Dimensional Breaking Soliton Equation Research Articles

Novel Analytical Approach for the Space-Time Fractional (2+1)-Dimensional Breaking Soliton Equation via Mathematical Methods

The aim of this work is to build novel analytical wave solutions of the nonlinear space-time fractional (2+1)-dimensional breaking soliton equations, with regards to the modified Riemann–Liouville derivative, by employing mathematical schemes, namely, the improved simple equation and modified F-expansion methods. We used the fractional complex transformation of the concern fractional differential equation to convert it for the solvable integer order differential equation. After the successful implementation of the presented methods, a comprehensive class of novel and broad-ranging exact and solitary travelling wave solutions were discovered, in terms of trigonometric, rational and hyperbolic functions. Hence, the present methods are reliable and efficient for solving nonlinear fractional problems in mathematics physics.

SITEM for the Conformable Space-Time Fractional (2+1)-Dimensional Breaking Soliton, Third-Order KdV and Burger's Equations

In the present paper, new analytical solutions for the conformable space-time fractional (2+1)-dimensional breaking soliton, third-order KdV and Burger's equations are obtained by using the simplified tan(ϕ(ξ)2)tan⁡(ϕ(ξ)2)-expansion method (SITEM). Here, fractional derivatives are described in conformable sense. The obtained traveling wave solutions are expressed by the trigonometric, hyperbolic, exponential and rational functions. Simulation of the obtained solutions are given at the end of the paper.

A study of Bogoyavlenskii’s (2+1)-dimensional breaking soliton equation: Lie symmetry, dynamical behaviors and closed-form solutions

This paper employs the Lie symmetry analysis to investigate novel closed-form solutions to a (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation. This Lie symmetry technique, used in combination with Maple’s symbolic computation system, demonstrates that the Lie infinitesimals are dependent on five arbitrary parameters and two independent arbitrary functions f1(t) and f2(t). The invariance criteria of Lie group analysis are used to construct all infinitesimal vectors, commutative relations of their examined vectors, a one-dimensional optimal system and then several symmetry reductions. Subsequently, Bogoyavlenskii’s breaking soliton (BBS) equation is reduced into several nonlinear ODEs by employing desirable Lie symmetry reductions through optimal system. Explicit exact solutions in terms of arbitrary independent functions and other constants are obtained as a result of solving the nonlinear ODEs. These established results are entirely new and dissimilar from the previous findings in the literature. The physical behaviors of the gained solutions illustrate the dynamical wave structures of multiple solitons, curved-shaped wave–wave interaction profiles, oscillating periodic solitary waves, doubly-solitons, kink-type waves, W-shaped solitons, and novel solitary waves solutions through 3D plots by selecting the suitable values for arbitrary functional parameters and free parameters based on numerical simulation. Eventually, the derived results verify the efficiency, trustworthiness, and credibility of the considered method.

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and (2+1)-dimensional breaking soliton equations via a generalized exponential rational function (GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons, double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons, elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.

Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the generalized (3+1)-dimensional breaking soliton equation using optimal system of Lie subalgebra

Nonlinear evolution equations (NLEEs) are primarily relevant to nonlinear complex physical systems in a wide range of fields, including ocean physics, plasma physics, chemical physics, optical fibers, fluid dynamics, biology physics, solid-state physics, and marine engineering. This paper investigates the Lie symmetry analysis of a generalized (3+1)-dimensional breaking soliton equation depending on five nonzero real parameters. We derive the Lie infinitesimal generators, one-dimensional optimal system, and geometric vector fields via the Lie symmetry technique. First, using the three stages of symmetry reductions, we converted the generalized breaking soliton (GBS) equation into various nonlinear ordinary differential equations (NLODEs), which have the advantage of yielding a large number of exact closed-form solutions. All established closed-form wave solutions include special functional parameter solutions, as well as hyperbolic trigonometric function solutions, trigonometric function solutions, dark-bright solitons, bell-shaped profiles, periodic oscillating wave profiles, combo solitons, singular solitons, wave-wave interaction profiles, and various dynamical wave structures, which we present for the first time in this research. Eventually, the dynamical analysis of some established solutions is revealed through three-dimensional sketches via numerical simulations. Some of the new solutions are often useful and helpful for studying the nonlinear wave propagation and wave-wave interactions of shallow water waves in many new high-dimensional 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.

Exact solutions of space-time fractional (2+1)-dimensional breaking soliton equation

This paper suggests a direct algebraic method for finding exact solutions of the space-time fractional (2+1)-dimensional breaking soliton equation. The solution procedure is reduced to solve a large system of algebraic equations, which is then solved by Wu?s method.

Novel solitons solutions of two different nonlinear PDEs appear in engineering and physics

Abstract In this piece of research, our aim is to investigate the novel solitons solutions of nonlinear (4+1)-dimensional Fokas equation (FE) and (2+1)-dimensional Breaking soliton equation (BSE) via new extended direct algebraic method. New acquired solutions are bright, singular, dark, periodic singular, combined dark-bright and combined dark-singular solitons along with hyperbolic and trigonometric functions solutions. We achieved different kinds of solitons solutions contain key applications in engineering and physics. By taking the appropriate values of these parameters, numerous novel structures are also plotted. These solutions define the wave performance of the governing models, actually. Furthermore, the physical understanding of the acquired solutions is revealed in forms of 3-D, 2-D and contour graphs for different appropriate parameters. From results, we conclude that the applied computational method is straight, talented and can be applied in more complex phenomena for such models.

SITEM for the conformable space-time fractional Boussinesq and (2 + 1)-dimensional breaking soliton equations

In the present paper, new analytical solutions for the space-time fractional Boussinesq and (2 + 1)-dimensional breaking soliton equations are obtained by using the simplified tan(ϕ(ξ)2)-expansion method. Here, fractional derivatives are defined in the conformable sense. To show the correctness of the obtained traveling wave solutions, residual error function is defined. It is observed that the new solutions are very close to the exact solutions. The solutions obtained by the presented method have not been reported in former literature.

Closed-Form Solutions and Conserved Vectors of a Generalized (3+1)-Dimensional Breaking Soliton Equation of Engineering and Nonlinear Science

In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the (G′/G)-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov’s theorem.

The extended auxiliary equation mapping method to determine novel exact solitary wave solutions of the nonlinear fractional PDEs

Abstract In this paper, some new nonlinear fractional partial differential equations (PDEs) have been considered.Three models are including the space-time fractional-order Boussinesq equation, space-time (2 + 1)-dimensional breaking soliton equations, and space-time fractional-order SRLW equation describe the behavior of these equations in the diverse applications. Meanwhile, the fractional derivatives in the sense of β-derivative are defined. Some fractional PDEs will convert to the considered ordinary differential equations by the help of transformation of β-derivative. These equations are analyzed utilizing an integration scheme, namely, the extended auxiliary equation mapping method. The different kinds of traveling wave solutions, solitary, topological, dark soliton, periodic, kink, and rational, fall out as a by-product of this scheme. Finally, the existence of the solutions for the constraint conditions is also shown. The outcome indicates that some fractional PDEs are used as a growing finding in the engineering sciences, mathematical physics, and so forth.

Deeper investigations of the (4 + 1)-dimensional Fokas and (2 + 1)-dimensional Breaking soliton equations

The main aim of this paper is to investigate the various dimensional nonlinear Fokas and Breaking soliton equations via a powerful analytical method, namely, sine-Gordon expansion method. Many new solutions such as complex combined dark-bright soliton solutions, singular and hyperbolic functions are derived. Choosing the suitable values of these parameters, various novel simulations are also plotted. Such results explain the wave behavior of the governing models, physically.

Some Nonlinear Fractional PDEs Involvingβ-Derivative by Using Rationalexp−Ωη-Expansion Method

In this article, some new nonlinear fractional partial differential equations (PDEs) (the space-time fractional order Boussinesq equation; the space-time (2 + 1)-dimensional breaking soliton equations; and the space-time fractional order SRLW equation) have been considered, in which the treatment of these equations in the diverse applications are described. Also, the fractional derivatives in the sense ofβ-derivative are defined. Some fractional PDEs will convert to consider ordinary differential equations (ODEs) with the help of transformationβ-derivative. These equations are analyzed utilizing an integration scheme, namely, the rationalexp−Ωη-expansion method. Different kinds of traveling wave solutions such as solitary, topological, dark soliton, periodic, kink, and rational are obtained as a by product of this scheme. Finally, the existence of the solutions for the constraint conditions is also shown. The outcome indicates that some fractional PDEs are used as a growing finding in the engineering sciences, mathematical physics, and so on.

Solitary wave solutions to some nonlinear fractional evolution equations in mathematical physics

The objective of this article is to construct new and further general analytical wave solutions to some nonlinear evolution equations of fractional order in the sense of the modified Riemann-Liouville derivative relating to mathematical physics, namely, the space-time fractional Fokas equation, the time fractional nonlinear model equation and the space-time fractional (2 + 1)-dimensional breaking soliton equation by exerting a rather new mechanism (G'/G,1/G) -expansion method. We use the fractional complex transformation and associate the fractional differential equations to the solvable integer order differential equations. A comprehensive class of new and broad-ranging exact traveling and solitary wave solutions are revealed in terms of trigonometric, rational and hyperbolic functions. The attained wave solutions are sketched graphically by using Mathematica and make a comparison to the results attained by the presented technique with other techniques in a comprehensive manner. It is notable that the method can be considered as a reduction of the reputed (G'/G) -expansion method commenced by Wang et al. It is noticeable that, the two variable (G'/G,1/G) -expansion method appears to be more reliable, straightforward, computerized and user-friendly.

Lie symmetries and invariant solutions of $$(2+1)$$-dimensional breaking soliton equation

The present article deals with the symmetry reductions and invariant solutions of breaking soliton equation by virtue of similarity transformation method. The equation represents the collision of a Riemann wave propagating along the y-axis with a long wave along the x-axis. The infinitesimal transformations under one parameter for the governing system have been derived by exploiting the invariance property of Lie group theory. Consequently, the number of independent variables is reduced by one and the system remains invariant. A repeated application transforms the governing system into systems of ordinary differential equations. These systems degenerate well-known soliton solutions under some limiting conditions. The obtained solutions are extended with numerical simulation resulting in dark solitons, lumps, compactons, multisolitons, stationary and parabolic profiles and are shown graphically.

Novel Localized Excitations Structures with Fusion and Fission Properties to a (2 + 1)-Dimensional Breaking Soliton Equation

Using suitable transformations of dependent variables, a separable of variables solution of the (2 + 1)-dimensional breaking soliton model has developed with two arbitrary functions. We determine lump solutions by picking trustworthy arbitrary functions from the separable of variables solution. Also, critical points of the lump solution have evaluated. We determine the non-elastic interaction of two solitons evolves that the offers interface subsequently collision moderates into a breather wave regiment. We regulate the most important phenomena of soliton solutions as ring type exciting solitons, soliton fission, fusion and annihilation phenomena from our achieved variables separable solution selecting appropriate functions. Likewise, profiles of achieved solutions are presented to visualize the properties of the solutions. These properties of the breaking soliton model are associated to physical phenomena: inner waves in a gyrating ocean and thoughtful of the influencing machinery twisted by the interface of Riemann waves.

Solitary Wave Solutions of Chafee-Infante Equation and (2+1)-Dimensional Breaking Soliton Equation by the Improved Kudryashov Method

In this paper, we apply the improved Kudryashov method for finding exact solution and then solitary wave solutions of the Chafee-Infante equation and (2+1)-dimensional breaking soliton equation, where mathematical software Maple-13 is used as an important mathematical tool for removing calculation complexity, justification of the solutions and its graphical representations.

Periodic type and periodic cross-kink wave solutions to the (2 + 1)-dimensional breaking soliton equation arising in fluid dynamics

In this paper, we have acquired the periodic type and cross-kink wave solutions. In this paper, we use the Hirota bilinear method. With the help of the symbolic calculation and applying the used method, we solve the (2[Formula: see text]+[Formula: see text]1)-dimensional Breaking Soliton (BS) equation. We obtain some periodic wave and cross-kink wave that have greatly enriched the existing literature on the BS equation. All solutions have been verified back into its corresponding equation with the aid of the Maple package program via the three-dimensional images and density images with the help of Maple, the physical characteristics of these waves are described very well. These will be widely used to describe many interesting physical phenomena in the fields of gas, plasma, optics, acoustics, heat transfer, fluid dynamics, classical mechanics and so on.

Some applications of the modified (G′/G2)-expansion method in mathematical physics

Abstract This study proposes a modified ( G ′ / G 2 ) -expansion method to seek new more general exact traveling wave solutions to nonlinear evolution equations (NLEEs). To demonstrate the validity of this method, two interesting NLEEs are considered: the (2 + 1)-dimensional typical breaking soliton equation and (1 + 1)-dimensional classical Boussinesq system of equations. Consequently, we construct some new exact solutions involving the free parameters of the considered equations, which are categorized into three unique forms including rational, periodic, and hyperbolic functions. Thus, this study illustrates the effectiveness and the simplicity of the proposed method with the aid of symbolically computational software such as MATHEMATICA.

Characteristics of integrability, bidirectional solitons and localized solutions for a ( $$3+1$$ 3 + 1 )-dimensional generalized breaking soliton equation

The characteristics of integrability, bidirectional solitons and localized solutions are investigated for a ( $$3+1$$ )-dimensional breaking soliton (GBS) equation with general forms. Firstly, starting from the GBS equation, we perform the singularity manifold analysis and obtain a new integrable model in the sense of Painleve property. Secondly, taking advantage of the Bell polynomial approach, we construct the Backlund transformation, Lax pair and an infinite sequence of conservation laws. Subsequently, this new equation is also found to allow bidirectional soliton solutions, and the head-on and overtaking collisions between solitons are illustrated by some illustrative graphs. Finally, some localized excitations, such as lump solution, multi-dromions, periodic solitary waves solution, are obtained.