The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

Abstract

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

1. Introduction

Nonlinear phenomena exist in all areas of science and engineering, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. It is well known that many nonlinear partial differential equations (NLPDEs) are widely used to describe these complex physical phenomena. The exact solution of a differential equation gives information about the construction of complex physical phenomena. Therefore, seeking exact solutions of NLPDEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages, like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balance method [1, 2], the auxiliary equation method [3], the sine-cosine method [4], the Jacobi elliptic function method [5], the exp-function method [6], the tanh-function method [7, 8], the Darboux transformation [9, 10], and the (𝐺^′/𝐺)-expansion method [11, 12].

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [13, 14] and used successfully by many authors for finding exact solutions of ODEs in mathematical physics [15–19].

In this paper, we first apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained.

2. Description of Methods

2.1. The Simplest Equation Method

Step 1. Suppose that we have a nonlinear partial differential equation (PDE) for 𝑢(𝑥,𝑡) in the form 𝑁𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,…=0, (1) where 𝑁 is a polynomial in its arguments.

Step 2. By taking 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥−𝑐𝑡, we look for traveling wave solutions of (1) and transform it to the ordinary differential equation (ODE) 𝑁𝑢,−𝑐𝑢^′,𝑢^′,𝑐²𝑢^′′,−𝑐𝑢^′′,𝑢^′′,…=0. (2)

Step 3. Suppose the solution 𝑢 of (2) can be expressed as a finite series in the form 𝑢=^𝑛_𝑖=0𝐴_𝑖(𝐻(𝜉))^𝑖, (3) where 𝐻(𝜉) satisfies the Bernoulli or Riccati equation, 𝑛 is a positive integer that can be determined by balancing procedure, and 𝐴_𝑖(𝑖=0,1,2,…,𝑛) are parameters to be determined.
The Bernoulli equation we consider in this paper is 𝐻^′(𝜉)=𝑎𝐻(𝜉)+𝑏𝐻²(𝜉), (4) where 𝑎 and 𝑏 are constants. Its solutions can be written as 𝐻(𝜉)=−𝑎𝐶₁𝑏𝐶₁+cosh𝑎𝜉+𝜉₀−sinh𝑎𝜉+𝜉₀,𝐻(𝜉)=−𝑎cosh𝑎𝜉+𝜉₀+sinh𝑎𝜉+𝜉₀𝑏𝐶₂+cosh𝑎𝜉+𝜉₀+sinh𝑎𝜉+𝜉₀, (5) where 𝐶₁,𝐶₂, and 𝜉₀ are constants.
For the Riccati equation 𝐻^′(𝜉)=𝑎𝐻²(𝜉)+𝑏𝐻(𝜉)+𝑠, (6) where 𝑎,𝑏, and 𝑠 are constants, we will use the solutions 𝐻(𝜉)𝑏=−−𝜃2𝑎𝜃2𝑎tanh2𝜉+𝜉₀,𝐻(𝜉)𝑏=−−𝜃2𝑎𝜃2𝑎tanh2𝜉+sech(𝜃𝜉/2)()−()(),𝐶cosh𝜃𝜉/22𝑎/𝜃sinh𝜃𝜉/2 (7) where 𝜃²=𝑏²−4𝑎𝑠.

Step 4. Substituting (3) into (2) with (4) (or (6)), then the left hand side of (2) is converted into a polynomial in 𝐻(𝜉); equating each coefficient of the polynomial to zero yields a set of algebraic equations for 𝐴_𝑖,𝑎,𝑏(𝑖=0,1,2,…,𝑛). Solving the algebraic equations by symbolic computation, we can determine those parameters explicitly.

Step 5. Assuming that the constants 𝐴_𝑖,𝑎,𝑏(𝑖=0,1,2,…,𝑛) can be obtained in Step 4 and substituting the results into (3), then we obtain the exact traveling wave solutions for (1).

2.2. The Modified Simplest Equation Method

In the modified version, one makes an ansatz for the solution 𝑢(𝜉) as 𝑢=^𝑛_𝑖=0𝑎_𝑖𝜓^′(𝜉)𝜓(𝜉)^𝑖, (8) where 𝑎_𝑖(𝑖=0,1,2,…,𝑛) are arbitrary constants to be determined, such that 𝑎_𝑛≠0 and 𝜓(𝜉) is an unspecified function to be determined afterward.

Substitute (8) into (2) and then we account the function 𝜓(𝜉). As a result of this substitution, we get a polynomial of 𝜓^′(𝜉)/𝜓(𝜉) and its derivatives. In this polynomial, we equate the coefficients of the same power of 𝜓^−𝑗(𝜉) to zero, where 𝑗≥0. This procedure yields a system of equations which can be solved to find 𝑎_𝑖(𝑖=0,1,2,…,𝑛),𝜓(𝜉), and 𝜓^′(𝜉). Then the substitution of the values of 𝑎_𝑖(𝑖=0,1,2,…,𝑛),𝜓(𝜉), and 𝜓^′(𝜉) into (8) completes the determination of exact solutions of (1).

3. Solutions of the Elliptic-Like Equation

Now, let us choose the following elliptic-like equation 𝐴𝜙^′′(𝜉)+𝐵𝜙(𝜉)+𝐶𝜙³(𝜉)=0, (9) where 𝐴,𝐵, and 𝐶 are arbitrary constants. Equation (9) is one of the most important auxiliary equations, because many nonlinear evolution equations can be converted to (9) using the travelling wave reduction.

3.1. Using Simplest Equation Method

3.1.1. Solutions of (9) Using the Bernoulli Equation as the Simplest Equation

Considering the homogeneous balance between 𝜙^′′(𝜉), and 𝜙³(𝜉) we get 𝑛=1, so the solution of (9) is the form 𝜙(𝜉)=𝐴₀+𝐴₁𝐻(𝜉). (10)

Substituting (10) into (9) and making use of the Bernoulli equation (4) and then equating the coefficients of the functions 𝐻^𝑖(𝜉) to zero, we obtain an algebraic system of equations in terms of 𝐴_𝑖(𝑖=0,1),𝑎, and 𝑏. Solving this system of algebraic equations, with the aid of Maple, we obtain 𝐴₀=±𝑖𝐵𝐶,𝐴₁=±𝑖2𝐴𝐶,𝑎=2𝐵𝐴,𝐴₀=±𝑖𝐵𝐶,𝐴₁=∓𝑖2𝐴𝐶,𝑎=−2𝐵𝐴. (11)

Therefore, using solutions (5) of (4) and ansatz (10), we obtain the following exact solution of (9): 𝜙₁(𝜉)=±𝑖𝐵𝐶1−2𝐶₁×𝐶₁+cosh2𝐵𝐴𝜉+𝜉₀−sinh2𝐵𝐴𝜉+𝜉₀⁻¹,𝜙₂(𝜉)=±𝑖𝐵𝐶1−2𝐶₁×𝐶₁+cosh2𝐵𝐴𝜉+𝜉₀+sinh2𝐵𝐴𝜉+𝜉₀⁻¹,𝜙₃(𝜉)=±𝑖𝐵𝐶1−2cosh2𝐵𝐴𝜉+𝜉₀+sinh2𝐵𝐴𝜉+𝜉₀×𝐶₂+cosh2𝐵𝐴𝜉+𝜉₀+sinh2𝐵𝐴𝜉+𝜉₀⁻¹,𝜙₄(𝜉)=±𝑖𝐵𝐶1−2cosh2𝐵𝐴𝜉+𝜉₀−sinh2𝐵𝐴𝜉+𝜉₀×𝐶₂+cosh2𝐵𝐴𝜉+𝜉₀−sinh2𝐵𝐴𝜉+𝜉₀⁻¹. (12)

3.1.2. Solutions of (9) Using Riccati Equation as the Simplest Equation

Suppose the solutions of (9) are the form 𝜙(𝜉)=𝐵₀+𝐵₁𝐻(𝜉). (13)

Substituting (13) into (9) and making use of the Riccati equation (6) and then equating the coefficients of the functions 𝐻^𝑖(𝜉) to zero, we obtain an algebraic system of equations in terms of 𝐵_𝑖(𝑖=0,1),𝑎,𝑏, and 𝑠. Solving this system of algebraic equations, with the aid of Maple, one possible set of values of 𝐵_𝑖(𝑖=0,1),𝑎,𝑏, and 𝑠 is 𝐵₀=±𝑖𝐴2𝐶𝑏,𝐵₁=±𝑖2𝐴𝐶𝑎,𝑠=𝐴𝑏²−2𝐵.4𝐴𝑎 (14)

Therefore, using solutions (7) of (6) and ansatz (13), we obtain the following exact solution of (9): 𝜙₅(𝜉)=∓𝑖𝐵𝐶tanh𝐵2𝐴𝜉+𝜉₀,𝜙₆(𝜉)=∓𝑖𝐵𝐶2𝐶2𝐵𝐴sinh𝐵𝜉2𝐴−2𝑎cosh𝐵𝜉×𝐶2𝐴𝐵𝐴cosh𝐵𝜉√2𝐴−𝑎2sinh𝐵𝜉2𝐴⁻¹. (15)

3.2. Using Modified Simplest Equation Method

Suppose the solution of (9) is the form 𝜙(𝜉)=𝑎₀+𝑎₁𝜓^′(𝜉)𝜓(𝜉), (16) where 𝑎₀ and 𝑎₁ are constants, such that 𝑎₁≠0, and 𝜙(𝜉) is an unspecified function to be determined. It is simple to calculate that 𝜙^′=𝑎₁𝜓^′′𝜓−𝜓^′𝜓²,𝜙^′′=𝑎₁𝜓^′′′𝜓−3𝑎₁𝜓^′′𝜓^′𝜓²+2𝑎₁𝜓^′𝜓³,𝜙³=𝑎³₁𝜓^′𝜓³+3𝑎²₁𝑎₀𝜓^′𝜓²+3𝑎₁𝑎²₀𝜓^′𝜓+𝑎³₀. (17)

Substituting the values of 𝜙,𝜙^′′, and 𝜙³ into (9) and equating the coefficients of 𝜓⁰,𝜓⁻¹,𝜓⁻², and 𝜓⁻³ to zero yield 𝜓⁰∶𝐶𝑎³₀+𝐵𝑎₀𝜓=0,₋₁∶3𝐶𝑎²₀+𝐵𝜓^′+𝐴𝜓^′′′𝜓=0,₋₂∶−3𝐴𝜓^′′+3𝐶𝑎₀𝑎₁𝜓^′𝜓=0,₋₃∶𝐶𝑎³₁+2𝐴𝑎₁𝜓^′3=0. (18)(19)(20)(21)

Solving (18), we obtain 𝑎₀=0,𝑎₀=±𝑖𝐵𝐶. (22)

And solving (21), we obtain 𝑎₁=±𝑖2𝐴𝐶,since𝑎₁≠0. (23)

Case 1. When 𝑎₀=0, we obtain trivial solution; therefore, the case is rejected.

Case 2. When 𝑎₀=±𝑖𝐵/𝐶and𝑎₁=±𝑖2𝐴/𝐶, (19) and (20) yield 𝜓^′′′𝜓^′′+2𝐵𝐴=0. (24) Integrating (24) with respect to 𝜉, we obtain 𝜓^′′=𝐶₁exp−2𝐵𝐴𝜉. (25) Using (25), from (20), we obtain 𝜓^′=−𝐴𝐶2𝐵¹exp−2𝐵𝐴𝜉. (26)

Upon integration, we obtain 𝜓=𝐴𝐶₁−2𝐵exp2𝐵𝐴𝜉+𝐶₂, (27) Where 𝐶₁ and 𝐶₂ are constants of integration. Therefore, the exact solution of (9) is 𝜙₇(𝜉)=±𝑖𝐵𝐶1−2𝐴𝐶₁exp−(2𝐵/𝐴)𝜉𝐴𝐶₁exp−(2𝐵/𝐴)𝜉+2𝐵𝐶₂. (28)

From (28), we obtain the exact solution of (9) which is 𝜙₈(𝜉)=±𝑖𝐵𝐶1−2𝐴𝐶₁cosh𝐵𝜉2𝐴−sinh𝐵𝜉×2𝐴2𝐵𝐶₂+𝐴𝐶₁cosh𝐵𝜉+2𝐴2𝐵𝐶₂−𝐴𝐶₁sinh𝐵𝜉2𝐴⁻¹. (29)

We can arbitrarily choose the parameters 𝐶₁ and 𝐶₂. Therefore, if we set 𝐶₁=2𝐵𝐶₂/𝐴, (29) reduces to 𝜙₉(𝜉)=±𝑖𝐵𝐶tanh𝐵𝜉.2𝐴 (30)

Again setting 𝐶₁=−(2𝐵𝐶₂/𝐴), (29) reduces to, 𝜙₁₀(𝜉)=±𝑖𝐵𝐶coth𝐵𝜉.2𝐴 (31)

Using hyperbolic function identities, from (30) and (31), we obtain the following periodic solutions 𝜙₁₁(𝜉)=±𝐵𝐶tan−𝐵𝜉,𝜙2𝐴₁₂(𝜉)=±𝐵𝐶cot−𝐵𝜉.2𝐴 (32)

4. Exact Solutions of Some Class of NLPDEs

4.1. The Perturbed Nonlinear Schrödinger's Equation (NLSE) in the Form [20]

Using 𝑖𝑢_𝑡+𝑢_𝑥𝑥+𝛼𝑢|𝑢|²+𝑖𝛾₁𝑢_𝑥𝑥𝑥+𝛾₂|𝑢|²𝑢_𝑥+𝛾₃𝑢|𝑢|²_𝑥=0, (33) where 𝛾₁ is the third order dispersion, 𝛾₂ is the nonlinear dispersion, while 𝛾₃ is also a version of nonlinear dispersion. We assume that (33) has exact solution in the form 𝑢(𝑥,𝑡)=𝜙(𝜉)exp(𝑖(𝜆𝑥−𝜔𝑡)),𝜉=𝑘(𝑥−𝑐𝑡), (34) where 𝜆,𝜔,𝑘, and 𝑐 are arbitrary constant to be determined. Substituting (34) into (33), removing the common factor exp(𝑖(𝜆𝑥−𝜔𝑡)), we have 𝑖𝛾₁𝑘³𝜙^′′′−𝑘𝑐−2𝜆+3𝛾₁𝜆²𝜙^′+𝑘𝛾₂+2𝛾₃𝜙²𝜙^′+𝑘²1−3𝛾₁𝜆𝜙^′′+𝜔−𝜆²+𝛾₁𝜆³𝜙+𝛼−𝛾₂𝜆𝜙³=0, (35) where 𝛾_𝑖(𝑖=1,2, and 3),𝛼,  and  𝑘 are positive constants and the prime means differentiation with respect to 𝜉. Then we have two equations as follows 𝛾₁𝑘²𝜙^′′′−𝑐−2𝜆+3𝛾₁𝜆²𝜙^′+𝛾₂+2𝛾₃𝜙²𝜙^′𝑘=0,₂1−3𝛾₁𝜆𝜙^′′+𝜔−𝜆²+𝛾₁𝜆³𝜙+𝛼−𝛾₂𝜆𝜙³=0. (36)(37)

Integrating (36) with respect to 𝜉 once and setting the integration constant to be zero, then we have 𝛾₁𝑘²𝜙^′′+2𝜆−𝑐−3𝛾₁𝜆²1𝜙+3𝛾₂+23𝛾₃𝜙³=0. (38)

As (37) and (38) have the same solutions, we have the following equation:𝛾₁1−3𝛾₁𝜆=2𝜆−𝑐−3𝛾₁𝜆²𝜔−𝜆²+𝛾₁𝜆³=𝐶𝛼−𝛾₂𝜆, (39) where 𝐶=(1/3)𝛾₂+(2/3)𝛾₃.

From (39), we can obtain 𝜔=𝛼−𝛾₂𝜆2𝜆−𝑐−3𝛾₁𝜆²𝐶+𝜆²−𝛾₁𝜆³,𝜆=𝐶−𝛼𝜆₁3𝐶𝛾₁−𝛾₁𝛾₂. (40)

Based on the conclusion just mentioned, we only solve (38) or (37), instead of both (37) and (38), provided that (37) and (36) are replaced by (40), respectively, we get 𝐴𝜙^′′(𝜉)+𝐵𝜙(𝜉)+𝐶𝜙³(𝜉)=0. (41)

Equation (41) is identical to (9) and 𝐴,𝐵, and 𝐶 are defined by 𝐴=𝛾₁𝑘²,𝐵=2𝜆−𝑐−3𝛾₁𝜆²1,𝐶=3𝛾₂+23𝛾₃. (42)

Then, solutions of (33) are defined as follows: 𝑢(𝑥,𝑡)=𝜙(𝜉)exp(𝑖(𝜆𝑥−𝜔𝑡)),𝜉=𝑘(𝑥−𝑐𝑡),𝜔=𝛼−𝛾₂𝜆2𝜆−𝑐−3𝛾₁𝜆²𝐶+𝜆²−𝛾₁𝜆³,𝜆=𝐶−𝛼𝜆₁3𝐶𝛾₁−𝛾₁𝛾₂, (43) where 𝜙(𝜉), appearing in these solutions, is given by relations (12), (15), and (28)–(32). 𝐴,𝐵, and 𝐶 are defined by (42).

4.2. The Klein-Gordon-Zakharov (KGZ) System [21]

Consider𝐸_𝑡𝑡−𝐸_𝑥𝑥+𝐸−𝛼𝑁𝐸−𝛽|𝐸|²𝑁𝐸=0,_𝑡𝑡−𝑁_𝑥𝑥=𝛾|𝐸|²_𝑥𝑥, (44) wherein the complex valued unknown function 𝐸=𝐸(𝑥,𝑡) denotes the fast time scale component of electric field raised by electrons, and the real valued unknown function 𝑁=𝑁(𝑥,𝑡) represents the deviation of ion density. 𝛼,𝛽 and 𝛾 are some real parameters.

We assume that 𝐸(𝑥,𝑡)=𝜙(𝜉)exp(𝑖(𝑘𝑥+𝜔𝑡)),𝑁=𝑁(𝜉),𝜉=𝜔𝑥+𝑘𝑡. (45)

Substituting (45) into (44), we have −𝛼𝑁𝜙−𝛽𝜙³+𝑘²−𝜔²+1𝜙+𝑘²−𝜔²𝜙^′′𝑘=0,₂−𝜔²𝑁^′′−𝛾𝜔²2𝜙^′2+2𝜙𝜙^′′=0. (46)(47)

Integrating (47) with respect to 𝜉 twice and setting the integration constant to be zero, then we have 𝑁=𝛾𝜔²𝑘²−𝜔²𝜙². (48)

Substituting (48) into (46), we have 𝐴𝜙^′′(𝜉)+𝐵𝜙(𝜉)+𝐶𝜙³(𝜉)=0. (49)

Equation (49) is identical to (9) and 𝐴,𝐵, and 𝐶 are defined by 𝐴=𝑘²−𝜔²,𝐵=𝑘²−𝜔²+1,𝐶=−𝛽+𝛼𝛾𝜔²𝑘²−𝜔². (50)

Then, solutions of the Klein-Gordon-Zakharov (KGZ) system are defined as follows: 𝐸(𝑥,𝑡)=𝜙(𝜉)exp(𝑖(𝑘𝑥+𝜔𝑡)),𝑁=𝛾𝜔²𝑘²−𝜔²𝜙²,𝜉=𝜔𝑥+𝑘𝑡, (51) where 𝜙(𝜉) appearing in these solutions is given by relations (12), (15), and (28)–(32). 𝐴,𝐵, and 𝐶 are defined by (50).

4.3. A Class of Nonlinear Partial Differential Equations (NPDEs)

We consider a class of NLPDEs with constant coefficients [22] 𝑖𝑢_𝑡+𝜇𝑢_𝑥𝑥+𝐷₁𝑢_𝑦𝑦+𝐸₁|𝑢|²𝑢+𝐴₁𝐷𝑢𝑛=0,₂𝑛_𝑡𝑡+𝑛_𝑥𝑥−𝐸₂𝑢_𝑦𝑦+𝐴₂|𝑢|²_𝑥𝑥=0, (52) where 𝜇,𝐷_𝑖,𝐸_𝑖,and𝐴_𝑖(𝑖=1,2) are real constants and 𝜇≠0,𝐷₁≠0,𝐴₁≠0,and𝐴₂≠0. Equations (52) are a class of physically important equations. In fact, if one takes 1𝜇=2𝜅²,𝐷₁=2𝜇,𝐸₁=𝛼,𝐴₁𝐷=−1,₂=0,𝐸₂=𝐷₁,𝐴₂=−2𝛼,𝜅²=±1, (53) then (52) represent the Davey-Stewartson (DS) equations [23] 𝑖𝑢_𝑡+12𝜅²𝑢_𝑥𝑥+𝜅²𝑢_𝑦𝑦+𝛼|𝑢|²𝑛𝑢−𝑢𝑛=0,_𝑥𝑥−𝜅²𝑛_𝑦𝑦−2𝛼|𝑢|²_𝑥𝑥=0. (54)

If one takes 𝑛=𝑛(𝑥,𝑡),thatis,𝑛_𝑦𝐷=0,𝜇=1,₁=0,𝐸₁𝐸=−2𝜆,₂=−1,𝐴₂=−1,𝐴₁=2, (55) then (52) become generalized Zakharov (GZ) equations [24] 𝑖𝑢_𝑡+𝑢_𝑥𝑥−2𝜆|𝑢|²𝑛𝑢+2𝑢𝑛=0,_𝑡𝑡−𝑛_𝑥𝑥+|𝑢|²_𝑥𝑥=0. (56)

Since 𝑢 is a complex function, we assume that 𝑢()(𝜉)𝑖,(𝜉)𝑥,𝑡=𝜙exp𝑘𝑥+𝑙𝑦−Ω𝑡𝑛=𝑛𝜉=𝑝𝑥+𝑞𝑦−𝜔𝑡, (57) where both 𝜙(𝜉) and 𝑛(𝜉) are real functions and 𝑘,𝑙,𝑝,𝑞,Ω, and 𝜔 are constants to be determined later. Substituting (57) into (52), we have the following ODE for 𝜙(𝜉) and 𝑛(𝜉): 𝐸₁𝜙³+Ω−𝜇𝑘²+𝐷₁𝑙²+𝐴₁𝑛𝜙+𝜇𝐷₁𝑞²+𝑝²𝜙^′′+𝑖2𝜇𝑝𝑘+𝐷₁𝑞𝑙−𝜔𝜙^′𝐷=0,₂𝜔²+𝑝²−𝐸₂𝑞²𝑛^′′+𝐴₂𝑝²2𝜙^′2+2𝜙𝜙^′′=0. (58)(59)

If we set 𝜔=2𝜇𝑝𝑘+𝐷₁𝑞𝑙, (60) then (58) reduces to 𝐸₁𝜙³+Ω−𝜇𝑘²+𝐷₁𝑙²+𝐴₁𝑛𝜙+𝜇𝐷₁𝑞²+𝑝²𝜙^′′=0. (61)

Integrating (59) twice to 𝜉, we get 𝐷𝑛=𝐷₂𝜔²+𝑝²−𝐸₂𝑞²−𝐴₂𝑝²𝐷₂𝜔²+𝑝²−𝐸₂𝑞²𝜙², (62) where 𝐷 is the integrating constant and Substituting (62) into (61) yields 𝐴𝜙^′′+𝐵𝜙+𝐶𝜙³=0. (63)

Equation (63) is identical to (9) and 𝐴,𝐵, and 𝐶 are defined by 𝐴=𝜇𝐷₁𝑞²+𝑝²,𝐵=Ω−𝜇𝑘²+𝐷₁𝑙²+𝐴₁𝐷𝐷₂𝜔²+𝑝²−𝐸₂𝑞²,𝐶=𝐸₁−𝐴₁𝐴₂𝑝²𝐷₂𝜔²+𝑝²−𝐸₂𝑞². (64)

Then, solutions of (52) are defined as follows: 𝑢𝑥,𝑦,𝑡=𝜙(𝜉)exp𝑖𝑘𝑥+𝑙𝑦−Ω𝑡,𝜉=𝑝𝑥+𝑞𝑦−𝜔𝑡,𝜔=2𝜇𝑝𝑘+𝐷₁,𝑛=𝐷𝑞𝑙𝑥,𝑦,𝑡𝐷₂𝜔²+𝑝²−𝐸₂𝑞²−𝐴₂𝑝²𝐷₂𝜔²+𝑝²−𝐸₂𝑞²𝜙², (65) where 𝜙(𝜉) appearing in these solutions is given by relations (12), (15), and (28)–(32) and 𝐴,𝐵, and 𝐶 are defined by (64).

We may obtain from (54) that 𝜔=𝜅²𝑝𝑘+𝜅²,𝑢(𝜉)𝑖,𝑛=𝐷𝑞𝑙𝑥,𝑦,𝑡=𝜙exp𝑘𝑥+𝑙𝑦−Ω𝑡𝑥,𝑦,𝑡𝜔²−𝜅²𝑞²+2𝛼𝑝²𝜔²−𝜅²𝑞²𝜙², (66) where 𝐷 is the integrating constant and then (54) reduce to 𝐴𝜙^′′+𝐵𝜙+𝐶𝜙³=0. (67)

This equation coincides also with (9), where 𝐴,𝐵, and 𝐶 are defined as follows: 1𝐴=2𝜅²𝑝²+𝜅²𝑞²,1𝐵=Ω−2𝜅²𝑘²+𝜅²𝑙²−𝐷𝜔²−𝜅²𝑞²,𝐶=𝛼−2𝛼𝑝²𝜔²−𝜅²𝑞², (68) where 𝜉=𝑝𝑥+𝑞𝑦−𝜔𝑡, 𝜙(𝜉) appearing in these solutions is given by relations (12), (15), and (28)–(32) and 𝐴,𝐵, and 𝐶 are defined by (68).

We may obtain from (56) that 𝑢()(𝜉)(𝑖(,𝑛()=𝐷𝜔=2𝑝𝑘,𝑥,𝑡=𝜙exp𝑘𝑥−Ω𝑡))𝑥,𝑡𝑝²−𝜔²+𝑝²𝑝²−𝜔²𝜙², (69) where 𝐷 is the integrating constant and then (56) reduce to 𝐴𝜙^′′+𝐵𝜙+𝐶𝜙³=0. (70)

This equation coincides also with (9), where 𝐴,𝐵, and 𝐶 are defined as follows: 𝐴=𝑝²,𝐵=Ω−𝑘²+2𝐷𝑝²−𝜔²,𝑝𝐶=2²𝑝²−𝜔²−𝜆, (71) where 𝜉=𝑝𝑥−𝜔𝑡, 𝜙(𝜉) appearing in these solutions is given by relations (12), (15), and (28)–(32) and 𝐴,𝐵, and 𝐶 are defined by (71).

5. Conclusions

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations, and the elliptic-like equation is one of the most important auxiliary equations because many nonlinear evolution equations, such as the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, the generalized Zakharov equations, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system, and the generalized ZK-BBM equation, can be converted to this equation using the travelling wave reduction.

In this paper, we apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation. The exact solutions of the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, and the generalized Zakharov equations are derived. Comparing the currently proposed method with other methods, such as the (𝐺^′/𝐺)-expansion method, the various extended hyperbolic methods, and the exp-function method, we might conclude that some exact solutions that we obtained can be investigated using these methods with the aid of the symbolic computation software, such as Matlab, Mathematica, and Maple to facilitate the complicated algebraic computations. But, by means of the simplest equation method and the modified simplest equation method the exact solutions to these equations have been gained in this paper without using the symbolic computation software since the computations are simple. This study shows that the simplest equation method and the modified simplest equation method are much more simple than the other methods and can be applied to many other nonlinear evolution equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11161020; 11361023), the Natural Science Foundation of Yunnan Province (2011FZ193; 2013FZ117), and the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452; 2013C079).