ZK-BBM Equation Research Articles

Exploring Wave Dynamics: Symmetry Reductions and Similarity Solutions to the (2+1) Dimensional ZK-BBM Equation

Exploring Wave Dynamics: Symmetry Reductions and Similarity Solutions to the (2+1) Dimensional ZK-BBM Equation

Travelling Wave Solutions to the Space-Time Fractional ZKBBM Equation in Mathematical Physics

Travelling Wave Solutions to the Space-Time Fractional ZKBBM Equation in Mathematical Physics

A new expansion auxiliary equation method for traveling wave solutions of the simplified MCH equation and the ZKBBM equation

Abstract In the paper, a new expansion auxiliary equation method is proposed. By this method, traveling wave solutions of the simplified MCH equation and the ZKBBM equation are constructed successfully, including soliton solutions, singular solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions and other new solutions, their characteristics are demonstrated by three-dimension and two-dimension profile figures and comparisons with previous references are made. It is shown that our proposed method can obtain new and more general solutions of nonlinear evolution equations and would be a powerful mathematical tool for solving nonlinear evolution equations.

Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach

This study investigates exact analytical solutions of some nonlinear partial differential equations arising in mathematical physics. To this reason, the Kudryashov-Sinelshchikov equation, the ZK-BBM equation and the Gardner equation have been considered. With the implementation of the trial solution algorithm, solitary wave, bright, dark and periodic exact traveling wave solutions of the considered equations have been attained. The solutions have been checked and graphs have been given via package programs to see the behavior of the waves.

Optimal system, symmetry reductions and group-invariant solutions of (2+1)-dimensional ZK-BBM equation

The present article intends to generate optimal system of one dimensional subalgebra and group–invariant solutions of ZK–BBM equation with the aid of Lie group theory. The ZK–BBM equation is long wave equation with large wavelength, which describes the water wave phenomena in nonlinear dispersive system. The infinitesimal vectors, commutative relations and invariant functions for optimal system of ZK–BBM equation are derived under invariance of Lie groups. The invariance property leads to the reduction of independent variable and leaves the system invariant. Based on the optimal system, ZK–BBM equation is transformed into ordinary differential equations by twice reductions. These ODEs are solved under parametric constraints and result into invariant solutions. The obtained solutions are analyzed physically based on their numerical simulation. Consequently, elastic multisoliton, dark and bright lumps, compacton and annihilation profiles of the solutions are well presented graphically.

Construction of Functional Closed Form Wave Solutions to the ZKBBM Equation and the Schrödinger Equation

The time fractional (1+1)-dimensional Schrodinger equation and the space–time fractional (1+1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation are important mathematical models to interpret signal processing in optical fibers, the fractional quantum mechanics, water waves due to gravity, the motion of turbulent flows, the wave of driving flow of fluid, the ion osculate waves in plasma, etc. In the present article, we have examined the closed form soliton solutions with the assistance of the modified simple equation method together with the fractional wave transformation. The form of the attained solutions is rational, trigonometry and hyperbolic functions. We have shown that the assigned method is further general, efficient, straightforward and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering. We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.

Dark and kink soliton solutions of the generalized ZK–BBM equation by iterative scheme

The generalized ZK–BBM equation is solved using iterative scheme of the Adomian decomposition method (ADM) and variational iteration method (VIM). A dark and a kink soliton solutions of the generalized ZK–BBM equation are obtained under initial conditions. The convergence analysis of the ADM and VIM solution shows that these solutions are convergent. The comparison of the ADM and VIM solutions with the exact solution shows that the solutions of the generalized ZK–BBM equation by the iterative methods are almost exact. The absolute errors show that the accuracy and efficiency of the ADM and VIM depend on the problem and its domain. It is found that the iterative scheme of Adomian decomposition method and variational iteration method are quite efficient for the soliton solution of the generalized ZK–BBM equation.

Invariant analysis and conservation laws of ([formula omitted]) dimensional time-fractional ZK–BBM equation in gravity water waves

This paper intends to make an in-depth study on the symmetry properties and conservation laws of the (2+1) dimensional time fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZK–BBM) equation with Riemann–Liouville fractional derivative. Symmetry properties have been investigated here via Lie symmetry analysis method. In view of Erdélyi-Kober fractional differential operator, the reduction of (2+1) dimensional time fractional ZK–BBM equation has been done into fractional ordinary differential equation. To analyse the conservation laws, new theorem of conservation law has been proposed here for constructing the new conserved vectors for (2+1) dimensional time fractional ZK–BBM equation with the help of formal Lagrangian.

Conservation laws, soliton solutions for modified Camassa–Holm equation and (2 + 1)-dimensional ZK–BBM equation

By using the multiplier approach, we construct the conservation laws and the corresponding conserved quantities for the modified Camassa–Holm equation and (2 + 1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahoney equation for each multipliers. We also deduce the soliton solutions for the equations using semi-inverse variational principle. We give the discussion of the properties of the soliton waves obtained numerically via some figures and the physical interpretation to complete these studies. Finally, we compare the solutions obtained with other solutions obtained in previous papers to prove that the results in this paper cannot appear anywhere.

Solitary wave solutions of some nonlinear PDEs arising in electronics

In the present article, an Extended Trial Equation method has been applied to derive the new exact solutions for generalized form of equations. We consider ZK equation and ZK-BBM equation to demonstrate the features and credibility of the suggested technique. As a result, many new exact soliton solutions are obtained, which includes rational solutions, soliton type solutions, and singular soliton solutions. These types of solutions might perform significant role in engineering domains.

The improved F-expansion method and its application to the MEE circular rod equation and the ZKBBM equation

AbstractIn the present article, we implement the improved F-expansion method combined with Riccati equation to attain traveling wave solutions to nonlinear evolution equations (NLEEs) via the Magneto-electro-elastic circular rod equation and the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. We establish three classes of explicit solutions-hyperbolic, trigonometric and rational solutions containing several free parameters. For particular values of the parameters, solitary wave solutions are emanated from the traveling wave solutions. It turns out that, the method is straightforward, concise and it can be applied to many other NLEEs in mathematical physics and engineering.

A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.

A Meshless Method Using Radial Basis Functions for the Numerical Solution of Two-Dimensional ZK–BBM Equation

The main idea of this article is obtain the numerical solution of the two-dimensional ZK–BBM equation. We construct the solution by using different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. we use a predictor–corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation Via the Modified Simple Equation Method

Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation Via the Modified Simple Equation Method

G/G'-expansion Method for ZK-BBM Equation

In this paper, we apply -expansion to find travelling wave solution of the ZK-BBM equation. The travelling wave solutions are expressed by functions which are hyperbolic, trigonometric and rational functions. Numerical results clearly reflect the efficiency of proposed scheme.

Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations

The fractional derivatives in the sense of the modified Riemann-Liouville derivative and Feng’s first integral method are employed to obtain the exact solutions of the nonlinear space-time fractional ZKBBM equation and the nonlinear space-time fractional generalized Fisher equation. The power of this manageable method is presented by applying it to the above equations. Our approach provides first integrals in polynomial form with high accuracy. Exact analytical solutions are obtained through establishing first integrals. The present method is efficient and reliable, and it can be used as an alternative to establish new solutions of different types of fractional differential equations applied in mathematical physics.

Exact Solitary Wave Solution in the ZK-BBM Equation

The traveling wave solution for the ZK-BBM equation is considered, which is governed by a nonlinear ODE system. The bifurcation structure of fixed points and bifurcation phase portraits with respect to the wave speed c are analyzed by using the dynamical system theory. Furthermore, the exact solutions of the homoclinic orbits for the nonlinear ODE system are obtained which corresponds to the solitary wave solution curve of the ZK-BBM equation.

On the stability of iteration methods for special solution of time-fractional generalized nonlinear ZK-BBM equation

An investigation of a possible description of non-linear wave was performed inside the scope of fractional order derivative. A clear motivation underpinning the extension of this study within the concept of fractional calculus was presented. The special solution of the extended equation was derived using an efficient iteration method called homotopy decomposition method. A theorem underpinning the uniqueness of the special solution was proposed and proved in detail. An investigation for stability of the iteration method for solving the extended equation was presented in detail. Numerical simulations were carried out using the proposed algorithm for different values of alpha and n.

Periodic wave solutions and their limits for the ZK–BBM equation

In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the ZK–BBM equation. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions, unbounded solutions and blow-up wave solutions.

Some Further Results on Traveling Wave Solutions for the ZK-BBM() Equations

We investigate the traveling wave solutions for the ZK-BBM() equations by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.