New Application of (𝐺^′/𝐺)-Expansion Method for Generalized (2+1)-Dimensional Nonlinear Evolution Equations

Abstract

We established (𝐺^′/𝐺)-expansion method for (2+1)-dimensional nonlinear evolution equations. This method was used to construct travelling wave solutions of (2+1)-dimensional nonlinear evolution equations. (2+1)-Dimensional breaking soliton equation, (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation, and (2+1)-dimensional Bogoyavlenskii’s Breaking soliton equation are chosen to illustrate the effectiveness of the method.

1. Introduction

In this work, we will study the generalized (2+1)-dimensional nonlinear evolution equations 𝑢_𝑥𝑡+𝑎𝑢_𝑥𝑢_𝑥𝑦+𝑏𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0, (1) where 𝑎 and 𝑏 are parameters, for example, namely, the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation for which 𝑎=4 and 𝑏=2, [1]: 𝑢_𝑥𝑡+4𝑢_𝑥𝑢_𝑥𝑦+2𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0, (2) and the (2+1)-dimensional breaking soliton equation for which 𝑎=−4 and 𝑏=−2, [2]: 𝑢_𝑥𝑡−4𝑢_𝑥𝑢_𝑥𝑦−2𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0, (3) and the (2+1)-dimensional Bogoyavlenskii’s Breaking soliton equation for which 𝑎=4 and 𝑏=4, [3]: 𝑢_𝑥𝑡+4𝑢_𝑥𝑢_𝑥𝑦+4𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0. (4) In this paper, we solve (1) by the (𝐺^′/𝐺)-expansion method and obtain some exact and new solutions for (2), (3), and (4).

2. The (𝐺^′/𝐺)-Expansion Method

In this section we describe the (𝐺^′/𝐺)-expansion method for finding traveling wave solutions of nonlinear evolution equations. Suppose that a nonlinear equation, say in two independent variables 𝑥, 𝑡 is given by 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑥,𝑢_𝑥𝑥,…=0, (5) where 𝑢=𝑢(𝑥,𝑡) and 𝑃 is a polynomial of 𝑢 and its derivatives, in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of the (𝐺^′/𝐺)-expansion method.

Firstly, suppose that 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥+𝑤𝑡. (6) The traveling wave variable (6) permits reducing (5) to an ODE for 𝑢=𝑢(𝜉)𝑃𝑢,𝑢^′,𝑢^′′,𝑢^′′′,…=0. (7)

Secondly, suppose that the solution of (7) can be expressed by a polynomial in (𝐺^′/𝐺) as follows: 𝑢(𝜉)=𝛼_𝑚𝐺^′𝐺^𝑚+⋯, (8) where 𝐺=𝐺(𝜉) satisfies the second-order LODE in the form 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (9)𝛼_𝑚,…,𝜆 and 𝜇 are constants to be determined later, 𝛼_𝑚≠0. The unwritten part in (8) is also a polynomial in (𝐺^′/𝐺), the degree of which is generally equal to or less than 𝑚−1. The positive integer 𝑚 can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (7).

Thirdly, substituting (8) into (7) and using the second-order linear ODE (9), collecting all terms with the same order of (𝐺^′/𝐺) together, the left-hand side of (7) is converted into another polynomial in (𝐺^′/𝐺). Equating each coefficient of this polynomial to zero yields a set of algebraic equations for 𝛼_𝑚,…,𝜆 and 𝜇.

Fourthly, assuming that the constants 𝛼_𝑚,…,𝜆 and 𝜇 can be obtained by solving the algebraic equations in Thirdly. Since the general solutions of the second-order LODE (9) have been well known for us, then substituting 𝛼_𝑚,…,𝑤 and the general solutions of (9) into (8) we have traveling wave solutions of the nonlinear evolution equation (5) (for more details see [4–7]).

3. New Application of (𝐺^′/𝐺)-Expansion Method

In this section we apply the (𝐺^′/𝐺)-expansion method to the generalized (2+1)-dimensional nonlinear evolution equations 𝑢_𝑥𝑡+𝑎𝑢_𝑥𝑢_𝑥𝑦+𝑏𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0. (10) We introduce a transformation 𝑢𝑥,𝑦,𝑡=𝑢(𝜉),𝜉=𝑥+𝑦−𝑐𝑡. (11) By substituting (11) into (10), we have −𝑐𝑢^′′+𝑎𝑢^′𝑢^′′+𝑏𝑢^′𝑢^′′+𝑢⁽⁴⁾=0, (12) or equivalently −𝑐𝑢^′′+(𝑎+𝑏)𝑢^′𝑢^′′+𝑢⁽⁴⁾=0, (13) where prime denotes the differential with respect to 𝜉. Integrating (13) with respect to 𝜉 and taking the integration constant as zero yields −𝑐𝑢^′+(𝑎+𝑏)2𝑢^′2+𝑢⁽³⁾=𝑔, (14) in which (14) is obtained by integrating and neglecting the constant of integration and where prime denotes the derivative with respect to the same variable 𝜉 and 𝑔 is the integration constant that can be determined later.

Suppose that the solutions of the O.D.E (14) can be expressed by a polynomial in (𝐺^′/𝐺) as follows: 𝑢(𝜉)=^𝑚_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖, (15) where 𝑎_𝑖 are constants, 𝐺=𝐺(𝜉) satisfies the second-order LODE in the form: 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (16) where 𝜆 and 𝜇 are constants.

By balancing the order of (𝑢^′)² and 𝑢⁽³⁾ in (11), we have 2𝑚+2=𝑚+3, then 𝑚=1. So we can write 𝑢(𝜉)=𝑎₁𝐺^′𝐺+𝑎₀,𝑎₁≠0, (17)where  𝑎₁, 𝑎₀ are constants to be determined later. Then it follows 𝑢^′(𝜉)=−𝑎₁𝐺^′𝐺²−𝑎₁𝜆𝐺^′𝐺−𝑎₁𝜇,𝑢^′′(𝜉)=2𝑎₁𝐺^′𝐺³+3𝑎₁𝜆𝐺^′𝐺²+𝑎₁𝜆²𝐺−2𝜇^′𝐺+𝑎₁𝜆𝜇,𝑢^′′′(𝜉)=−6𝑎₁𝐺^′𝐺⁴−12𝑎₁𝜆𝐺^′𝐺³−7𝑎₁𝜆²+8𝑎₁𝜇𝐺^′𝐺²−𝑎₁𝜆³+8𝑎₁𝐺𝜆𝜇^′𝐺−𝑎₁𝜆²𝜇−2𝑎₁𝜇². (18)(19)(20) Substituting (18), (19), and (20) into (14) and collecting all the terms with the same power of (𝐺^′/𝐺) together, equating each coefficient to zero yields a set of simultaneous algebraic equations as follows: 𝐺^′𝐺⁴∶𝑎₁()𝐺𝑎+𝑏−12=0,_′𝐺³∶2𝑎₁()𝐺𝑎+𝑏𝜆−24𝜆=0,_′𝐺²∶2𝑐+2𝑎₁(𝑎+𝑏)𝜇−14𝜆²+𝑎₁(𝑎+𝑏)𝜆²𝐺−16𝜇=0,_′𝐺¹∶2𝑎₁(𝑎+𝑏)𝜆𝜇−2𝜆³𝐺+2𝑐𝜆−16𝜆𝜇=0,_′𝐺⁰∶𝑎₁(𝑎+𝑏)𝜇²+2𝑐𝜇−2𝜆²𝜇−4𝜇²−𝑔=0. (21) Solving the algebraic equations above by using the Maple, we get 𝑎₀=𝑎₀,𝑎₁=12𝑎+𝑏,𝑔=0,𝑐=𝜆²−4𝜇, (22) where 𝑐,𝜆,𝜇 are arbitrary constants.

Substituting (22) into (17) and by using (11), we have 𝑢(𝜉)=12𝐺𝑎+𝑏𝐺^′+𝑎₀, (23) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡.

Substituting the general solutions of (16) into (23), we obtain the following.

Case 1. When 𝜆²−4𝜇>0𝑢₁(𝜉)=6𝜆²−4𝜇×𝐶𝑎+𝑏₁sinh(1/2)𝜆²−4𝜇𝜉+𝐶₂cosh(1/2)𝜆²−4𝜇𝜉𝐶₁cosh(1/2)𝜆²−4𝜇𝜉+𝐶₂sinh(1/2)𝜆²−4𝜇𝜉−𝜆+𝑎₀, (24) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 2. When 𝜆²−4𝜇<0𝑢₂(𝜉)=64𝜇−𝜆²×𝑎+𝑏−𝐶₁sin(1/2)4𝜇−𝜆²𝜉+𝐶₂cos(1/2)4𝜇−𝜆²𝜉𝐶₁cos(1/2)4𝜇−𝜆²𝜉+𝐶₂sin(1/2)4𝜇−𝜆²𝜉−𝜆+𝑎₀, (25) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 3. When 𝜆²−4𝜇=0𝑢₃(𝜉)=12𝐶𝑎+𝑏𝐶₁+²𝐶₂𝜉−𝜆2+𝑎₀, (26) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

In particular, if we take 𝐶₁≠0 and 𝐶₂=0, then 𝑢₁(𝜉) becomes 𝑢₁₁(𝜉)=6𝜆²−4𝜇1𝑎+𝑏tanh2𝜆²−4𝜇𝜉−𝜆+𝑎₀ (27) and 𝑢₂(𝜉) becomes 𝑢₂₁(𝜉)6=−4𝜇−𝜆²1𝑎+𝑏tan24𝜇−𝜆²𝜉+𝜆+𝑎₀. (28) When we take 𝐶₁=0 and 𝐶₂≠0, then 𝑢₁(𝜉) becomes 𝑢₁₂(𝜉)=6𝜆²−4𝜇1𝑎+𝑏coth2𝜆²−4𝜇𝜉−𝜆+𝑎₀ (29) and 𝑢₂(𝜉) becomes 𝑢₂₂(𝜉)=64𝜇−𝜆²𝛼+𝑏cot124𝜇−𝜆²𝜉−𝜆+𝑎₀. (30) In the following Sections 4–6, we will apply solutions 𝑢_𝑖(𝜉),𝑖=1,2,3 to (2+1)-dimensional breaking soliton equation and (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and (2+1)-dimensional Bogoyavlenskii’s Breaking soliton equation.

4. Exact Solution of (2)

In this section, we investigate explicit formula of solutions of the following (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation given in [1] 𝑢_𝑥𝑡+4𝑢_𝑥𝑢_𝑥𝑦+2𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0. (31) By using Section 3, we have following exact solution.

Case 1. When 𝜆²−4𝜇>0𝑢₁(𝜉)=𝜆²×𝐶−4𝜇₁sinh(1/2)𝜆²−4𝜇𝜉+𝐶₂cosh(1/2)𝜆²−4𝜇𝜉𝐶₁cosh(1/2)𝜆²−4𝜇𝜉+𝐶₂sinh(1/2)𝜆²−4𝜇𝜉−𝜆+𝑎₀, (32) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 2. When 𝜆²−4𝜇<0𝑢₂(𝜉)=𝜆²×−4𝜇−𝐶₁sin(1/2)4𝜇−𝜆²𝜉+𝐶₂cos(1/2)4𝜇−𝜆²𝜉𝐶₁cos(1/2)4𝜇−𝜆²𝜉+𝐶₂sin(1/2)4𝜇−𝜆²𝜉−𝜆+𝑎₀, (33) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 3. When 𝜆²−4𝜇=0𝑢₃(𝜉)𝐶=2²𝐶₁+𝐶₂𝜉−𝜆2+𝑎₀, (34) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

In particular, if we take 𝐶₁≠0 and 𝐶₂=0, then 𝑢₁(𝜉) becomes 𝑢₁₁(𝜉)=𝜆²1−4𝜇tanh2𝜆²−4𝜇𝜉−𝜆+𝑎₀, (35) and 𝑢₂(𝜉) becomes 𝑢₂₁(𝜉)=−𝜆²1−4𝜇tan24𝜇−𝜆²𝜉+𝜆+𝑎₀ (36) when we take 𝐶₁=0 and 𝐶₂≠0, then 𝑢₁(𝜉) becomes 𝑢₁₂(𝜉)=𝜆²1−4𝜇coth2𝜆²−4𝜇𝜉−𝜆+𝑎₀ (37) and 𝑢₂(𝜉) becomes 𝑢₂₂(𝜉)=𝜆²−4𝜇cot124𝜇−𝜆²𝜉−𝜆+𝑎₀. (38)

5. Exact Solution of (3)

In this section, we investigate explicit formula of solutions of the following (2+1)-dimensional Breaking soliton equation given in [2] 𝑢_𝑥𝑡−4𝑢_𝑥𝑢_𝑥𝑦−2𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0. (39) By using Section 3, we have following exact solution:

Case 1. When 𝜆²−4𝜇>0𝑢₁(𝜉)=−𝜆²×𝐶−4𝜇₁sinh(1/2)𝜆²−4𝜇𝜉+𝐶₂cosh(1/2)𝜆²−4𝜇𝜉𝐶₁cosh(1/2)𝜆²−4𝜇𝜉+𝐶₂sinh(1/2)𝜆²−4𝜇𝜉−𝜆+𝑎₀, (40) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 2. When 𝜆²−4𝜇<0𝑢₂(𝜉)=−𝜆²×−4𝜇−𝐶₁sin(1/2)4𝜇−𝜆²𝜉+𝐶₂cos(1/2)4𝜇−𝜆²𝜉𝐶₁cos(1/2)4𝜇−𝜆²𝜉+𝐶₂sin(1/2)4𝜇−𝜆²𝜉−𝜆+𝑎₀, (41) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 3. When 𝜆²−4𝜇=0𝑢₃(𝜉)𝐶=−2²𝐶₁+𝐶₂𝜉−𝜆2+𝑎₀, (42) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

In particular, if we take 𝐶₁≠0 and 𝐶₂=0, then 𝑢₁(𝜉) becomes 𝑢₁₁(𝜉)=−𝜆²1−4𝜇tanh2𝜆²−4𝜇𝜉−𝜆+𝑎₀, (43) and 𝑢₂(𝜉) becomes 𝑢₂₁(𝜉)=𝜆²1−4𝜇tan24𝜇−𝜆²𝜉+𝜆+𝑎₀. (44) When we take 𝐶₁=0 and 𝐶₂≠0, then 𝑢₁(𝜉) becomes 𝑢₁₂(𝜉)=−𝜆²1−4𝜇coth2𝜆²−4𝜇𝜉−𝜆+𝑎₀, (45) and 𝑢₂(𝜉) becomes 𝑢₂₂(𝜉)=−𝜆²−4𝜇cot124𝜇−𝜆²𝜉−𝜆+𝑎₀. (46)

6. Exact Solution of (4)

In this section, we investigate explicit formula of solutions of the following (2+1)-dimensional Bogoyavlenskii’s Breaking soliton equation given in [3]: 𝑢_𝑥𝑡+4𝑢_𝑥𝑢_𝑥𝑦+4𝑢_𝑥𝑥𝑢_𝑦+𝑢_{𝑥𝑥𝑥𝑦}=0. (47) By using Section 3, we have following exact solutions.

Case 1. When 𝜆²−4𝜇>0𝑢₁(𝜉)=3𝜆²−4𝜇4×𝐶₁sinh(1/2)𝜆²−4𝜇𝜉+𝐶₂cosh(1/2)𝜆²−4𝜇𝜉𝐶₁cosh(1/2)𝜆²−4𝜇𝜉+𝐶₂sinh(1/2)𝜆²−4𝜇𝜉−𝜆+𝑎₀, (48) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 2. When 𝜆²−4𝜇<0𝑢₂(𝜉)=34𝜇−𝜆²4×−𝐶₁sin(1/2)4𝜇−𝜆²𝜉+𝐶₂cos(1/2)4𝜇−𝜆²𝜉𝐶₁cos(1/2)4𝜇−𝜆²𝜉+𝐶₂sin(1/2)4𝜇−𝜆²𝜉−𝜆+𝑎₀, (49) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

Case 3. When 𝜆²−4𝜇=0𝑢₃(𝜉)=32𝐶₂𝐶₁+𝐶₂𝜉−𝜆2+𝑎₀, (50) where 𝜉=𝑥+𝑦−(𝜆²−4𝜇)𝑡 and 𝐶₁, 𝐶₂, 𝑐, 𝜆, 𝜇 are arbitrary constant.

In particular, if we take 𝐶₁≠0 and 𝐶₂=0, then 𝑢₁(𝜉) becomes 𝑢₁₁(𝜉)=3𝜆²−4𝜇41tanh2𝜆²−4𝜇𝜉−𝜆+𝑎₀, (51) and 𝑢₂(𝜉) becomes 𝑢₂₁(𝜉)3=−4𝜇−𝜆²41tan24𝜇−𝜆²𝜉+𝜆+𝑎₀. (52) When we take 𝐶₁=0 and 𝐶₂≠0, then 𝑢₁(𝜉) becomes 𝑢₁₂(𝜉)=3𝜆²−4𝜇41coth2𝜆²−4𝜇𝜉−𝜆+𝑎₀, (53) and 𝑢₂(𝜉) becomes 𝑢₂₂(𝜉)=34𝜇−𝜆²4cot124𝜇−𝜆²𝜉−𝜆+𝑎₀. (54)

7. Conclusions

In this paper, by using the (𝐺^′/𝐺)-expansion method, we obtained some explicit formulas of solutions for the generalized (2+1)-dimensional nonlinear evolution equations. We chose (2+1)-dimensional breaking soliton equation and (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and (2+1)-dimensional Bogoyavlenskii’s Breaking soliton equation to illustrate the effectiveness of the method. Those solutions were similar to the solutions obtained in other paper. The study reveals the power of the method.