Painlevé Analysis and Abundant Meromorphic Solutions of a Class of Nonlinear Algebraic Differential Equations

Abstract

In this paper, a class of nonlinear algebraic differential equations (NADEs) is studied. The Painlevé analysis of the NADEs is considered. Abundant meromorphic solutions of the NADEs are obtained by means of the complex method. Then, meromorphic exact solutions of the Schamel-Korteweg-de Vries (S-KdV) equation and (2+1)-dimensional sine-Gordon equation are derived via the applications of the NADEs.

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].…”

A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

“…where the left and right side solutions y l (x, λ) and y r (x, λ) are given by (16) and (19) respectively. Substituting (19) in the formula…”

“…Recently a great deal of interest has been focused on the application of different types of approximation methods for the solutions of linear and nonlinear problems (See, [13][14][15][16][17]).…”