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

Abstract

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.

Keywords

• Combined-(ZK-mZK) equation
• Complete discrimination system for polynomial
• Solitary wave
• Shock wave
• Periodic wave
• Jacobean elliptic functions