Complete Discrimination System For Polynomial Method Research Articles

Complete discrimination System method for finding exact solutions, dynamical properties of combined Zakharsov-Kuznetsov-modified Zakarsov-Kuznetsov equation

The (2+1)-dimensional combined Zakharsov-Kuznetsov-modified Zakarsov-Kuznetsov (ZK-mZK) equation with constant co-efficients are solved by employing the complete discrimination system for polynomial Method (CDSPM). Use of it reduces the key Nonlinear Differential Equation (NLPDE) to an Ordinary Differential Equation (ODE) in the integral form under the traveling wave transformation, and the solutions are classified. Exact traveling wave solutions are derived using CDSPM of order four. Based on CDSPM, Liu’s approach provides a set of new solutions including Jacobi elliptic functions for the Combined-(ZK-mZK) equation. These analytic solutions exhibit different nonlinear wave structures viz., solitary wave, shock wave, and periodic wave. Among them, the finite amplitude periodic solutions are very important wave features for Combined-(ZK-mZK) equation and their existence is confirmed from the phase portrait of the said equation. Especially the finite amplitude periodic solutions are acquired from the system with the inclination of Jacobean elliptic functions. Numerical results confirm the reliability and effectiveness of the method. Finally, the phase portrait of the dynamical system ensures the entity and exactness of different topological structures of the solutions. Based on this analysis, we conclude that this method is appropriate one to classify the solutions, and can be expanded throughout the vast areas of application.

Abundant variant wave patterns by coupled Boussinesq–Whitham–Broer–Kaup equations

This paper focus on searching for abundant wave patterns to time-fractional coupled Boussinesq–Whitham–Broer–Kaup equations, which plays a crucial role in obtaining the novel insights for propagational dynamics of shallow water. Through the innovative application of complete discrimination system for polynomial method (CDSPM) and conformable fractional transformation, the whole types of solutions for the fractional equations presented in the existing literatures are obtained. In particular, we obtain the solitary wave type solutions and Jacobian elliptic functions type solutions, and the later are initially founded. The existence and applicability of the obtained solutions are proved by concrete examples. Finally, the nature of wave propagation and the fractional characteristics are intuitively revealed by graphical simulations.

Exact solutions and dynamic properties of a nonlinear fourth-order time-fractional partial differential equation

This paper considers a fourth-order time-fractional partial differential equation with Riemann–Liouville definition. We first use the general method of separation of variables to transform the original equation into an ordinary differential equation and subsequently apply the trial equation method to obtain its integral form. The complete discrimination system for polynomial method(CDSPM) is also adopted herein. By applying this method, dynamic properties such as phase portraits are determined. The results suggest that the soliton solution coexists with the periodic solution as long as the homoclinic orbits exist. Moreover, to directly show our conclusions, the corresponding exact solutions to this equation are presented using this method.

Exact solutions and dynamic properties of Ito-Type coupled nonlinear wave equations

In this paper, we conduct qualitative and quantitative analysis to Ito system by the complete discrimination system for polynomial method (CDSPM). Through the traveling wave transformation, the original equations could be transformed into a dynamic system. A conserved quantity, namely the Hamiltonian, is established, and then the qualitative properties such as equilibrium points are gotten. Especially, a typical phase portrait is emphasized to present prior estimate of existences of smooth soliton, cusped soliton and periodic solutions. Concrete examples of these solutions are all constructed to verify our conclusion directly.

Qualitative and quantitative fractional low-pass electrical transmission line model

In this paper, the fractional low-pass electrical transmission line model is considered, whose fractional definition is defined by conformal fractional derivative. We first apply the complex fractional traveling wave transformation to the original equation, then the dynamic properties such as Hamiltonian and topological properties to its traveling wave system are presented via the complete discrimination system for polynomial method (CDSPM). From these results, the prior estimate of periodic solution and soliton solution is established. Moreover, quantitative results of all traveling wave solutions are also given. To the best of our knowledge, this is the first time that the Hamiltonian is constructed to this model, and all single traveling wave solutions are obtained.

Abundant traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line

The main idea of this paper is to search for all traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line, which plays a essential role in obtaining the new insights in nonlinear voltage dynamics. We first give a brief introduction how to transform the discrete system into a continuous one, which is described by a fractional-order partial differential equation. After that, this equation is handled by conformable fractional transformation and the complete discrimination system for polynomial method (CDSPM). By applying the advanced method, the whole of the exact traveling wave solutions emerged in existing articles are obtained, especially we obtain the solitary wave solutions and elliptic functions solutions which are hardly founded by other methods. Notably, the elliptic functions solutions in rational form are discovered for the first time. Finally, the electrical characteristics and the fractional nature are revealed via graphical represents. By the depicted graphs, we intuitively observe the existence of the phenomena for periodic wave and solitary wave, and the time-fractional derivative is proved do has important influence on the behaviors of the solutions. Considering the significance of the nonlinear electrical transmission line, the acquired results would have wide application in electrical engineering and nonlinear voltage dynamics, liking analyzing and predicting the complex voltage wave propagation phenomenon in realistic electrical transmission system.