Exact Solutions of the Vakhnenko-Parkes Equation with Complex Method

Abstract

We derive exact solutions to the Vakhnenko-Parkes equation by means of the complex method, and then we illustrate our main results by some computer simulations. We can apply the idea of this study to related nonlinear differential equation.

Searching for Analytical Solutions of the (2+1)‐Dimensional KP Equation by Two Different Systematic Methods

“…For instance, the singular behaviors [12,13] and impulsive phenomena [14,15] often show some blow-up properties [16,17] which happen in lots of complex physical processes. In order to solve various differential equations, some analytical tools as well as symbolic calculation techniques were established, such as fixed-point theorems [18,19], variational methods [20,21], topological degree method [22][23][24][25], iterative techniques [26,27], bilinear method [28][29][30][31], modified simple equation method [32], exp(− ( ))-expansion method [33][34][35][36][37][38], Lie group method [39,40], and complex method [41][42][43][44][45][46][47][48][49][50].…”

Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation

“…In recent years, many mathematicians and physicists studied the nonlinear integrable systems that occur in various fields such as biology, fluid dynamics, quantum and plasma physics, thermal engineering and optics. Plenty of methods have been developed for getting exact solutions to nonlinear differential equations such as the modified extended tanh method [1,2], the improved F-expansion method [3], the modified simple equation method [4], the complex method [5][6][7][8], the generalized ( ′/ ) G G -expansion method [9][10][11], the exp(− ( )) ψ z -expansion method [12][13][14][15][16], the ( + / ′) m G 1 -expansion method [17], the sine-Gordon expansion method [18][19][20][21][22][23][24], the extended sine-Gordon expansion method [25][26][27], the extended rational sinh-cosh method [28], the modified Kudryashov method [29] and other methods [30][31][32].…”

Application of the modified exponential function method to Vakhnenko-Parkes equation

“…In the other study, a variable separation solution with two arbitrary functions is obtained and the soliton-type, instanton-type and rogue wave-type structures are presented [61]. The many scientist's studies on the Vakhnenko--Parkes (VP) equation via various methods can be seen at [29,36,42,44,49]. This paper is organized as follows: In the 2nd Section, we present the general structure of the modified exponential function method.…”