Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

Abstract

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 𝑣_𝑠,3(𝑧) 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.

1. Introduction

In this work, we aim to further study the Drinfeld-Sokolov system (Drinfeld-Sokolov equations (DS system of equations)) [1–3] 𝑢_𝑡+𝑣²_𝑥𝑣=0,_𝑡−𝑎𝑣_𝑥𝑥𝑥+3𝑑𝑣𝑢_𝑥+3𝑘𝑢𝑣_𝑥=0, (1) where 𝑎, 𝑑, and 𝑘 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 [1].

Wazwaz [4] used the sine-cosine method and the tanh method to stress the power of these methods to nonlinear equations. Wazwaz [4] 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 𝑢(𝑥,𝑡)=𝑢(𝑧), 𝑧=𝑥−𝑐𝑡 (where 𝑐 is a complex constant number). Then in (1), the system is carried to a system of ordinary differential equation: −𝑐𝑢^′+𝑣^2′=0,𝑐𝑣^′+𝑎𝑣^′′′−3𝑑𝑣𝑢^′−3𝑘𝑢𝑣²=0. (2)(3) By integrating the first equation in the system and neglecting the constant of integration we find 𝑐𝑢=𝑣². (4) By inserting (4) into (3) and integration we find 𝑐²𝑣−(2𝑑+𝑘)𝑣³+𝑎𝑐𝑣^′′=0. (5)

In 2007, El-Wakil and Abdou [5] got two solutions of (5) via tanh-function method for finding exact solutions of (5).

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 𝑤(𝑧) means that 𝑤(𝑧) is holomorphic in the complex plane ℂ except for poles.℘(𝑧;𝑔₂,𝑔₃) is the Weierstrass elliptic function with invariants 𝑔₂ and 𝑔₃. We say that a meromorphic function 𝑓 belongs to the class 𝑊 if 𝑓 is an elliptic function, or a rational function of 𝑒^𝛼𝑧, 𝛼∈ℂ, or a rational function of 𝑧.

Our main result is the following theorem.

Theorem 1. All meromorphic solutions 𝑢 of (5) belong to the class 𝑊. Furthermore, (5) has the following two forms of solutions:(i) the elliptic general solutions 𝑣_𝑑(𝑧)=±𝑎𝑐×4𝑑+2𝑘−℘+𝐴4℘𝐴²+4℘²𝐴+2℘^′𝐵−℘𝑔₂−𝐴𝑔₂×12𝐴²−𝑔₂℘+4𝐴³−3𝐴𝑔₂℘^′+4𝐵℘³+12𝐴𝐵℘²−3𝐵𝑔₂℘−𝐴𝐵𝑔₂⁻¹, (6)where 𝑔₃=0, 𝐵²=4𝐴³−𝑔₂𝐴, 𝑔₂ and 𝐴 are arbitrary constants;(ii) the simply periodic solutions, where 𝜉=𝑒^𝛼𝑧 are obtained, for 𝑧₀∈ℂ, by 𝑣_𝑠,1(𝜉)1=±𝑐√𝛼2𝑑+𝑘coth2𝑧−𝑧₀, (7)where 𝑎=2𝑐/𝛼²; 𝑣_𝑠,2(𝜉)=±√2𝑐−112𝑑+𝑘sinh𝛼𝑧−𝑧₀, (8) where 𝑎=−𝑐/𝛼²; 𝑣_𝑠,3(𝜉)=±𝑐112𝑑+𝑘sinh𝛼𝑧−𝑧₀+1, (9) where 𝑎=𝑐/2𝛼².

Remark 2. Compared with the results of Wazwaz [4], we find that 𝑣_𝑠,3(𝑧) 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 𝑚∈ℕ∶={1,2,3,…}, 𝑟_𝑗∈ℕ₀=ℕ∪{0}, 𝑟=(𝑟₀,𝑟₁,…,𝑟_𝑚), and 𝑗=0,1,…,𝑚. We define a differential monomial denoted by 𝑀_𝑟[𝑤](𝑧)∶=[𝑤(𝑧)]^𝑟0𝑤^′(𝑧)^𝑟1𝑤^′′(𝑧)^𝑟2⋯𝑤^(𝑚)(𝑧)^𝑟𝑚. (10)𝑝(𝑟)∶=𝑟₀+2𝑟₁+⋯+(𝑚+1)𝑟_𝑚 and deg(𝑀) are called the weight and degree of 𝑀_𝑟[𝑤], respectively.

A differential polynomial 𝑃(𝑤,𝑤^′,…,𝑤^(𝑚)) is defined as follows: 𝑃𝑤,𝑤^′,…,𝑤^(𝑚)∶=_𝑟∈𝐼𝑎_𝑟𝑀_𝑟[𝑤], (11) where 𝑎_𝑟 are complex constants and 𝐼 is a finite index set. The total weight and degree of 𝑃(𝑤,𝑤^′,…,𝑤^(𝑚)) are defined by 𝑊(𝑃)∶=max_𝑟∈𝐼{𝑝(𝑟)} and deg(𝑃)∶=max_𝑟∈𝐼{deg(𝑀_𝑟)}, respectively.

We will consider the following complex ordinary differential equations: 𝑃𝑤,𝑤^′,…,𝑤^(𝑚)=𝑏𝑤^𝑛+𝑐, (12) where 𝑏≠0, 𝑐 are complex constants, 𝑛∈ℕ.

Let 𝑝, 𝑞∈ℕ. Suppose that (12) has a meromorphic solution 𝑤 with at least one pole; we say that (12) satisfies weak ⟨𝑝,𝑞⟩ condition if it substitutes Laurent series 𝑤(𝑧)=^∞_𝑘=−𝑞𝑐_𝑘𝑧^𝑘,𝑞>0,𝑐_−𝑞≠0 (13) into (12). We can determine 𝑝 distinct Laurent singular parts as follows: ⁻¹_𝑘=−𝑞𝑐_𝑘𝑧^𝑘. (14)

Lemma 3 (see [6–8]). Let 𝑝,𝑙,𝑚,𝑛∈ℕ, deg𝑃(𝑤,𝑤^(𝑚))<𝑛. Suppose that an 𝑚-order Briot-Bouquet equation 𝑃𝑤^(𝑚),𝑤=𝑏𝑤^𝑛+𝑐 (15) satisfies weak ⟨𝑝,𝑞⟩ condition; then all meromorphic solutions of (15) belong to the class 𝑊. If for some values of parameters such solution 𝑤 exists, then other meromorphic solutions form a one-parametric family 𝑤(𝑧−𝑧₀),𝑧₀∈ℂ. Furthermore, each elliptic solution with pole at 𝑧=0 can be written as 𝑤(𝑧)=^𝑙−1𝑞_𝑖=1^𝑖_𝑗=2(−1)^𝑗𝑐_−𝑖𝑗!𝑑𝑗−1𝑑𝑧^{𝑗−2𝑗−2}14℘^′(𝑧)+𝐵_𝑖℘(𝑧)−𝐴_𝑖²(𝑧)+−℘_{𝑙−1𝑖=1}𝑐_−𝑖12℘^′(𝑧)+𝐵_𝑖℘(𝑧)−𝐴_𝑖+^𝑞𝑙_𝑗=2(−1)^𝑗𝑐_−𝑙𝑗!𝑑𝑗−1𝑑𝑧^{𝑗−2𝑗−2}℘(𝑧)+𝑐₀, (16) where 𝑐_−𝑖𝑗 are given by (13), 𝐵²_𝑖=4𝐴³_𝑖−𝑔₂𝐴_𝑖−𝑔₃, and ∑^𝑙_𝑖=1𝑐_−𝑖1=0.

Each rational function solution 𝑤∶=𝑅(𝑧) is of the form 𝑅(𝑧)=^𝑙𝑞_𝑖=1𝑗=1𝑐_𝑖𝑗𝑧−𝑧_𝑖^𝑗+𝑐₀, (17) with 𝑙(≤𝑝) distinct poles of multiplicity 𝑞.

Each simply periodic solution is a rational function 𝑅(𝜉) of 𝜉=𝑒^𝛼𝑧(𝛼∈ℂ). 𝑅(𝜉) has 𝑙(≤𝑝) distinct poles of multiplicity 𝑞 and is of the form 𝑅(𝜉)=^𝑙𝑞_𝑖=1𝑗=1𝑐_𝑖𝑗𝜉−𝜉_𝑖^𝑗+𝑐₀. (18)

In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [9].

Let 𝜔₁, 𝜔₂ be two given complex numbers such that Im(𝜔₁/𝜔₂)>0, and let 𝐿=𝐿[2𝜔₁,2𝜔₂] be a discrete subset 𝐿[2𝜔₁,2𝜔₂]={𝜔∣𝜔=2𝑛𝜔₁+2𝑚𝜔₂,𝑛,𝑚∈ℤ}, which is isomorphic to ℤ×ℤ. The discriminant Δ=Δ(𝑐₁,𝑐₂)∶=𝑐³₁−27𝑐²₂ and 𝑠_𝑛=𝑠_𝑛(𝐿)∶=_{𝜔∈𝐿⧵{0}}1𝜔^𝑛. (19)

Weierstrass elliptic function ℘(𝑧)∶=℘(𝑧,𝑔₂,𝑔₃) is a meromorphic function with double periods 2𝜔₁, 2𝜔₂ and satisfying the equation ℘^′(𝑧)²=4℘(𝑧)³−𝑔₂℘(𝑧)−𝑔₃, (20) where 𝑔₂=60𝑠₄, 𝑔₃=140𝑠₆, and Δ(𝑔₂,𝑔₃)≠0.

If we changed (18) to the form ℘^′(𝑧)²=4℘(𝑧)−𝑒₁℘(𝑧)−𝑒₂℘(𝑧)−𝑒₂, (21) we will have 𝑒₁=℘(𝜔₁), 𝑒₂=℘(𝜔₂), and 𝑒₃=℘(𝜔₁+𝜔₂).

Inversely, given two complex numbers 𝑔₂ and 𝑔₃ such that Δ(𝑔₂,𝑔₃)≠0, then there exists double periods 2𝜔₁, 2𝜔₂ Weierstrass elliptic function ℘(𝑧) such that the previous results hold.

Lemma 4 (see [9, 10]). Weierstrass elliptic functions ℘(𝑧)∶=℘(𝑧,𝑔₂,𝑔₃) have two successive degeneracies and addition formula:(i) degeneracy to simply periodic functions (i.e., rational functions of one exponential 𝑒^𝑘𝑧) according to ℘𝑧,3𝑑²,−𝑑³=2𝑑−3𝑑2coth²3𝑑2𝑧, (22)if one root 𝑒_𝑗 is double (Δ(𝑔₂,𝑔₃)=0);(ii) degeneracy to rational functions of 𝑧 according to ℘()=1𝑧,0,0𝑧² (23)if one root 𝑒_𝑗 is triple (𝑔₂=𝑔₃=0);(iii) addition formula ℘𝑧−𝑧₀=−℘(𝑧)−℘𝑧₀+14℘^′(𝑧)+℘^′𝑧₀℘(𝑧)−℘𝑧₀². (24)

By the previous lemma and results, we can give a new method next say complex method, to find exact solutions of some PDEs.

Step 1. Substituting the transform 𝑇∶𝑢(𝑥,𝑡)→𝑤(𝑧),(𝑥,𝑡)→𝑧 into a given PDE gives a nonlinear ordinary differential equations (12) or (15).

Step 2. Substitute (13) into (12) or (15) to determine that weak ⟨𝑝,𝑞⟩ condition holds.

Step 3. By determinant relation (16)–(18), we find the elliptic rational and simply periodic solutions 𝑢(𝑧) of (12) or (15) with pole at 𝑧=0, respectively.

Step 4. By Lemmas 3 and 4, we obtain the all meromorphic solutions 𝑤(𝑧−𝑧₀).

Step 5. By substituting the inverse transform 𝑇⁻¹ into these meromorphic solutions 𝑤(𝑧−𝑧₀), we get all the exact solutions 𝑢(𝑥,𝑡) of the original given PDE.

3. Proof of Theorem 1

By substituting (13) into (5), we have 𝑞=1. 𝑝=2, 𝑐₋₁=±2𝑎𝑐/(2𝑑+𝑘), 𝑐₀=0, 𝑐₁=√2𝑐²/6𝑎𝑐(2𝑑+𝑘), and 𝑐₂=0. Hence, (5) satisfies weak ⟨2,1⟩ 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 𝑊. 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 𝑧=0 that 𝑣_𝑟(𝑧)=𝑐₁₁𝑧+𝑐₁₂𝑧−𝑧₁+𝑐₁₀. (25) By substituting 𝑣_𝑟(𝑧) into (5), we get two distinct forms as follows: 𝑣_𝑟,1(𝑧)𝑣=0,_𝑟,2(𝑧)=±𝑐1.2𝑑+𝑘 (26) We omit the constant solutions, and we obtain that there does not exist rational function solution.

In order to have simply periodic solutions, set 𝜉=exp(𝛼𝑧) and put 𝑣=𝑣(𝜉) into (5), then 𝑎𝑐𝛼²𝜉²𝑣^′′+𝜉𝑣^′−(2𝑑+𝑘)𝑣³+𝑐²𝑣=0. (27) By substituting 𝑣₂(𝜉)=𝑐₂+𝑐𝜉−1𝜉−¹𝜉₁+𝑐₀ (28) into (5), we obtain the indeterminate simply periodic solutions of (27) with pole at 𝜉=1 that 𝑣_𝑠1,1(𝜉)=±𝑐122𝑑+𝑘,𝜉−1+1 (29) where 𝑎=2𝑐/𝛼²; 𝑣_𝑠1,2(𝜉)=±√2𝑐−112𝑑+𝑘+1𝜉−1,𝜉+1 (30) where 𝑎=−𝑐/𝛼²; 𝑣_𝑠1,3(𝜉)=±𝑐112𝑑+𝑘+1𝜉−1,𝜉+1+1 (31) where 𝑎=𝑐/2𝛼².

By substituting 𝜉=𝑒^𝛼𝑧 into the previous six relations, we get all simply periodic solutions of (5) with pole at 𝑧=0: 𝑣_𝑠0,1(𝑧)=±𝑐122𝑑+𝑘𝑒^𝛼𝑧,−1+1 (32) where 𝑎=2𝑐/𝛼²; 𝑣_𝑠0,2(𝑧)=±√2𝑐−112𝑑+𝑘𝑒^𝛼𝑧+1−1𝑒^𝛼𝑧,+1 (33) where 𝑎=−𝑐/𝛼²; 𝑣_𝑠0,3(𝑧)=±𝑐112𝑑+𝑘𝑒^𝛼𝑧+1−1𝑒^𝛼𝑧,+1+1 (34) where 𝑎=𝑐/2𝛼².

So all simply periodic solutions of (5) are obtained, for 𝑧₀∈ℂ, by 𝑣_𝑠,1(𝑧)=±𝑐122𝑑+𝑘𝑒^{𝛼(𝑧−𝑧0)}1−1+1=±𝑐√𝛼2𝑑+𝑘coth2𝑧−𝑧₀, (35) where 𝑎=2𝑐/𝛼²; 𝑣_𝑠,2(𝑧)=±√2𝑐−112𝑑+𝑘𝑒^{𝛼(𝑧−𝑧0)}+1−1𝑒^{𝛼(𝑧−𝑧0)}√+1=±2𝑐−112𝑑+𝑘sinh𝛼𝑧−𝑧₀, (36) where 𝑎=−𝑐/𝛼²; 𝑣_𝑠,3(𝑧)=±𝑐112𝑑+𝑘𝑒^{𝛼(𝑧−𝑧0)}+1−1𝑒^{𝛼(𝑧−𝑧0)}+1+1=±𝑐112𝑑+𝑘sinh𝛼𝑧−𝑧₀+1, (37) where 𝑎=𝑐/2𝛼².

From (15) in Lemma 3, we have indeterminant relations of elliptic solutions of (5) with pole at 𝑧=0: 𝑣_𝑑0(𝑧)=±𝑎𝑐℘4𝑑+2𝑘^′(𝑧)+𝐹℘(𝑧)−𝐸+𝑐₃₀, (38) where 𝐹²=4𝐸³−𝑔₂𝐸−𝑔₃. By applying the conclusion (ii) of Lemma 4 to 𝑢_𝑑0(𝑧) and noting that the results of rational solutions obtained previously, we deduce that 𝑐₃₀=0, 𝐸=𝐹=0, and 𝑔₃=0. Then, we get that 𝑣_𝑑0(𝑧)=±𝑎𝑐℘4𝑑+2𝑘^′(𝑧)℘(𝑧). (39)

Therefore, all elliptic function solutions of (5) are as follows 𝑣_𝑑0(𝑧)=±𝑎𝑐℘4𝑑+2𝑘^′𝑧−𝑧₀℘𝑧−𝑧₀. (40) Here 𝑧₀∈ℂ,𝑔₃=0. Making use of the addition of Lemma 3 we rewrite it to the form 𝑣_𝑑(𝑧)=±𝑎𝑐×4𝑑+2𝑘−℘+𝐴4℘𝐴²+4℘²𝐴+2℘^′𝐵−℘𝑔₂−𝐴𝑔₂×12𝐴²−𝑔₂℘+4𝐴³−3𝐴𝑔₂℘^′+4𝐵℘³+12𝐴𝐵℘²−3𝐵𝑔₂℘−𝐴𝐵𝑔₂⁻¹. (41) Here 𝑔₃=0, 𝐵²=4𝐴³−𝑔₂𝐴, 𝑔₂, and 𝐴 are arbitrary constants.

This completes the proof of Theorem 1.

4. Computer Graphs for New Solutions

In this section, we give some computer graphs to illustrate our main results. Here, we take the simple periodic solutions 𝑣_𝑠,1−3(𝑧) by Figures 1, 2, and 3.

(a)

(b)

(c)

Figure 1 

The solution of (DS) corresponding to 𝑣𝑠,1: take 𝑐=1, 𝑧0=0, 𝑑=1/2, 𝑘=0, 𝛼=2, (a) 𝑡=−1/10, (b) 𝑡=0, and (c) 𝑡=1/10.

(a)

(b)

(c)

Figure 2 

The solution of (DS) corresponding to 𝑣𝑠,2: take 𝑐=1, 𝑧0=0, 𝑑=−1/2, 𝑘=0, 𝛼=1, (a) 𝑡=−1/3, (b) 𝑡=0, and (c) 𝑡=1/3.

(a)

(b)

(c)

Figure 3 

The solution of (DS) corresponding to 𝑣𝑠,3: take 𝑐=1, 𝑧0=0, 𝑑=1/2, 𝑘=0, 𝛼=1, (a) 𝑡=−1/3, (b) 𝑡=0, and (c) 𝑡=1/3.

5. Conclusions

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, [5]. 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 𝑣_𝑠,3(𝑧) 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.

Acknowledgments

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.