Research Article | Open Access
Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations
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 which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.
In this work, we aim to further study the Drinfeld-Sokolov system (Drinfeld-Sokolov equations (DS system of equations)) [1–3] where , , and are constants. This system was introduced by Drinfeld and Sokolov as an example of a system of nonlinear equations possessing Lax pairs of a special form .
Wazwaz  used the sine-cosine method and the tanh method to stress the power of these methods to nonlinear equations. Wazwaz  also investigated the traveling wave solutions with compact and noncompact structures for the Drinfeld-Sokolov equations (DS system of equations).
To look for the traveling wave solution of (1), we use the transformation , (where is a complex constant number). Then in (1), the system is carried to a system of ordinary differential equation: By integrating the first equation in the system and neglecting the constant of integration we find By inserting (4) into (3) and integration we find
In this paper, we employ the complex method which was introduced by Yuan et al. [6–8] to obtain the general solutions and some new solutions of (5). In order to state our results, we need some concepts and notations.
A meromorphic function means that is holomorphic in the complex plane except for poles. is the Weierstrass elliptic function with invariants and . We say that a meromorphic function belongs to the class if is an elliptic function, or a rational function of , , or a rational function of .
Our main result is the following theorem.
Theorem 1. All meromorphic solutions of (5) belong to the class . Furthermore, (5) has the following two forms of solutions:(i) the elliptic general solutions where , , and are arbitrary constants;(ii) the simply periodic solutions, where are obtained, for , by where ; where ; where .
Remark 2. Compared with the results of Wazwaz , we find that is the new solution of this Drinfeld-Sokolov equations (DS system of equations).
2. Preliminary Lemmas and the Complex Method
In order to give complex method and the proof of Theorem 1, we need some notations and results.
Set , , , and . We define a differential monomial denoted by and are called the weight and degree of , respectively.
A differential polynomial is defined as follows: where are complex constants and is a finite index set. The total weight and degree of are defined by and , respectively.
We will consider the following complex ordinary differential equations: where , are complex constants, .
Let , . Suppose that (12) has a meromorphic solution with at least one pole; we say that (12) satisfies weak condition if it substitutes Laurent series into (12). We can determine distinct Laurent singular parts as follows:
Lemma 3 (see [6–8]). Let , . Suppose that an -order Briot-Bouquet equation satisfies weak condition; then all meromorphic solutions of (15) belong to the class . If for some values of parameters such solution exists, then other meromorphic solutions form a one-parametric family . Furthermore, each elliptic solution with pole at can be written as where are given by (13), , and .
Each rational function solution is of the form with distinct poles of multiplicity .
Each simply periodic solution is a rational function of . has distinct poles of multiplicity and is of the form
In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function .
Let , be two given complex numbers such that , and let be a discrete subset , which is isomorphic to . The discriminant and
Weierstrass elliptic function is a meromorphic function with double periods , and satisfying the equation where , , and .
If we changed (18) to the form we will have , , and .
Inversely, given two complex numbers and such that , then there exists double periods , Weierstrass elliptic function such that the previous results hold.
Lemma 4 (see [9, 10]). Weierstrass elliptic functions have two successive degeneracies and addition formula:(i) degeneracy to simply periodic functions (i.e., rational functions of one exponential ) according to if one root is double ;(ii) degeneracy to rational functions of according to if one root is triple ;(iii) addition formula
By the previous lemma and results, we can give a new method next say complex method, to find exact solutions of some PDEs.
Step 5. By substituting the inverse transform into these meromorphic solutions , we get all the exact solutions of the original given PDE.
3. Proof of Theorem 1
By substituting (13) into (5), we have . , , , , and . Hence, (5) satisfies weak condition and is a 2-order Briot-Bouquet differential equation. Obviously, (5) satisfies the dominant condition. So, by Lemma 3, we know that all meromorphic solutions of (5) belong to . Now, we will give the forms of all meromorphic solutions of (5).
By (16), we infer the indeterminate rational solutions of (5) with pole at that By substituting into (5), we get two distinct forms as follows: We omit the constant solutions, and we obtain that there does not exist rational function solution.
In order to have simply periodic solutions, set and put into (5), then By substituting into (5), we obtain the indeterminate simply periodic solutions of (27) with pole at that where ; where ; where .
By substituting into the previous six relations, we get all simply periodic solutions of (5) with pole at : where ; where ; where .
So all simply periodic solutions of (5) are obtained, for , by where ; where ; where .
From (15) in Lemma 3, we have indeterminant relations of elliptic solutions of (5) with pole at : where . By applying the conclusion (ii) of Lemma 4 to and noting that the results of rational solutions obtained previously, we deduce that , , and . Then, we get that
This completes the proof of Theorem 1.
4. Computer Graphs for New Solutions
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, . Our results show that simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there does not exist any rational solutions and there exist simply periodic solutions which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role in finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.
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.
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