^{1}

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We employ the complex method to obtain all
meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of
equations). The idea introduced in this paper can be applied to other nonlinear evolution equations.
Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutions

In this work, we aim to further study the Drinfeld-Sokolov system (Drinfeld-Sokolov equations (DS system of equations)) [

Wazwaz [

To look for the traveling wave solution of (

In 2007, El-Wakil and Abdou [

In this paper, we employ the complex method which was introduced by Yuan et al. [

A meromorphic function

Our main result is the following theorem.

All meromorphic solutions

the elliptic general solutions

the simply periodic solutions, where

Compared with the results of Wazwaz [

In order to give complex method and the proof of Theorem

Set

A differential polynomial

We will consider the following complex ordinary differential equations:

Let

Let

Each rational function solution

Each simply periodic solution is a rational function

In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [

Let

Weierstrass elliptic function

If we changed (

Inversely, given two complex numbers

Weierstrass elliptic functions

degeneracy to simply periodic functions (i.e., rational functions of one exponential

degeneracy to rational functions of

addition formula

By the previous lemma and results, we can give a new method next say

Substituting the transform

Substitute (

By determinant relation (

By Lemmas

By substituting the inverse transform

By substituting (

By (

In order to have simply periodic solutions, set

By substituting

So all simply periodic solutions of (

From (

Therefore, all elliptic function solutions of (

This completes the proof of Theorem

In this section, we give some computer graphs to illustrate our main results. Here, we take the simple periodic solutions

The solution of (DS) corresponding to

The solution of (DS) corresponding to

The solution of (DS) corresponding to

Complex method is a very important tool in finding the exact solutions of nonlinear evolution equations, and the Drinfeld-Sokolov equations (DS system of equations) are a classic and simplest case of the nonlinear reaction-diffusion equation. In this paper, we employ the complex method to obtain the general meromorphic solutions of the Drinfeld-Sokolov equations (DS system of equations), which improves the corresponding result obtained by El-Wakil and Abdou, [

This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University where the second and third authors worked as visiting scholars. This work was supported by the NNSF of China (nos. 11271090, 11171184, and 11001057), the NSF of Guangdong Province (S2012010010121), and Shanghai University Young Teacher Training Program (ZZSDJ12020) and supported by projects 10XKJ01, 12C401, and 12C104 from the Leading Academic Discipline Project of Shanghai Dianji University.