Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

Abstract

In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order α. Phase portraits and simulations are given to show the validity of the obtained solutions.

Iterative solution of the fractional Wu-Zhang equation under Caputo derivative operator

“…To obtain the exact solution, this is one of the most critical topics in mathematical physics. Several authors have recently used various numerical and analytical approaches to find the numerical and analytical solution to the WZ equations, for example, using the first integral method [21], the modified Adomian decomposition method [22], the homotopy perturbation method [23], the extended Tanh method and the exp-function method [24], the exponential rational function method [25], the successive approximation method [26], the modified variation iteration method [27], the extended trial equation method [28] and the dynamic system method [29]. In addition, more solitonic solutions were extracted using the mapping method.…”

Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

“…This is called Wang-He's spatiotemporal fractional relationship (for more details see [32]). Due to the substantial importance of WZ systems, many scholars have attempted to solve and analyze these systems through variety of methodologies like mVIM [33], ADM [34,35], extended tanh and exp-function method [36], and dynamical analysis method [37]. Recently, for more generalized solutions and predictions, the WZ systems are also attempted fractionally by few of the scientists.…”

An analytical approach to solve the fractional‐order (2 + 1)‐dimensional Wu–Zhang equation

“…It is one of the most significant topics for mathematical physics to obtain the exact solutions. In the recent years, many authors find the numerical and analytical solution of WZ equation by using various methods, for illustration, the first integral method, 1 Adomian decomposition, 2 modified adomian decomposition method, 3 quintic method, 2 septic spline methods, 2 homotopy perturbation method (HPM), 4 extended tanh method, 5 exp–function method, 5 the exponential rational function method, 6 successive approximation method, 7 modified variation iteration method, 8 extended trial equation method, 9 and dynamical system method 10 …”

“…In the recent years, many authors find the numerical and analytical solution of WZ equation by using various methods, for illustration, the first integral method, 1 Adomian decomposition, 2 modified adomian decomposition method, 3 quintic method, 2 septic spline methods, 2 homotopy perturbation method (HPM), 4 extended tanh method, 5 exp-function method, 5 the exponential rational function method, 6 successive approximation method, 7 modified variation iteration method, 8 extended trial equation method, 9 and dynamical system method. 10 The nonlinear phenomenon plays a vital role in physics and applied mathematics. In recent years, great interest has been developed in analyzing fractional-order partial differential equations (FPDEs) appeared in mathematical modeling of physical systems.…”