On elliptic solutions of nonlinear ordinary differential equations

At present there exists a lot of methods for finding exact elliptic solutions of autonomous nonlinear ordinary differential equations. Let us name only a few: the Weierstrass function method [1,2], the Jacobi elliptic-function method [3][4][5] and their different extensions and modifications [6][7][8][9][10][11][12]. Making use of such a method one can not but come across the following questions. Whether all families of elliptic solutions are found. Whether an equation does not have elliptic solutions at all, if a method fails to find any. This questions are addressed very seldom (nevertheless see [13][14][15]).

The aim of this article is to present an algorithm, which enables one to find all families of doubly periodic meromorphic solutions satisfying an autonomous nonlinear ordinary differential equation. We will use an approach suggested in [14,15]. As an example we take the following third order differential equation w zzz + 3ηww zz + ηw 2 z + βw zz + 2αw 3 + δww z + γw z + σw 2 + µw + ν = 0 (1) with η = 0, α = 0. We find all conditions for elliptic solutions to exist and construct elliptic solutions in explicit form.

This article is organized as follows. In section 2 we present our method and give explicit expressions for elliptic solutions of an autonomous nonlinear ordinary differential equation. In section 3 we consider an example and classify elliptic solutions of equation (1).

Consider an algebraic autonomous nonlinear ordinary differential equation

Let us look for its elliptic solutions. If w(z) is such a solution, then equation ( 2) has the family of elliptic solutions w(zz 0 ) with z 0 being an arbitrary constant. Equation ( 2) necessary possesses an elliptic solution, if it admits at least one Laurent expansion in a neighborhood of the pole z = z 0 . Without loss of generality we may build Laurent series in a neighborhood of the point z = 0

Here p is the order of the pole z = 0. Proposition. Suppose Laurent series (3) with uniquely determined coefficients satisfies equation (2), then this equation admits at most one meromorphic solution having a pole z = 0 with Laurent series (3).

This proposition follows from the propertied of Laurent series and uniqueness of analytic continuation. As a consequence, equation (2) may have at most one elliptic solution possessing the pole z = 0 with Laurent series (3). The order of an elliptic function is defined as the number of poles in a parallelogram of periods, counting multiplicity.

Our algorithm for finding elliptic solutions of equation ( 2) in closed form is the following. Note that we omit arbitrary constant z 0 .

Step 1. Perform local singularity analysis around movable singular points for solutions of equation (2).

Step 2. Select the order M of w(z) and take K distinct Laurent series

from those, found at step 1, in such a way that the following conditions

hold.

Step 3. Define the general expression for the elliptic solution w(z) possessing K poles a 1 , . . ., a K in a parallelogram of periods (see [14][15][16]). The Laurent expansion in a neighborhood of the point z = a i is w (i) (za i ). In other words, take the following expression for w(z)

Here ℘(z) is the Weierstrass function satisfying the equation

ζ(z) is the Weierstrass ζ-function, h0 is a constant.

Step 4. Find the Laurent series for w(z) given by ( 6) around its poles a 1 , . . ., a K . Without loss of generality set a 1 = 0. Introduces notation

With the help of addition formulae for the functions ℘ and ζ (see [14][15][16]) rewrite expression (6) as

Step 5. Compare coefficients of the series found at the second and the fourth steps. Form a system of algebraic equations. Add to this system equations

The number of equations in the system is slightly more than the number of parameters of elliptic solution (9) and equation (2). Solve the algebraic system for the parameters of the elliptic solution w(z), i.e. find h 0 , g 2 , g 3 , A i , B i , i = 2, . . ., K. In addition correlations for the parameters of equation (2) may arise. If this system is inconsistent, then equation (2) does not possess elliptic solutions with supposed Laurent expansions around poles.

Step 6. Check-up obtained solutions, substituting them into original equation.

In the case g 3 2 -27g 2 3 = 0 the elliptic function ℘(z) degenerates and consequently elliptic solution (9) degenerates. The invariants g 2 , g 3 are related with the half-period ω 1 , ω 2 by means of equalities

With the help of presented algorithm one can construct any elliptic solution of equation (2). Note that if equation ( 2) possesses only N distinct Laurent series in a neighborhood of poles, then the orders of its elliptic solutions is not more than N i=1 p i , where p i (i = 1, . . ., N) are the orders of poles [14,15]. Thus we see that our approach may be used, if one need to classify families of elliptic solutions satisfying equation (2).

As an example let us classify doubly periodic meromorphic solutions of the third order non

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