Zakharov-Kuznetsov-Benjamin-Bona-Mahony Research Articles

Dynamic nature of analytical soliton solutions of the nonlinear ZKBBM and GZKBBM equations

A solitary wave, characterized as a localized perturbation in a medium, emerges as a result of a delicate equilibrium between nonlinear and dispersive phenomena. Solitons, a subtype of solitary waves, exhibit persistent shape and velocity during propagation, representing a fundamental phenomenon observed widely in natural systems and possessing various applications in nonlinear dynamics. This investigation focuses on two nonlinear evolution equations (NEEs), specifically the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZKBBM) equation, which find relevance in domains such as fluid dynamics and ocean engineering. Utilizing an ansatz-based methodology, soliton solutions of both bright and dark characteristics are derived, alongside exploration of rogue wave-type solutions. Notably, the manifestation of dark, bright, and rogue waves aligns with the physical interpretation of the generated solitons. Computational simulations conducted using Wolfram Mathematica aim to provide a comprehensive description of the physical phenomena. The novelty of this study lies in its unreported investigation, contributing new insights into the solitonic dynamics within the considered models.

Stable and effective traveling wave solutions to the non-linear fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony equations

The space–time fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equations are used to explain the transmission of shallow water waves inside a water channel of uniform speed and constant depth. It moreover simulates water waves in harbors and shallow water, which is useful for oceanography. These equations are significant nonlinear model equations to illustrate numerous physical structures namely in marine and coastal science, water wave mechanics, control theory, plasma engineering, fluid flow, optical fibers, phenomena of fission and fusion, and so on. In this study, we employed the extended tanh-function approach to generate some fresh and further general closed-form traveling waves solutions of those equations in the lite of conformal derivatives. The achieve results are more applicable to resolve the above mentioned phenomena properly. The fractional differential transform simplifies generate ordinary differential equations from fractional order differential equations. We discovered many types of solutions using the maple, including solitons, kink types, bell types, and other types of solutions that are illustrated using 3D and contour plots. It is important to note that all derived solutions are checked for accuracy by being directly replaced with the original equation. We proposed that the technique be revised to be more realistic, effective, and trustworthy and that we explore more generalized exact results of traveling waves, like solitary wave solutions.

Abundant analytical closed-form solutions and various solitonic wave forms to the ZK-BBM and GZK-BBM equations in fluids and plasma physics

Solitary wave profiles of various forms can be found in many disciplines, including oceanography, marine engineering, plasma physics, optical fibers, fluid dynamics, and mathematical sciences. In this work, we use the generalized exponential rational function (GERF) method to find the closed-form wave solutions to two nonlinear evolution equations, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZK-BBM) equation, and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZK-BBM) equation. The GERF method under the wave transformation is a very simple, efficient, robust, and straightforward technique for solving various nonlinear partial differential equations. By utilizing this technique, we obtain thirty different kinds of exact solitary wave solutions for governing equations. These newly solitary wave solutions are generated in terms of the rational function, trigonometric function, hyperbolic function, and exponential function forms, all of which play crucial roles in many areas of nonlinear sciences and engineering. Furthermore, we present the two-dimensional and three-dimensional wave profiles of some achieved solutions under the best suitable choices of involved arbitrary constants with dependable range space to understand the dynamical wave behaviors of the solutions, which makes this research more commendable. The GZK-BBM equation demonstrates how the model behaves when long waves propagate, which has a nonlinear and dissipative impact.

New traveling solutions of the fractional nonlinear KdV and ZKBBM equations with 𝒜ℬℛ fractional operator

This research paper investigates novel explicit wave solutions of the fractional Korteweg–de Vries (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. These models are used as gravity models in water and an interaction model between the long waves. The Atangana–Baleanu ([Formula: see text]) fractional operator is utilized for the first time to convert the fractional form of both models into nonlinear partial differential equations with an integer order. The extended simplest equation method is employed to construct some distinct types of solitary wave solutions such as exponential, rational, hyperbolic and trigonometric functions. For more illustration of our obtained solutions, some figures for them are given. The power and practical properties of the used method are tested.

Adequate soliton solutions to the time fractional Zakharov-Kuznetsov equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

The time fractional (2, 2, 2) Zakharov-Kuznetsov (ZK) equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation demonstrate the characteristic of shallow water waves, turbulent motion, waves of electro-hydro-dynamics in the local electric field, sound wave, waves of driving flow of fluid, ion acoustic waves in plasmas, traffic flow, financial mathematics, etc. The time-fractional (2, 2, 2) ZK equation is the particular case of the general time-fractional ZK equation, where represent the space coordinate and represents the temporal coordinate. Hereinto to evade the complexity and to ascertain soliton solutions of this model, we accept and in this case, the general ZK equation is called the time-fractional (2, 2, 2) ZK equation. In this article by making use of the concept of fractional complex transformation, the auxiliary equation method is put in use to search the closed form soliton solutions to the above indicated fractional nonlinear equations (FNLEs).The ascertained solutions are in the form of exponential, rational, hyperbolic and trigonometry functions with significant precision. We illustrate the soliton solutions relating to physical concern by setting the definite values of the free parameters through depicting diagram and interpreted the physical phenomena. The developed solutions assert that the method is effective, able to measure NLEEs, influential, powerful and offer vast amount of travelling wave solutions of nonlinear evolution equations in the area of mathematical sciences and engineering.

On abundant new solutions of two fractional complex models

We use an analytical scheme to construct distinct novel solutions of two well-known fractional complex models (the fractional Korteweg–de Vries equation (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation). A new fractional definition is used to covert the fractional formula of these equations into integer-order ordinary differential equations. We obtain solitons, rational functions, the trigonometric functions, the hyperbolic functions, and many other explicit wave solutions. We illustrate physical explanations of these solutions by different types of sketches.

Exact solitary wave solutions for a system of some nonlinear space–time fractional differential equations

We have enumerated new and exact general wave solutions, along with multiple exact soliton solutions of space–time nonlinear fractional differential equations (FDE), namely Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM), foam drainage and symmetric regularised long-wave (SRLW) equations, by employing a relatively new technique called ( $$G^\prime /G, 1/G)$$ -expansion method. Also, based on fractional complex transformation and the properties of the modified Riemann–Liouville fractional-order operator, the fractional partial differential equations transform into a form of ordinary differential equation (ODE). This method is a recollection of the commutation of the well-appointed ( $$G^\prime /G$$ )-expansion method introduced by Wang et al, Phys. Lett. A 372, 417 (2008) In this paper, it is mentioned that the two-variable ( $$G^\prime /G, 1/G$$ )-expansion method is more legitimate, modest, sturdy and effective in the sense of theoretical and pragmatical point of view. Lastly, the peculiarities of these analytic solutions are illustrated graphically by utilising the computer symbolic programming Wolfram Mathematica.

Exp-function method for the conformable space-time fractional STO, ZKBBM and coupled Boussinesq equations

In the present paper, new analytical solutions for the conformable space-time fractional Sharma-Tasso-Olever (STO), Zakharov Kuznetsov Benjamin Bona Mahony (ZKBBM) and coupled Boussinesq equations are obtained by using the Exp-function method. The obtained traveling wave solutions are presented by exponential functions. Simulations of the obtained solutions are given at the end of the paper.

Exact solutions for nonlinear fractional differential equations using [formula omitted]-expansion method

A relatively new technique which is named as G′G2-expansion method is applied to attain exact solution of nonlinear fractional differential equations (NLFDEs). Exact solutions include hyperbolic solutions, trigonometric solutions and rational solutions. Fractional complex transform is used to convert fractional differential equations into ordinary differential equations. We apply it to the space-time fractional Zakharov Kuznetsov Benjamin Bona Mahony (ZKBBM) and fractional coupled Burgers equation. By using our proposed method some innovative solutions can be derived.

The investigation of exact solutions of nonlinear partial differential equations by using exp(−Φ(ξ)) method

In this work, the methodology of the exp(−Φ(ξ)) method is given. The effectiveness of the method is demonstrated by applications to two nonlinear partial differential equations (NPDEs), namely, the Zakharov Kuznetsov–Benjamin Bona Mahony (ZK-BBM) equation and ill-posed Boussinesq equation. The obtained exact solutions are consist of trigonometric, hyperbolic and rational functions.

New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for ZKBBM equation

Abstract By means of two extended ( G ′/ G )-expansion methods, new approach of ( G ′/ G )-expansion method and new approach of generalized ( G ′/ G )-expansion method, we have successfully performed for analytical solutions on the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. In methods, the nonlinear auxiliary equation is implemented for constructing a rich class of new travelling wave solutions. Moreover, obtained solutions reveal that, these methods are very effective and powerful to handle various nonlinear evolution equations which frequently arise in mathematical physics, engineering sciences and many scientific real time application fields.

Further extension of the generalized and improved (G′/G)-expansion method for nonlinear evolution equation

In this article, the generalized and improved (G′/G)-expansion method has been proposed for further extension to generate many new travelling wave solutions. In addition, nonlinear ordinary differential equation is implemented as auxiliary equation including many parameters instead of linear ordinary differential equation. Moreover, the presentation of the travelling wave solutions is quiet new. The effectiveness and reliability of the method are shown by its application to the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. Some of our generated solutions turned into some known solutions, when parameters consider specific values and others are new.

Application of Classification of Traveling Wave Solutions to the Zakhrov-Kuznetsov-Benjamin-Bona-Mahony Equation

In order to get the traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation, it is reduced to an ordinary differential equation (ODE) under the travelling wave transformation first. Then complete discrimination system for polynomial is applied to the ZK-BBM equation. The traveling wave solutions of the equation can be obtained.

Conservation Laws and Traveling Wave Solutions of a Generalized Nonlinear ZK-BBM Equation

We study a generalized two-dimensional nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation, which is in fact Benjamin-Bona-Mahony equation formulated in the ZK sense. Conservation laws for this equation are constructed by using the new conservation theorem due to Ibragimov and the multiplier method. Furthermore, traveling wave solutions are obtained by employing the(G'/G)-expansion method.

Application of the new approach of generalized (G' /G) -expansion method to find exact solutions of nonlinear PDEs in mathematical physics

The exact solutions of nonlinear evolution equations (NLEEs) play a crucial role to make known the internal mechanism of complex physical phenomena. In this article, we construct the traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation by means of the new approach of generalized (G′ /G) -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G′ /G) -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations. BIBECHANA 10 (2014) 58-70 DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9312

Fractional Sub-Equation Method and its Applications to the Space–Time Fractional Differential Equations in Mathematical Physics

In the present paper, we construct the analytical solutions of somenonlinear evolution equations involving Jumaries modified RiemannLiouville derivative in mathematical physics; namely the spacetime fractional modified Benjamin -Bona-Mahony(mBBM) equation and the spacetime fractional Zakharov -Kuznetsov-Benjamin-Bona-Mahony(ZKBBM) equation by using a simple method which is called the fractional sub -equation method. As a result, three types of exact analytical solutions are obtained. This method is more powerful and will be used in further works to establish more entirel y new solutions for other kinds of nonlinear fractional PDEs arising in mathematical physics.

A Generalized and Improved(G′/G)-Expansion Method for Nonlinear Evolution Equations

A generalized and improved(G′/G)-expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering.