are investigated. Optimization of the commutative product for the nonlocal symmetry is constructed and two successive symmetry reductions reduced the equations to ordinary differential equations (ODEs). Hidden symmetries are used in the second reduction step. The singular manifold method (SMM) is then applied to the ordinary differential equations. This results in a Schwarzian derivative whose solution leads to Cuspon and Kink waves.

A (2+1)-dimensional shallow water equation and its explicit lump solutions

We introduce a new (2+1)-dimensional equation by modifying the potential form of the Calogero–Bogoyavlenskii–Schiff (CBS) equation. By applying the Hirota bilinear method, we construct explicit lump solutions to this new equation and establish necessary and sufficient conditions that guarantee that the solutions are analytic and rationally localized in all directions in space. We also depict the evolution of the profiles of some selected lump solutions with three-dimensional and contour plots. It is immediately observed that the lump solutions generated are solitary wave type solutions as is the case with the KP equation.

Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair

In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Backlund transformation of the Calogero-Bogoyavlenskii-Schiff equation are derived by using the singular manifold method (SMM). The optimal Lie infinitesimals of the Lax pair are obtained. The detected Lie infinitesimals contain eight unknown functions. These functions are optimized through the commutator table. The eight unknown functions are evaluated through the solution of a set of linear differential equations, in which solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ordinary differential equations (ODEs). The solitary wave solutions of Calogero-Bogoyavlenskii-Schiff equation Lax pair’s show soliton and kink waves. The obtained similarity solutions are plotted for different arbitrary functions and compared with previous analytical solutions. The comparison shows that we derive new solutions of Calogero-Bogoyavlenskii-Schiff equation by using the combination of two methods, which is different from the previous findings.

New exact solutions for the conformable space-time fractional KdV, CDG, (2+1)-dimensional CBS and (2+1)-dimensional AKNS equations

ABSTRACTIn the present paper, expansion method is applied to the space-time fractional third order Korteweg-De Vries (KdV) equation, space-time fractional Caudrey-Dodd-Gibbon (CDG) equation, space-time fractional (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and space-time fractional (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation. Here, the fractional derivatives are described in conformable sense. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. The graphs for some of these solutions have been presented by choosing suitable values of parameters to visualize the mechanism of the given nonlinear fractional evolution equations.

Lie symmetry analysis and invariant solutions of $$\varvec{(3+1)}$$ ( 3 + 1 ) -dimensional Calogero–Bogoyavlenskii–Schiff equation

It is shown that the novel Lie group of transformations method is a competent and prominent tool in solving nonlinear partial differential equations(PDEs) in mathematical physics. Lie group analysis is used to carry out the similarity reduction and exact solutions of the (3 + 1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. This research deals with the similarity solutions of CBS equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reduction for the CBS equation. For the different Lie algebra, Lie symmetry method reduced (3 + 1)-dimensional CBS equation into new (2 + 1)-dimensional partial differential equations and again using Lie symmetry method these PDEs are reduced into various ordinary differential equations(ODEs).