New Exact Solutions of Some Nonlinear Systems of Partial Differential Equations Using the First Integral Method

Abstract

The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE), (2 + 1)-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.

1. Introduction

Over the four decades or so, nonlinear partial differential equations (NPDEs) have been the subject of extensive studies in various branches of nonlinear sciences.

A special class of analytical solutions, the so-called traveling waves, for NPDEs is of fundamental importance because lots of mathematical-physical models are often described by such wave phenomena.

Therefore, the investigation of traveling wave solutions is becoming more and more attractive in nonlinear sciences nowadays. However, not all equations posed of these models are solvable. As a result, many new techniques have been successfully developed by diverse groups of mathematicians and physicists, such as the Exp-function method [1–3], the sine-cosine method [4–6], the extended tanh-function method [7, 8], the modified extended tanh-function method [9–11], the 𝐹-expansion method [12], and the first integral method (or the algebraic curve method) [13]. Of these, the first integral method, which is based on the ring theory of commutative algebra, was first established by Feng [14–23].

This method was further developed by some other mathematicians [11, 24–33]. The method is reliable, effective, precise, and does not require complicated and tedious computations. The main idea of the first integral method is to find first integrals of nonlinear differential equations in polynomial form. Taking the polynomials with unknown polynomial coefficients into account, the method provides exact and explicit solutions. The interest in the present work is to implement the first integral method to stress its power in handling nonlinear partial differential equations, so that we can apply it for solving various types of these equations.

In Section 2, we describe this method for finding exact travelling wave solutions of nonlinear evolution equations. In Section 3, we illustrate this method in detail with the classical Drinfel’d-Sokolov-Wilson system (DSWE), the (2+1)-dimensional Davey-Stewartson system, and the generalized Hirota-Satsuma coupled KdV system. In Section 4, we give some conclusions.

2. The First Integral Method

Hosseini et al. in [30] have summarized the first integral method in the following steps.

Step 1. Consider the following nonlinear system of partial differential equations with independent variables 𝑥 and 𝑡 and dependent variables 𝑢 and 𝑣, 𝐹₁𝑢,𝑣,𝑢_𝑡,𝑣_𝑡,𝑢_𝑥,𝑣_𝑥,𝑢_𝑡𝑡,𝑣_𝑡𝑡,𝑢_𝑥𝑥,𝑣_𝑥𝑥𝐹,…=0,₂𝑢,𝑣,𝑢_𝑡,𝑣_𝑡,𝑢_𝑥,𝑣_𝑥,𝑢_𝑡𝑡,𝑣_𝑡𝑡,𝑢_𝑥𝑥,𝑣_𝑥𝑥,…=0. (1) Applying the transformations 𝑢(𝑥,𝑡)=𝑢(𝜉) and 𝑣(𝑥,𝑡)=𝑣(𝜉), where 𝜉=𝑥−𝑐𝑡+𝜍, where 𝜍 is an arbitrary constant, converts (1) into a system of ordinary differential equations (ODEs) 𝐺₁𝑢,𝑣,𝑢^′,𝑣^′𝐺,…=0,₂𝑢,𝑣,𝑢^′,𝑣^′,…=0, (2) where the prime denotes the derivatives with respect to the same variable 𝜉.

Step 2. Using some mathematical operations, the system (2) is converted into a second-order ODE 𝐷𝑢,𝑢^′,𝑢^′′=0. (3)

Step 3. By introducing new variables 𝑋=𝑢(𝜉) and 𝑌=𝑢^′(𝜉), (3) changes into a system of ODEs as the following system:𝑋^′𝑌=𝑌,_′=𝐻(𝑋,𝑌). (4a)(4b)

Step 4. Now, the Division Theorem which is based on ring theory of commutative algebra is adopted to obtain one first integral to (4a) and (4b), which reduces (3) to a first-order integrable ordinary differential equation. Finally, an exact solution to (1) is then established, through solving the resulting first-order integrable differential equation.

Let us now recall the Division Theorem for two variables in the complex domain 𝐶(𝑤,𝑧).

Theorem 1 (Division Theorem). Suppose that 𝑃(𝑤,𝑧), 𝑄(𝑤,𝑧) are polynomials in 𝐶(𝑤,𝑧) and 𝑃(𝑤,𝑧) is irreducible in (𝑤,𝑧). If 𝑄(𝑤,𝑧) vanishes at all zero points of (𝑤,𝑧), then there exists a polynomial 𝐺(𝑤,𝑧) in 𝐶(𝑤,𝑧) such that 𝑄(𝑤,𝑧)=𝑃(𝑤,𝑧)𝐺(𝑤,𝑧). (5) The Division Theorem follows immediately from the Hilbert-Nullstellensatz Theorem [34], but it can also be proved by using the complex analysis [35].

Theorem 2 (Hilbert-Nullstellensatz Theorem). Let 𝑘 be a field and 𝐿 an algebraic closure of 𝑘.(1)Every ideal 𝛾 of 𝑘[𝑋₁,…,𝑋_𝑛] not containing 1 admits at least one zero in 𝐿^𝑛(2)Let 𝑥=(𝑥₁,…,𝑥_𝑛),  𝑦=(𝑦₁,…,𝑦_𝑛) be two elements of 𝐿^𝑛; for the set of polynomials of 𝑘[𝑋₁,…,𝑋_𝑛] zero at 𝑥 to be identical with the set of polynomials of 𝑘[𝑋₁,…,𝑋_𝑛] zero at 𝑦, it is necessary and sufficient that there exists a 𝑘-automorphism 𝑠 of 𝐿 such that 𝑦_𝑖=𝑠(𝑥_𝑖) for 1≤𝑖≤𝑛.(3)For an ideal 𝛼 of 𝑘[𝑋₁,…,𝑋_𝑛] to be maximal, it is necessary and sufficient that there exists an 𝑥 in 𝐿^𝑛 such that 𝛼 is the set of polynomials of 𝑘[𝑋₁,…,𝑋_𝑛] zero at 𝑥.(4)For a polynomial 𝑄 of 𝑘[𝑋₁,…,𝑋_𝑛] to be zero on the set of zeros in 𝐿^𝑛 of an ideal  𝛾 of [𝑋₁,…,𝑋_𝑛], it is necessary and sufficient that there exists an integer 𝑚≻0 such that 𝑄^𝑚∈𝛾.

3. Applications

In this section, we investigate three NPDEs by using the first integral method.

3.1. Classical Drinfel’d-Sokolov-Wilson System

Consider the classical Drinfel’d-Sokolov-Wilson system [36] 𝑢_𝑡+𝑝𝑣𝑣_𝑥𝑣=0,_𝑡+𝑞𝑣_𝑥𝑥𝑥+𝑟𝑢𝑣_𝑥+𝑠𝑢_𝑥𝑣=0, (6) where 𝑝, 𝑞, 𝑟, 𝑠  are some nonzero parameters.

Recently, DSWE and the coupled DSWE, a special case of the classical DSWE, have been studied by several authors [36] and the references therein.

Using a complex variation 𝜂 defined as 𝜂=𝑘(𝑥−𝑐𝑡)+𝛾, we can convert (6) into ODEs, which read −𝑐𝑢^′+𝑝𝑣𝑣^′=0,−𝑐𝑣^′+𝑞𝑘²𝑣^′′′+𝑟𝑢𝑣^′+𝑠𝑢^′𝑣=0, (7)(8) where the prime denotes the derivative with respect to 𝜂.

Integrating (7), we obtain 𝑢=𝑝𝑣²2𝑐+𝑐₁, (9) where 𝑐₁  is an arbitrary integration constant.

Substituting 𝑢 into (8) yields 2𝑐𝑞𝑘²𝑣^′′′+𝑝(𝑟+2𝑠)𝑣²𝑣^′+2𝑐𝑟𝑐₁−𝑐𝑣^′=0. (10) Integrating (10), we get 2𝑐𝑞𝑘²𝑣^′′()𝑣+𝑝𝑟+2𝑠³3+2𝑐𝑟𝑐₁−𝑐𝑣=𝑐₂, (11) where 𝑐₂ is an arbitrary integration constant.

By introducing new variables 𝑋=𝑣(𝜉)and 𝑌=𝑣^′(𝜉), (11) changes into a system of ODEs𝑋^′𝑌=𝑌,_′=−𝑝(𝑟+2𝑠)6𝑐𝑞𝑘²𝑋³−𝑟𝑐₁−𝑐𝑞𝑘²𝑐𝑋−²2𝑐𝑞𝑘². (12a)(12b)According to the first integral method, we suppose that 𝑋(𝜉) and 𝑌(𝜉) are nontrivial solutions of (12a) and (12b), and 𝑃(𝑋,𝑌)=∑^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖 is an irreducible polynomial in the complex domain 𝐶[𝑋,𝑌] such that 𝑃𝑋(𝜉),𝑌(𝜉)=^𝑚_𝑖=0𝑎_𝑖(𝑋(𝜉))𝑌^𝑖(𝜉)=0, (13) where 𝑎_𝑖(𝑋), (𝑖=0,1,2,…,𝑚) are polynomials of 𝑋  and 𝑎_𝑚(𝑋)≠0.

Equation (13) is called the first integral to (12a) and (12b). Due to the Division Theorem, there exists a polynomial ℎ(𝑋)+𝑔(𝑋)𝑌 in the complex domain 𝐶[𝑋,𝑌] such that 𝑑𝑃=𝑑𝜉𝜕𝑃𝜕𝑋𝑑𝑋+𝑑𝜉𝜕𝑃𝜕𝑌𝑑𝑌=ℎ(𝑋)(𝑋)𝑌𝑑𝜉+𝑔^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖. (14)

Here, we have considered one case only, assuming that 𝑚=1  in (13).

Suppose that 𝑚=1, by equating the coefficients of 𝑌^𝑖(𝑖=2,1,0) on both sides of (14), we have𝑎^′₁(𝑋)=𝑔(𝑋)𝑎₁(𝑋),𝑎^′₀(𝑋)=ℎ(𝑋)𝑎₁(𝑋)+𝑔(𝑋)𝑎₀(𝑋),𝑎₁(𝑋)−𝑝(𝑟+2𝑠)6𝑐𝑞𝑘²𝑋³−𝑟𝑐₁−𝑐𝑞𝑘²𝑐𝑋−²2𝑐𝑞𝑘²=ℎ(𝑋)𝑎₀(𝑋). (15a)(15b)(15c)Since 𝑎_𝑖(𝑋)(𝑖=0,1) are polynomials, then from (15a) we have deduced that 𝑎₁(𝑋) is constant and 𝑔(𝑋)=0. For simplicity, take 𝑎₁(𝑋)=1.

Balancing the degrees of ℎ(𝑋) and 𝑎₀(𝑋), we have concluded that deg(ℎ(𝑋))=1 only. Suppose that ℎ(𝑋)=𝐴𝑋+𝐵, and 𝐴≠0, then we find 𝑎₀(𝑋)𝑎₀(𝑋)=𝐴2𝑋²+𝐵𝑋+𝐷, (16) where 𝐷 is an arbitrary integration constant.

Substituting 𝑎₀(𝑋),  𝑎₁(𝑋)  and ℎ(𝑋) for (15c) and setting all the coefficients of powers 𝑋 to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we have obtained𝑐₂=0,𝑐₁=3𝑐−√3𝑘𝐷𝑞−𝑝(𝑟+2𝑠)/√𝑐𝑞,3𝑟𝐴=−𝑝(𝑟+2𝑠)√3𝑘√𝑐𝑐𝑞,𝐵=0,₂=0,𝑐₁=3𝑐+𝑘𝐷𝑞−𝑝(𝑟+2𝑠)/√3√𝑐𝑞𝑟,𝐴=−−𝑝(𝑟+2𝑠)√3𝑘√𝑐𝑞,𝐵=0. (17a)(17b)Setting (17a) and (17b) in (13) leads to 𝑌𝜂+−𝑝(𝑟+2𝑠)2√3𝑘√𝑋𝑐𝑞²𝜂𝑌𝜂+−+𝐷=0,−𝑝(𝑟+2𝑠)2√3𝑘√𝑋𝑐𝑞²𝜂+𝐷=0. (18) Combining (18) with (12a), a first-order ordinary differential equation is derived, then by solving this derived equation and considering 𝑋=𝑣(𝜉) and 𝑣(𝑥,𝑡)=𝑣(𝜉), we have obtained 𝑣₁(𝑥,𝑡)=𝑖(−3)^1/4𝑐𝑞^1/4√𝑘√𝐷−1×tan3^1/4√𝐷𝑝^1/4(𝑟+2𝑠)^1/4×𝑘(𝑥−𝑐𝑡)+𝛾−3√𝑐𝑞𝑘𝜉₀×𝑐𝑞^1/4√𝑘⁻¹×𝑝^1/4(𝑟+2𝑠)^1/4−1,𝑣₂(𝑥,𝑡)=(−3)^1/4𝑐𝑞^1/4√𝑘√𝐷×tan(−1)^3/4√𝐷𝑝^1/4(𝑟+2𝑠)^1/4×𝑘(𝑥−𝑐𝑡)+𝛾−3√𝑐𝑞𝑘𝜉₀×(3)^1/4𝑐𝑞^1/4√𝑘⁻¹×𝑝^1/4(𝑟+2𝑠)^1/4−1, (19)(20) respectively, where 𝜉₀ is an arbitrary integration constant.

Also, by considering the solution 𝑢  given by the relations (9), we have obtained 𝑢₁()=𝑝𝑥,𝑡×𝑖()2𝑐−3^1/4𝑐𝑞^1/4√𝑘√𝐷−1×tan3^1/4√𝐷𝑝^1/4(𝑟+2𝑠)^1/4×𝑘(𝑥−𝑐𝑡)+𝛾−3√𝑐𝑞𝑘𝜉₀×𝑐𝑞^1/4√𝑘⁻¹×𝑝^1/4(𝑟+2𝑠)^1/4−12+3𝑐−√3𝑘𝐷𝑞−𝑝(𝑟+2𝑠)/√𝑐𝑞,𝑢3𝑟₂()=𝑝𝑥,𝑡×)2𝑐(−3^1/4𝑐𝑞^1/4√𝑘√𝐷×tan(−1)^3/4√𝐷𝑝^1/4(𝑟+2𝑠)^1/4×𝑘(𝑥−𝑐𝑡)+𝛾−3√𝑐𝑞𝑘𝜉₀×(3)^1/4𝑐𝑞^1/4√𝑘⁻¹×𝑝^1/4(𝑟+2𝑠)^1/4−12+3𝑐+𝑘𝐷𝑞−𝑝(𝑟+2𝑠)/√3√𝑐𝑞𝑟, (21)(22) respectively, where 𝜉₀ is an arbitrary integration constant.

Thus, two solutions (𝑢₁,𝑣₁) and (𝑢₂,𝑣₂)  have been obtained for the system (6).

Comparing these results with the results obtained in [36], it can be seen that the solutions here are new.

3.2. (2+1)-Dimensional Davey-Stewartson System

The (2+1)-dimensional Davey-Stewartson system [37] reads 𝑖𝑢_𝑡+𝑢_𝑥𝑥−𝑢_𝑦𝑦−2|𝑢|²𝑣𝑢−2𝑢𝑣=0,_𝑥𝑥+𝑣_𝑦𝑦+2|𝑢|²_𝑥𝑥=0. (23) This equation is completely integrable and used to describe the long-time evolution of a two-dimensional wave packet.

Using the wave variables 𝑢=𝑒^𝑖𝜃𝑢(𝜉),𝑣=𝑣(𝜉),𝜃=𝑝𝑥+𝑞𝑦+𝑟𝑡+𝜀,𝜉=𝑘𝑥+𝑐𝑦+𝑑𝑡+𝛾, (24) where 𝑝,  𝑞,  𝑟,  𝑘,  𝑐, and 𝑑 are real constants, converts (23) into the ODE 𝑞²−𝑝²−𝑟𝑢+𝑘²−𝑐²𝑢^′′−2𝑢³𝑘−2𝑢𝑣=0,₂+𝑐²𝑣^′′+𝑢^2′′=0. (25)(26) Integrating (26) in the system and neglecting constants of integration, we have found 𝑢𝑣=−²𝑘²+𝑐². (27) Substituting (27) into (25) of the system and integrating we find 𝑞²−𝑝²−𝑟𝑢+𝑘²−𝑐²𝑢^′′−2𝑢³+2𝑢³𝑘²+𝑐²=0. (28) By introducing new variables 𝑋=𝑢(𝜉)and 𝑌=𝑢^′(𝜉), (28) changes into a system of ODEs𝑋^′𝑌=𝑌,_′=2−2𝑘²−2𝑐²𝑐⁴−𝑘⁴𝑋³+𝑞²−𝑝²−𝑟𝑐²−𝑘²𝑋. (29a)(29b)According to the first integral method, we suppose that 𝑋(𝜉) and 𝑌(𝜉) are nontrivial solutions of (29a) and (29b), and 𝑃(𝑋,𝑌)=∑^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖 is an irreducible polynomial in the complex domain 𝐶[𝑋,𝑌] such that 𝑃𝑋(𝜉),𝑌(𝜉)=^𝑚_𝑖=0𝑎_𝑖(𝑋(𝜉))𝑌^𝑖(𝜉)=0, (30) where 𝑎_𝑖(𝑋), (𝑖=0,1,2,…,𝑚) are polynomials of 𝑋  and 𝑎_𝑚(𝑋)≠0.

Equation (30) is called the first integral to (29a) and (29b). Due to the Division Theorem, there exists a polynomial ℎ(𝑋)+𝑔(𝑋)𝑌 in the complex domain 𝐶[𝑋,𝑌] such that 𝑑𝑃=𝑑𝜉𝜕𝑃𝜕𝑋𝑑𝑋+𝑑𝜉𝜕𝑃𝜕𝑌𝑑𝑌=ℎ(𝑋)(𝑋)𝑌𝑑𝜉+𝑔^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖. (31)

Here, we have considered two different cases, assuming that 𝑚=1 and 𝑚=2 in (30).

Case 1. Suppose that 𝑚=1, by equating the coefficients of 𝑌^𝑖  (𝑖=2,1,0) on both sides of (31), we have𝑎^′₁(𝑋)=𝑔(𝑋)𝑎₁(𝑋),𝑎^′₀(𝑋)=ℎ(𝑋)𝑎₁(𝑋)+𝑔(𝑋)𝑎₀(𝑋),𝑎₁(𝑋)2−2𝑘²−2𝑐²𝑐⁴−𝑘⁴𝑋³+𝑞²−𝑝²−𝑟𝑐²−𝑘²𝑋=ℎ(𝑋)𝑎₀(𝑋). (32a)(32b)(32c)Since 𝑎_𝑖(𝑋)(𝑖=0,1) are polynomials, then from (32a) it can be deduced that 𝑎₁(𝑋) is constant and 𝑔(𝑋)=0. For simplicity, take 𝑎₁(𝑋)=1.
Balancing the degrees of ℎ(𝑋) and 𝑎₀(𝑋), it can be concluded that deg(ℎ(𝑋))=1 only. Suppose that ℎ(𝑋)=𝐴𝑋+𝐵, and 𝐴≠0, then we find 𝑎₀(𝑋)𝑎₀(𝑋)=𝐴2𝑋²+𝐵𝑋+𝐷. (33) Substituting 𝑎₀(𝑋), 𝑎₁(𝑋), and ℎ(𝑋) for (32c) and setting all the coefficients of powers 𝑋 to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain 𝐷=∓−−1+𝑐²+𝑘²/𝑐²−𝑘²𝑐²+𝑘²𝑝²−𝑞²+𝑟2−1+𝑐²+𝑘²,2𝐵=0,𝐴=∓−−1+𝑐²+𝑘²/𝑐²−𝑘²𝑐²+𝑘². (34) Using the conditions (34) in (30), we obtain 𝑌(𝜉)=±−−1+𝑐²+𝑘²/𝑐²−𝑘²𝑐²+𝑘²𝑋²(𝜉)±𝐷, (35) respectively.
Combining (35) with (29a), we obtain the exact solutions to (28), and considering the solution 𝑣 given by the relation (27), thus the exact traveling wave solutions to the (2+1)-dimensional Davey-Stewartson system (23) were obtained and can be written as 𝑢_1,2𝑖𝑥,𝑦,𝑡=±𝑝²−𝑞²+𝑟2−2/𝑐²+𝑘²×tanh𝑝²−𝑞²×√+𝑟𝑘𝑥+𝑐𝑦+𝑑𝑡+𝛾∓2𝑖𝑐−𝑘√×𝑐+𝑘−1+𝑐²+𝑘²𝑐²+𝑘²𝜉₀×√2√𝑐−𝑘√𝑐+𝑘⁻¹𝑖,𝑣×exp𝑝𝑥+𝑞𝑦+𝑟𝑡+𝜀_1,2=−1𝑥,𝑦,𝑡𝑐²+𝑘²×±𝑖𝑝²−𝑞²+𝑟2−2/𝑐²+𝑘²×tanh𝑝²−𝑞²×√+𝑟𝑘𝑥+𝑐𝑦+𝑑𝑡+𝛾∓2𝑖𝑐−𝑘√×𝑐+𝑘−1+𝑐²+𝑘²×𝑐²+𝑘²𝜉₀×√2√𝑐−𝑘√𝑐+𝑘⁻¹×exp𝑖𝑝𝑥+𝑞𝑦+𝑟𝑡+𝜀², (36)(37) respectively, where, 𝜉₀ is an arbitrary integration constant.

Case 2. Suppose that 𝑚=2, by equating the coefficients of 𝑌^𝑖(𝑖=3,2,1,0) on both sides of (31), we have𝑎^′₂(𝑋)=𝑔(𝑋)𝑎₂(𝑋),𝑎^′₁(𝑋)=ℎ(𝑋)𝑎₂(𝑋)+𝑔(𝑋)𝑎₁(𝑋),𝑎^′₀(𝑋)+2𝑎₂(𝑋)2−2𝑘²−2𝑐²𝑐⁴−𝑘⁴𝑋³+𝑞²−𝑝²−𝑟𝑐²−𝑘²𝑋=ℎ(𝑋)𝑎₁(𝑋)+𝑔(𝑋)𝑎₀(𝑋),𝑎₁(𝑋)2−2𝑘²−2𝑐²𝑐⁴−𝑘⁴𝑋³+𝑞²−𝑝²−𝑟𝑐²−𝑘²𝑋=ℎ(𝑋)𝑎₀(𝑋). (38a)(38b)(38c)(38d)Since, 𝑎_𝑖(𝑋)(𝑖=0,1,2) are polynomials, then from (38a) it can be deduced that 𝑎₂(𝑋) is a constant and 𝑔(𝑋)=0. For simplicity, we take 𝑎₂(𝑋)=1. Balancing the degrees of ℎ(𝑋)  and 𝑎₀(𝑋) it can be concluded that deg(ℎ(𝑋))=1 only.
In this case, it was assumed that ℎ(𝑋)=𝐴𝑋+𝐵, and 𝐴≠0, then we find 𝑎₁(𝑋) and 𝑎₀(𝑋) as follows:𝑎₁(𝑋)=𝐴2𝑋²𝑎+𝐵𝑋+𝐷,₀(𝑋)=𝐴²8−1−𝑘²−𝑐²𝑐⁴−𝑘⁴𝑋⁴+𝐴𝐵2𝑋³+𝐴𝐷+𝐵²2−𝑞²−𝑝²−𝑟𝑐²−𝑘²𝑋²+𝐵𝐷𝑋+𝐹, (39a)(39b)where 𝐴,  𝐵,  𝐷, and 𝐹 are arbitrary constants.
Substituting 𝑎₀(𝑋), 𝑎₁(𝑋), 𝑎₂(𝑋), and ℎ(𝑋) for (38d) and setting all the coefficients of powers 𝑋 to be zero, a system of nonlinear algebraic equations was obtained and by solving it, we got𝑐𝐹=−²+𝑘²𝑝²−𝑞²+𝑟²4−𝑐²+𝑐⁴+𝑘²−𝑘⁴𝑝,𝐵=0,𝐷=−²−𝑞²+𝑟(𝑐−𝑘)√𝑐+𝑘−−1+𝑐²+𝑘²/(𝑐−𝑘)𝑐²+𝑘²,4𝐴=−−1+𝑐²+𝑘²/(𝑐−𝑘)𝑐²+𝑘²√,𝑐𝑐+𝑘𝐹=−²+𝑘²𝑝²−𝑞²+𝑟²4−𝑐²+𝑐⁴+𝑘²−𝑘⁴𝑝,𝐵=0,𝐷=²−𝑞²+𝑟(𝑐−𝑘)√𝑐+𝑘−−1+𝑐²+𝑘²/(𝑐−𝑘)𝑐²+𝑘²,4𝐴=−−−1+𝑐²+𝑘²/(𝑐−𝑘)𝑐²+𝑘²√.𝑐+𝑘 (40a)(40b)Using the conditions (40a) and (40b) in (30), we obtain𝑌(𝜉)=−−−1+𝑐²+𝑘²(𝑐−𝑘)𝑐²+𝑘²×∓4(𝑐+𝑘)²−1+𝑐²+𝑘²³𝑐²+𝑘²𝑝²−𝑞²+𝑟𝑋²(𝜉)+(𝑐−𝑘)−1+𝑐²+𝑘²×−2𝑋²(𝜉)+𝑐²+𝑘²𝑝²−𝑞²+𝑟+2𝑋²(𝜉)×2(𝑐+𝑘)^3/2−1+𝑐²+𝑘²²⁻¹,𝑌(𝜉)=−−−1+𝑐²+𝑘²(𝑐−𝑘)𝑐²+𝑘²×∓4(𝑐+𝑘)²−1+𝑐²+𝑘²³𝑐²+𝑘²𝑝²−𝑞²+𝑟𝑋²(𝜉)−(𝑐−𝑘)−1+𝑐²+𝑘²×−2𝑋²(𝜉)+𝑐²+𝑘²𝑝²−𝑞²+𝑟+2𝑋²(𝜉)×2(𝑐+𝑘)^3/2−1+𝑐²+𝑘²²⁻¹. (41a)(41b)Combining (41a) and (41b) with (29a) we have obtained the exact solutions to (28), and considering the solution 𝑣 given by the relation (27), thus the exact traveling wave solutions to the (2+1)-dimensional Davey-Stewartson system (23) can be written as 𝑢_3,4=𝑥,𝑦,𝑡𝑝²−𝑞²+𝑟21−1/𝑐²+𝑘²×±2∓√2×tanh𝑝²−𝑞²×+𝑟∓𝑖−1+𝑐²+𝑘²𝑐²+𝑘²×√𝑘𝑥+𝑐𝑦+𝑑𝑡+𝛾±2𝑐−𝑘√×𝑐+𝑘−1+𝑐²+𝑘²𝑐²+𝑘²𝜉₀×√2√𝑐−𝑘√𝑐+𝑘−1+𝑐²+𝑘²×𝑐²+𝑘²⁻¹𝑖,𝑣×exp𝑝𝑥+𝑞𝑦+𝑟𝑡+𝜀_3,41𝑥,𝑦,𝑡=−𝑐²+𝑘²×𝑝²−𝑞²+𝑟21−1/𝑐²+𝑘²×±2∓√2×tanh𝑝²−𝑞²×+𝑟∓𝑖−1+𝑐²+𝑘²𝑐²+𝑘²×√𝑘𝑥+𝑐𝑦+𝑑𝑡+𝛾±2𝑐−𝑘√×𝑐+𝑘−1+𝑐²+𝑘²𝑐²+𝑘²𝜉₀×√2√𝑐−𝑘√𝑐+𝑘−1+𝑐²+𝑘²×𝑐²+𝑘²⁻¹×exp𝑖𝑝𝑥+𝑞𝑦+𝑟𝑡+𝜀²,𝑢_5,6=𝑥,𝑦,𝑡𝑝²−𝑞²+𝑟21−1/𝑐²+𝑘²×∓2±√2×tanh𝑝²−𝑞²×+𝑟∓𝑖−1+𝑐²+𝑘²𝑐²+𝑘²×√𝑘𝑥+𝑐𝑦+𝑑𝑡+𝛾∓2𝑐−𝑘√×𝑐+𝑘−1+𝑐²+𝑘²𝑐²+𝑘²𝜉₀×√2√𝑐−𝑘√𝑐+𝑘−1+𝑐²+𝑘²×𝑐²+𝑘²⁻¹𝑖,𝑣×exp𝑝𝑥+𝑞𝑦+𝑟𝑡+𝜀_5,61𝑥,𝑦,𝑡=−𝑐²+𝑘²×𝑝²−𝑞²+𝑟21−1/𝑐²+𝑘²×∓2±√2×tanh𝑝²−𝑞²×+𝑟∓𝑖−1+𝑐²+𝑘²𝑐²+𝑘²×√𝑘𝑥+𝑐𝑦+𝑑𝑡+𝛾∓2𝑐−𝑘√×𝑐+𝑘−1+𝑐²+𝑘²𝑐²+𝑘²𝜉₀×√2√𝑐−𝑘√𝑐+𝑘−1+𝑐²+𝑘²×𝑐²+𝑘²⁻¹×exp𝑖𝑝𝑥+𝑞𝑦+𝑟𝑡+𝜀², (42)(43)(44)(45) respectively, where 𝜉₀ is an arbitrary integration constant.

Equations (36)-(37) and (42)–(45) are new types of exact traveling wave solutions to the (2+1)-dimensional Davey-Stewartson system (23). It could not be obtained by the methods presented in [37].

3.3. Generalized Hirota-Satsuma Coupled KdV System

Consider the generalized Hirota-Satsuma coupled KdV system [38] 𝑢_𝑡=14𝑢_𝑥𝑥𝑥+3𝑢𝑢_𝑥+3𝑤−𝑣²_𝑥,𝑣_𝑡1=−2𝑣_𝑥𝑥𝑥−3𝑢𝑣_𝑥,𝑤_𝑡1=−2𝑤_𝑥𝑥𝑥−3𝑢𝑤_𝑥. (46)(47)(48) When 𝑤=0, (46)–(48) reduce to be the well-known Hirota-Satsuma coupled KdV system. We seek traveling wave solutions for (46)–(48) in the form 𝑢()(𝜉)()(𝜉),𝑤()(𝜉)()𝑥,𝑡=𝑢,𝑣𝑥,𝑡=𝑣𝑥,𝑡=𝑤,𝜉=𝑘𝑥−𝑐𝑡+𝜍, (49) where 𝜍 is an arbitrary constant.

Substituting (49) into (46)-(47) yields an ODE −𝑐𝑘𝑢^′=14𝑘³𝑢^′′′+3𝑘𝑢𝑢^′+3𝑘𝑤−𝑣^2′,−𝑐𝑘𝑣^′1=−2𝑘³𝑣^′′′−3𝑘𝑢𝑣^′,−𝑐𝑘𝑤^′1=−2𝑘³𝑤^′′′−3𝑘𝑢𝑤^′. (50)(51)(52) Let 𝑢=𝛼𝑣²+𝛽𝑣+𝛾,𝑤=𝐴₀𝑣+𝐵₀, (53) where 𝛼, 𝛾, 𝛽, 𝐴₀, and 𝐵₀ are constants [38]. Inserting (53) into (50) and (51) integrating once we know that (50) and (51) give rise to the same equation 𝑘²𝑣^′′=−2𝛼𝑣³−3𝛽𝑣²+2𝑐−3𝛾𝑣+𝑐₁, (54) where 𝑐₁is an integration constant. Integrating (54) once again we have 𝑘²𝑣^′2=−𝛼𝑣⁴−2𝛽𝑣³+2𝑐−3𝛾𝑣²+2𝑐₁𝑣+𝑐₂, (55) where 𝑐₂is an integration constant. By means of (53)–(55) we get 𝑘²𝑢^′′=2𝛼𝑘²𝑣^′2+𝑘²2𝛼𝑣+𝛽𝑣^′′=2𝛼−𝛼𝑣⁴−2𝛽𝑣³+2𝑐−3𝛾𝑣²+2𝑐₁𝑣+𝑐₂+2𝛼𝑣+𝛽−2𝛼𝑣³−3𝛽𝑣²+2𝑐−3𝛾𝑣+𝑐₁. (56) Integrating (50) once we have 14𝑘²𝑢^′′+32𝑢²+𝑐𝑢+3𝑤−𝑣²+𝑐₃=0, (57) where 𝑐₃  is an integration constant. Inserting (53) and (56) into (57) gives 33𝛼𝑐−3𝛼𝛾+4𝛽²1−3=0,2𝛼𝑐₁+𝛽𝑐+𝛾𝛽+𝐴₀1=0,42𝛼𝑐₂+𝛽𝑐₁+32𝛾²+𝑐𝛾+3𝐵₀+𝑐₃=0. (58) Let 𝑐₁=12𝛼²𝛽³,𝑣(𝜉)(𝜉)−𝛽+2𝑐𝛼𝛽−6𝛼𝛽𝛾=𝑎𝑃.2𝛼 (59) Therefore from (58), we have 𝑘²𝑃^′′(𝜉)−𝑎3𝛽²𝑃(𝜉)2𝛼+2𝑐−6𝛾+2𝛼𝑎³𝑃³(𝜉)=0. (60) By introducing new variables 𝑋=𝑃(𝜉)and 𝑌=𝑃^′(𝜉), (60) changes into a system of ODEs𝑋^′𝑌=𝑌,_′=−2𝛼𝑎³𝑘²𝑋³+𝑎𝑘²3𝛽²2𝛼+2𝑐−6𝛾𝑋. (61a)(61b)According to the first integral method, we suppose that 𝑋(𝜉) and 𝑌(𝜉) are nontrivial solutions of (61a) and (61b), and 𝑃(𝑋,𝑌)=∑^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖 is an irreducible polynomial in the complex domain 𝐶[𝑋,𝑌] such that 𝑃𝑋(𝜉),𝑌(𝜉)=^𝑚_𝑖=0𝑎_𝑖(𝑋(𝜉))𝑌^𝑖(𝜉)=0, (62) where 𝑎_𝑖(𝑋), (𝑖=0,1,2,…,𝑚) are polynomials of 𝑋 and 𝑎_𝑚(𝑋)≠0.

Equation (62) is called the first integral to (61a) and (61b). Due to the Division Theorem, there exists a polynomial ℎ(𝑋)+𝑔(𝑋)𝑌 in the complex domain 𝐶[𝑋,𝑌] such that 𝑑𝑃=𝑑𝜉𝜕𝑃𝜕𝑋𝑑𝑋+𝑑𝜉𝜕𝑃𝜕𝑌𝑑𝑌=ℎ(𝑋)(𝑋)𝑌𝑑𝜉+𝑔^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖. (63)

Here, we have considered two different cases, assuming that 𝑚=1 and 𝑚=2 in (62).

Case 1. Suppose that 𝑚=1, by equating the coefficients of 𝑌^𝑖  (𝑖=2,1,0) on both sides of (63), we have𝑎^′₁(𝑋)=𝑔(𝑋)𝑎₁(𝑋),𝑎^′₀(𝑋)=ℎ(𝑋)𝑎₁(𝑋)+𝑔(𝑋)𝑎₀(𝑋),𝑎₁(𝑋)−2𝛼𝑎³𝑘²𝑋³+𝑎𝑘²3𝛽²𝑋(𝑋)𝑎2𝛼+2𝑐−6𝛾=ℎ₀(𝑋). (64a)(64b)(64c)Since 𝑎_𝑖(𝑋)(𝑖=0,1) are polynomials, then from (64a) it was deduced that 𝑎₁(𝑋) is constant and 𝑔(𝑋)=0. For simplicity, take 𝑎₁(𝑋)=1.
Balancing the degrees of ℎ(𝑋) and 𝑎₀(𝑋), it was concluded that deg(ℎ(𝑋))=1 only. Suppose that ℎ(𝑋)=𝐴𝑋+𝐵, and 𝐴≠0, then we find 𝑎₀(𝑋)=𝐴2𝑋²+𝐵𝑋+𝐷, (65) where 𝐷 is an arbitrary integration constant.
Substituting 𝑎₀(𝑋),  𝑎₁(𝑋), and ℎ(𝑋) for (64c) and setting all the coefficients of powers 𝑋 to be zero, then we have obtained a system of nonlinear algebraic equations and by solving it, we obtain 𝐴=∓2𝑖𝑎^3/2√𝛼𝑘√,𝐵=0,𝑐=∓4𝑖𝑎𝐷𝑘𝛼^3/2−3𝛽²+12𝛼𝛾.4𝛼 (66) Using the conditions (66) in (62), we obtain 𝑌(𝜉)=±𝑖𝑎^3/2√𝛼𝑘𝑋²(𝜉)−𝐷, (67) respectively. Combining (67) with (61a), the exact solutions to (60) were obtained, and considering the solutions given by the relation (53), then the exact traveling wave solutions to the generalized Hirota-Satsuma coupled KdN system (46)–(48) are obtained and can be written as 𝑢₁(𝑥,𝑡)=𝛼𝑎(1−𝑖)√𝐷√𝑘1×tanh2+𝑖2𝑎^3/2√𝐷𝛼^1/4×−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀×√𝑘⁻¹×𝑎^3/2𝛼^1/4−1−𝛽2𝛼²+𝛽𝑎(1−𝑖)√𝐷√𝑘1×tanh2+𝑖2𝑎^3/2√𝐷𝛼^1/4×[−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀√𝑘⁻¹×𝑎^3/2𝛼^1/4−1−𝛽𝑣2𝛼+𝛾,₁(𝑥,𝑡)=𝑎(1−𝑖)√𝐷√𝑘1×tanh2+𝑖2𝑎^3/2√𝐷𝛼^1/4×−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀×√𝑘⁻¹𝑎^3/2𝛼^1/4−1−𝛽,𝑤2𝛼₁(𝑥,𝑡)=𝐴₀𝑎(1−𝑖)√𝐷√𝑘1×tanh2+𝑖2𝑎^3/2√𝐷𝛼^1/4×−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀×√𝑘⁻¹×𝑎^3/2𝛼^1/4−1−𝛽2𝛼+𝐵₀,𝑢₂(𝑥,𝑡)=𝛼𝑎(1−𝑖)√𝐷√𝑘1×tan2+𝑖2𝑎^3/2√𝐷𝛼^1/4×−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀×√𝑘⁻¹×𝑎^3/2𝛼^1/4−1−𝛽2𝛼²+𝛽𝑎(1−𝑖)√𝐷√𝑘1×tanh2+𝑖2𝑎^3/2√𝐷𝛼^1/4×−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀×√𝑘⁻¹×𝑎^3/2𝛼^1/4−1−𝛽𝑣2𝛼+𝛾,₂(𝑥,𝑡)=𝑎(1−𝑖)√𝐷√𝑘1×tan2+𝑖2𝑎^3/2√𝐷𝛼^1/4×−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀×√𝑘⁻¹×𝑎^3/2𝛼^1/4−1−𝛽,𝑤2𝛼₂(𝑥,𝑡)=𝐴₀𝑎(1−𝑖)√𝐷√𝑘1×tan2+𝑖2𝑎^3/2√𝐷𝛼^1/4×−(𝑘(𝑥−𝑐𝑡)+𝜍)+2𝑘𝜉₀×√𝑘⁻¹×𝑎^3/2𝛼^1/4−1−𝛽2𝛼+𝐵₀, (68)(69)(70)(71)(72)(73) respectively, where 𝜉₀ is an arbitrary constant.

Case 2. Suppose that 𝑚=2, by equating the coefficients of 𝑌^𝑖(𝑖=3,2,1,0) on both sides of (63), we have𝑎^′₂(𝑋)=𝑔(𝑋)𝑎₂(𝑋),𝑎^′₁(𝑋)=ℎ(𝑋)𝑎₂(𝑋)+𝑔(𝑋)𝑎₁(𝑋),𝑎^′₀(𝑋)+2𝑎₂(𝑋)−2𝛼𝑎³𝑘²𝑋³+𝑎𝑘²3𝛽²𝑋(𝑋)𝑎2𝛼+2𝑐−6𝛾=ℎ₁(𝑋)+𝑔(𝑋)𝑎₀(𝑋),𝑎₁(𝑋)−2𝛼𝑎³𝑘²𝑋³+𝑎𝑘²3𝛽²𝑋(𝑋)𝑎2𝛼+2𝑐−6𝛾=ℎ₀(𝑋). (74a)(74b)(74c)(74d)Since 𝑎_𝑖(𝑋)(𝑖=0,1,2) are polynomials, then from (74a) it can be deduced that 𝑎₂(𝑋) is a constant and 𝑔(𝑋)=0. For simplicity, we take 𝑎₂(𝑋)=1. Balancing the degrees of ℎ(𝑋)   and 𝑎₀(𝑋) it can be concluded that deg(ℎ(𝑋))=1, only.
In this case, it was assumed that ℎ(𝑋)=𝐴𝑋+𝐵, and 𝐴≠0, then we find 𝑎₁(𝑋) and 𝑎₀(𝑋) as follows:𝑎₁(𝑋)=𝐴2𝑋²𝑎+𝐵𝑋+𝐷,₀(𝑋)=𝐴²8+𝛼𝑎³𝑘²𝑋⁴+12(𝐴𝐵)𝑋³+𝐴𝐷+𝐵²2+−𝑎𝑘²3𝛽²𝑋2𝛼+2𝑐−6𝛾²+𝐵𝐷𝑋+𝐹, (75a)(75b)where 𝐴,  𝐵,  𝐷, and 𝐹 are arbitrary constants.
Substituting 𝑎₀(𝑋),  𝑎₁(𝑋),  𝑎₂(𝑋),  and ℎ(𝑋) for (74d) and setting all the coefficients of powers 𝑋 to be zero, a system of nonlinear algebraic equations was obtained and by solving it, we got3𝐹=0,𝑐=−𝛽²+4𝛼𝛾4𝛼,𝐵=0,𝐷=0,𝐴=−4𝑖𝑎^3/2√𝛼𝑘,3𝐹=0,𝑐=−𝛽²+4𝛼𝛾4𝛼,𝐵=0,𝐷=0,𝐴=4𝑖𝑎^3/2√𝛼𝑘. (76a)(76b)Using the conditions (76a) in (62), we obtain 𝑌(𝜉)=𝑖𝑎³𝑘𝑋²(𝜉)√𝛼∓−𝑎³−2+𝑎³𝑘²𝑋⁴(𝜉)𝛼2𝑘², (77) respectively. Combining (77) with (61a), the exact solutions to (60) were obtained, and considering the solutions given by the relation (53), then the exact traveling wave solutions to the generalized Hirota-Satsuma coupled KdN system (46)–(48) are obtained and can be written as 𝑢₃()𝑎−()×𝑥,𝑡=𝛼2𝑘𝑖𝑎^3/2𝑎^3/2−√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼²𝑎−()×+𝛽2𝑘𝑖𝑎^3/2𝑎^3/2−√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽𝑣2𝛼+𝛾,₃()−()×𝑥,𝑡=𝑎2𝑘𝑖𝑎^3/2𝑎^3/2−√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽,𝑤2𝛼₃(𝑥,𝑡)=𝐴₀𝑎−()×2𝑘𝑖𝑎^3/2𝑎^3/2−√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼+𝐵₀,𝑢₄()𝑎−()×𝑥,𝑡=𝛼2𝑘𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼²𝑎−()×+𝛽2𝑘𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼²𝑣+𝛾,₄()−()×𝑥,𝑡=𝑎2𝑘𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽,𝑤2𝛼₄(𝑥,𝑡)=𝐴₀𝑎−()×2𝑘𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼+𝐵₀. (78)(79)(80)(81)(82)(83) Similarly, as for the case of (76b) from (62) we obtain 𝑌(𝜉)=∓±𝑖𝑎³𝑘𝑋²(𝜉)√𝛼+−𝑎³−2+𝑎³𝑘²𝑋⁴(𝜉)𝛼2𝑘², (84) respectively. Combining (84) with (61a), the exact solutions to (60) were obtained, and considering the solutions given by the relation (53), then the exact traveling wave solutions to the generalized Hirota-Satsuma coupled KdN system (46)–(48) are obtainred and can be written as 𝑢₅()𝑎−()×𝑥,𝑡=𝛼2𝑘−𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼²,𝑎−()×+𝛽2𝑘−𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽𝑣2𝛼+𝛾,₅()−()×𝑥,𝑡=𝑎2𝑘−𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽,𝑤2𝛼₅(𝑥,𝑡)=𝐴₀𝑎−()×2𝑘−𝑖𝑎^3/2𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼+𝐵₀,𝑢₆()𝑎−()×𝑥,𝑡=𝛼2𝑘𝑖𝑎^3/2−𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼²𝑎−()×+𝛽2𝑘𝑖𝑎^3/2−𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽𝑣2𝛼+𝛾,₆()−()×𝑥,𝑡=𝑎2𝑘𝑖𝑎^3/2−𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽,𝑤2𝛼₆(𝑥,𝑡)=𝐴₀𝑎−()×2𝑘𝑖𝑎^3/2−𝑎^3/2+√−2+𝑎³×√𝛼[𝑘(𝑥−𝑐𝑡)+𝜍]+2𝑘𝜉₀⁻¹−𝛽2𝛼+𝐵₀, (85)(86)(87)(88)(89)(90) respectively, where 𝜉₀ is an arbitrary integration constant.
Comparing the results (68)–(73), (78)–(83), and (85)–(90) with the results in [38, 39], it can be seen that the solutions here are new.

Remark 3. We note that our results were based on the assumptions 𝑚=1 and 𝑚=2. The discussion becomes more complicated for 𝑚=3 and 𝑚=4 because the hyper-elliptic integrals, the irregular singular point theory, and the elliptic integrals of the second kind are involved. Also, we do not need to consider the case 𝑚≥5 because an algebraic equation with its degree greater than or equal to 5 is generally not solvable.

4. Conclusion

In this paper, the first integral method was applied successfully to obtain solutions of some important nonlinear systems, namely, the classical Drinfel’d-Sokolov-Wilson system (DSWE), the (2+1)-dimensional Davey-Stewartson system, and the generalized Hirota-Satsuma coupled KdV system. Also, we conclude that the proposed method is powerful, effective, and can be extended to solve more other nonlinear evolution equations as well as linear ones, and this will be done in a future work.

Acknowledgment

The author would like to thank the referees for their useful comments which led to some improvements of the current paper.