In order to give our complex method and the proof of Theorem 1, we need some lemmas and results.
Lemma 1 [29, 30]
Let , and w be a meromorphic solution of k order, and the Briot-Bouquet equations be
where are polynomials with constant coefficients. If w has at least one pole, then w belongs to the class W.
Set , , , . A differential monomial is defined by
is called the degree of . A differential polynomial is defined by
where are constants, and I is a finite index set. The total degree of is defined by .
We will consider the following complex ordinary differential equations:
where , c are constants, .
Let . Suppose that (3) has a meromorphic solution w with at least one pole, we say that (3) satisfies the weak condition if on substituting the Laurent series
into (3) we can determine p distinct Laurent singular parts:
Lemma 2 [15, 31, 32]
Let , . Suppose that an m order Briot-Bouquet equation
satisfies the weak condition, then all meromorphic solutions w belong to the class W. If for some values of the parameters such a solution w exists, then the other meromorphic solutions form a one-parametric family , . Furthermore each elliptic solution with pole at can be written as
where are given by (4), , and .
Each rational function solution
is of the form
with l (≤p) distinct poles of multiplicity q.
Each simply periodic solution is a rational function of (). has l (≤p) distinct poles of multiplicity q, and is of the form
In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic functions .
Let , be two given complex numbers such that , are a discrete subset , which is isomorphic to . The discriminant and
The Weierstrass elliptic function is a meromorphic function with double periods , and satisfies the equation
where , , and .
If changing (9) to the form
we have , , .
Inversely, given two complex numbers and such that , then there exists a double period , Weierstrass elliptic function such that the above relations hold.
Lemma 3 [31, 33]
The Weierstrass elliptic functions have two successive degeneracies, and we have the following addition formula.
Degeneracy to simply periodic functions (i.e., rational functions of one exponential ) is according to
if one root is double ().
Degeneracy to rational functions of
is according to
if one root is triple ().
The addition formula is
By the above lemmas, we can give a new method below, called the complex method, to find exact solutions of some PDEs.
Step 1. Substituting the transform , into a given PDE gives a nonlinear ordinary differential equations (3) or (5).
Step 2. Substitute (4) into (3) or (5) to determine that the weak condition holds.
Step 3. By the indeterminate relation (6)-(8) we find the elliptic, rational, and simply periodic solutions of (3) or (5) with a pole at , respectively.
Step 4. By Lemmas 1 and 2 we obtain all meromorphic solutions .
Step 5. Substituting the inverse transform into these meromorphic solutions , we get all exact solutions of the originally given PDE.