Meromorphic solutions of nonlinear ordinary differential equations

Abstract

Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations.

Some Remarks on Exp-Function Method and Its Applications

“…(32), then −v(−x, t) is also a solution of Eq. (32). Taking the last remark and the issues outlined in the previous section into account, we first assume a function of the form…”

Modified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics

“…Because of the increased concentration in the theory of solitary waves, a large variety of analytic and computational methods have been established in the analysis of the nonlinear models. For example the inverse scattering transformation method [1], the Hirota bilinear transform method [2], the Painleve integration method [3][4][5][6], the Backlund transformation method [7,8], the exp-function method [9][10][11][12][13], the tanh-function method [14][15][16][17], the Jacobi-elliptic function expantion method [18][19][20], the (G'/G)-expansion method [21][22][23][24][25][26][27][28][29], the (G'/G,1/G)-expansion method [30,31], the first integral method [32], the variational iteration method [33], the homotopy perterbation method [34], the modified simple equation method [35][36][37][38][39] and so on. Recently, Jawad et al [35], Zayed [36] and Zayed et al [37][38][39] have employed the modified simple equatio...…”

Comment on “Application of -expansion method to travelling-wave solutions of three nonlinear evolution equation” [Comput Fluids 2010;39:1957–63]

“…The foregoing illustrative discussion exemplifies two deficiencies of the ( ′ / )-expansion method: firstly, that the method delivers solutions in a cumbersome form (see (11) with (12), or (13), for example) and secondly, that the solutions appear to contain more free parameters than is actually the case. Typically, each solution can be manipulated into a neater form which displays the correct number of free parameters.…”

“…In [13], periodic-wave solutions to Eq. (1) in terms of the Jacobi elliptic cn function were derived by direct integration.…”

“…However, in [11] we showed that all twenty eight solutions may be reduced to 11 or 12 . In [13], the solutions 11 , 21 and 3 were derived via a method involving Laurent series.…”