Calogero Bogoyavlenskii Schiff Equation Research Articles

Residual Symmetry and Interaction Solutions of the (2+1)-Dimensional Generalized Calogero–Bogoyavlenskii–Schiff Equation

In this paper, we investigate the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (gCBS) equation by using the residual symmetry analysis and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the gCBS equation is localized into a Lie point symmetry in a prolonged system and a new Bäcklund transformation of this equation is obtained. By applying the standard Lie symmetry method to the prolonged gCBS system, new symmetry reduction solutions of the gCBS equation are obtained. The gCBS equation is proved to be CRE integrable and new Bäcklund transformations of it are obtained, from which interaction solutions between solitons and periodic waves are generated and analyzed.

Complexiton and interaction solutions to a specific form of extended Calogero–Bogoyavlenskii–Schiff equation via its bilinear form

Abstract In this article, we are focusing on an extended Calogero–Bogoyavlenskii–Schiff equation which was altered originally from a new generalized fourth-order nonlinear differential equation that obtained from Calogero–Bogoyavlenskii–Schiff equation. We apply simplified Hirota method, which is an exclusive form of the direct Hirota bilinear method, to a specific form of a new generalized fourth-order nonlinear differential equation. The key point in applicability of referred method is attainability proper forms of dispersion relations and phase shifts. Through this procedure, we present different types of solutions for three different cases. We also give constrictions for each solution type in this work so that readers can distinguish differences among types of solutions. In addition, we introduce some graphical representations for obtained solutions, even for existence of complexiton and interaction solutions.

Numerical calculation and characteristics of N-periodic waves of a (4+1)-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff equation in fluid physics and plasma physics

Numerical calculation and characteristics of N-periodic waves of a (4+1)-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff equation in fluid physics and plasma physics

Solitons, nonlinear wave transitions and characteristics of quasi-periodic waves for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics and plasma physics

A (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation describing many nonlinear phenomena in fluid dynamics and plasma physics is considered. The N-solitons and breathers are obtained by basing on its Hirota’s bilinear form and taking the complex conjugate condition on parameters of N-solitons. What is more, breathers can be transformed into a series of nonlinear localized waves by the mechanism of breather transformation. Then through the multi-dimensional Riemann-theta function and the bilinear method, the high-dimensional complex three-periodic wave solutions are constructed systematically, which are the generalization of one-periodic wave and two-periodic wave solutions. By a limiting procedure, the asymptotic relations between the quasi-periodic waves and solitons are strictly established. Additionally, a novel analytical method of characteristic line is introduced to analyze statistically the dynamical characteristics of the quasi-periodic waves. The analytical method employed in this paper can be further extended to investigate the other complex high-dimensional nonlinear integrable equations.

Investigation of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation by generalized Kudryashov method and two variable $$\big(\frac{G'}{G},\frac{1}{G}\big)$$-expansion method

Investigation of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation by generalized Kudryashov method and two variable $$\big(\frac{G'}{G},\frac{1}{G}\big)$$-expansion method

Exact traveling wave solutions of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation using extended trial equation method and modified auxiliary equation method

Exact traveling wave solutions of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation using extended trial equation method and modified auxiliary equation method

Solitary travelling wave profiles to the nonlinear generalized Calogero–Bogoyavlenskii–Schiff equation and dynamical assessment

Solitary travelling wave profiles to the nonlinear generalized Calogero–Bogoyavlenskii–Schiff equation and dynamical assessment

The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics

Abstract The generalized (2+1)-dimensional stochastic Calogero–Bogoyavlenskii Schiff equation (SCBSE) driven by a multiplicative Brownian motion is taken into consideration. The Riccati equation mapping and He’s semi-inverse methods are utilized to obtain the rational function, hyperbolic function, and trigonometric function for SCBSE. We expand some solution from previous studies. The acquired solutions of SCBSE may explain many exciting physical phenomena because it is widely used in plasma physics and fluid dynamics. Also, it explains the relationship between the Riemann y-axis propagating wave and the long x-axis propagating wave. Using a variety of 2D and 3D graphs, we illustrate how the Brownian motion influences the exact solutions of SCBSE.

Rational and semi-rational solutions of a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Rational and semi-rational solutions of a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Exploring the wave solutions of a nonlinear non-local fractional model for ocean waves

In this research, analytical and semi-analytical soliton solutions for the nonlinear fractional (2 + 1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation (FCBSE) in the non-local form are obtained using recent computational and numerical methods. The FCBSE is a significant model for investigating various phenomena, such as internal ocean waves, tsunamis, river tidal waves, and magneto-sound waves in plasma. The constructed solution helps in understanding the interaction between a long wave moving along the x-axis and a Riemann wave propagating along the y-axis. Various analytical solutions, such as exponential, trigonometric, and hyperbolic, have been formulated differently for this model, which is a specific derivation of the well-known Korteweg–de Vries equation. Density charts in two and three dimensions are used to visualize the behavior of a single soliton in reality through simulations. The results demonstrate the effectiveness of the employed numerical scheme and various methods to ensure the consistency of computational and approximation answers. Overall, this study demonstrates the potential of recent computational and numerical techniques for solving nonlinear mathematical and physical problems.

The Propagating Exact Solitary Waves Formation of Generalized Calogero–Bogoyavlenskii–Schiff Equation with Robust Computational Approaches

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures.

NEW PROMISING AND CHALLENGES OF THE FRACTIONAL CALOGERO–BOGOYAVLENSKII–SCHIFF EQUATION

The Calogero–Bogoyavlenskii–Schiff equation is an important nonlinear evolution model to describe the propagation of Riemann waves. A fractional Calogero–Bogoyavlenskii–Schiff is described based on the conformable derivative for the first time. Some new soliton solutions are acquired with the aid of the extended fractional [Formula: see text] function method and fractional variable method. The two novel mathematical methods are very efficient and concise, which can also be utilized to solve other fractional evolution equations. Furthermore, these derived soliton solutions are illustrated by some 3D and 2D graphs with different fractal parameters and fractal dimensions, which might be helpful to study in plasma physics.

Symmetry analysis and soliton–cnoidal solutions of the negative-order Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics

In this paper, a special integrable negative-order Calogero–Bogoyavlenskii–Schiff equation (nCBS) in fluid mechanics is studied by means of the symmetry reduction method and consistent tanh expansion method. The Painlevé integrability is investigated to confirm the compatibility conditions. This integrable nCBS equation has been transformed into different reduction equations and the corresponding invariant solutions with arbitrary functions are obtained. The corresponding structures of the invariant solutions for the nCBS equation are also shown graphically. At last, new types of soliton–cnoidal interaction solutions for the nCBS equation are presented through the consistent tanh expansion method on the basis of the truncated Painlevé expansion.

Construction of complexiton-type solutions using bilinear form of Hirota-type

Abstract In this paper, based on the Hirota bilinear form and the extended transformed rational function method, complexiton solutions have been found of the Hirota–Satsuma–Ito (HSI) equation and generalized Calogero–Bogoyavlenskii–Schiff equation through a direct symbolic computation with Maple. This method is the improved form of the transformed rational function method. The obtained complexiton solutions, includes trigonometric and hyperbolic trigonometric solutions, have verified utilizing Hirota bilinear forms. Also, a graphical representation of the obtained solutions is given.

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.

Soliton solutions of some nonlinear evolution equations in shallow water theory

This paper proposes some soliton solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation, a (3+1) dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov and a Variant Boussinesq equations. We presented a exponential function method (EFM) for solving the considered evolution equations. The solitary, cuspon and periodic wave solutions are acquired and presented graphically. The soliton solutions of considered equations play a typical role for expressing kinds of wave transmission in any natural instance, particularly in shallow wave kinetics. The results suggest that EFM is effective and useful technique to handle non-linear engineering problems in oceans.

Symmetries and dynamic wave solutions for ([formula omitted])-dimensional potential Calogero–Bogoyavlenskii–Schiff equation

In this paper, the symmetry method and generalized Kudryashov method are utilized for constructing new dynamic exact solutions to a (3+1)-dimensional potential Calogero–Bogoyavlenskii–Schiff equation (CBS) equation. The symmetry method was employed to obtain symmetry vectors and symmetries to use it for reducing the governing equation for various types of ordinary differential equations. The obtained ordinary differential equations are solved by using generalized Kudryashov method. The obtained exact solutions include solitary wave solutions, kink-wave and anti-kink wave solutions, and parabolic wave solutions. Using numerical computing, with proper selection of the constants, the dynamical behaviors of the solutions were shown through 3-dimensional and 2-dimensional figures.

Investigation of wave solutions and conservation laws of generalized Calogero–Bogoyavlenskii–Schiff equation by group theoretic method

This work is focused to analyze the generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) by the Lie symmetry method. GCBS equation has been utilized to explain the wave profiles in soliton theory. GCBSE was constructed by Bogoyavlenskii and Schiff in different ways (explained in the introduction section). With the aid of Lie symmetry analysis, we have computed the symmetry generators of the GCBSE and commutation relation. We observed from the commutator table, translational symmetries make an Abelian algebra. Then by using the theory of Lie, we have discovered the similarity variables, which are used to convert the supposed nonlinear partial differential equation (NLPDE) into a nonlinear ordinary differential equation (NLODE). Using the new auxiliary method (NAM), we have to discover some new wave profiles of GCBSE in the type of few trigonometric functions. These exits some parameters which we give to some suitable values to attain the different diagrams of some obtained solutions. Further, the GCBSE is presented by non-linear self-adjointness, and conserved vectors are discovered corresponding to each generator.

Abundant fission and fusion solutions in the ($$2+1$$)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Abundant fission and fusion solutions in the ($$2+1$$)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation