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

and Applied Analysis 3 P(X, Y) = ∑ m i=0 a i (X)Y i is an irreducible polynomial in the complex domain C[X, Y] such that P [X (ξ) , Y (ξ)] = m ∑ i=0 a 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 ∂X dX dξ + ∂P ∂Y dY dξ = [h (X) + g (X)Y] m ∑ i=0 a 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 = 2, 1, 0) on both sides of (14), we have a 󸀠 1 (X) = g (X) a 1 (X) , (15a) a 󸀠 0 (X) = h (X) a 1 (X) + g (X) a 0 (X) , (15b) a 1 (X) ((− p (r + 2s) 6cqk )X 3 − ( rc 1 − c qk )X − c 2 2cqk ) = h (X) a 0 (X) . (15c) Since a i (X) (i = 0, 1) are polynomials, then from (15a) we have deduced that a 1 (X) is constant and g(X) = 0. For simplicity, take a 1 (X) = 1. Balancing the degrees of h(X) and a 0 (X), we have concluded that deg(h(X)) = 1 only. Suppose that h(X) = AX + B, and A ̸ = 0, then we find a 0 (X) a 0 (X) = A 2 X 2 + BX + D, (16) whereD is an arbitrary integration constant. Substituting a 0 (X), a 1 (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 obtained c 2 = 0, c 1 = 3c − (√3kDq√−p (r + 2s)) /√cq 3r , A = √−p (r + 2s) √3k√cq , B = 0, (17a) c 2 = 0, c 1 = 3c + (kDq√−p (r + 2s)) /√3√cq r , A = − √−p (r + 2s) √3k√cq , B = 0. (17b) Setting (17a) and (17b) in (13) leads to Y (η) + ( √−p (r + 2s) 2√3k√cq X 2 (η) + D) = 0, Y (η) + (− √−p (r + 2s) 2√3k√cq X 2 (η) + D) = 0. (18) Combining (18) with (12a), a first-order ordinary differential equation is derived, then by solving this derived equation and consideringX = V(ξ) and V(x, t) = V(ξ), we have obtained V 1 (x, t) = i(−3) 1/4 c(q) 1/4 √k√D × tan[((−1 3 ) 1/4 √D(p) 1/4 (r + 2s) 1/4 × [(k (x − ct) + γ) − 3√cqkξ0] ) × (c(q) 1/4 √k) −1 ] × ((p) 3/4 (r + 2s) 1/4 ) −1 , (19) V 2 (x, t) = (−3) 1/4 c(q) 1/4 √k√D × tan [((−1)√D(p)1/4(r + 2s)1/4 × [(k (x − ct) + γ) − 3√cqkξ0] ) × ((3) 1/4 c(q) 1/4 √k) −1 ] × ((p) 1/4 (r + 2s) 1/4 ) −1 , (20) respectively, where ξ 0 is an arbitrary integration constant. Also, by considering the solution u given by the relations (9), we have obtained u 1 (x, t) = ( p 2c ) × [i(−3) 1/4 c(q) 1/4 √k√D × tan[((−1 3 ) 1/4 √D(p) 1/4 (r + 2s) 1/4 × [(k (x − ct) + γ) − 3√cqkξ0] ) × (c(q) 1/4 √k) −1 ] 4 Abstract and Applied Analysis ×((p) 1/4 (r + 2s) 1/4 ) −1 ] 2 + 3c − (√3kDq√−p (r + 2s)) /√cq 3r , (21) u 2 (x, t) = ( p 2c ) × [(−3) 1/4 c(q) 1/4 √k√D × tan [((−1)√D(p)1/4(r + 2s)1/4 × [(k (x − ct) + γ) − 3√cqkξ0] ) × ((3) 1/4 c(q) 1/4 √k) −1 ] ×((p) 1/4 (r + 2s) 1/4 ) −1 ] 2 + 3c + (kDq√−p (r + 2s)) /√3√cq r , (22) respectively, where ξ 0 is an arbitrary integration constant. Thus, two solutions (u 1 , V 1 ) and (u 2 , V 2 ) 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| 2 u − 2uV = 0, V 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 iθ u (ξ) , V = V (ξ) , θ = px + qy + rt + ε, ξ = kx + cy + dt + γ, (24) wherep, q, r, k, c, andd are real constants, converts (23) into the ODE (q 2 − p 2 − r) u + (k 2 − c 2 ) u 󸀠󸀠 − 2u 3 − 2uV = 0, (25) (k 2 + c 2 ) V 󸀠󸀠 + (u 2 ) 󸀠󸀠 = 0. (26) Integrating (26) in the system and neglecting constants of integration, we have found


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.
This method was further developed by some other mathematicians [11,[24][25][26][27][28][29][30][31][32][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.

The First Integral Method
Hosseini et al. in [30] have summarized the first integral method in the following steps.
Step 2. Using some mathematical operations, the system ( 2) is converted into a second-order ODE  (,   ,   ) = 0. (3) Step 3. By introducing new variables  = () and  =   (), (3) changes into a system of ODEs as the following system: 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 firstorder 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 (, ).

Applications
In this section, we investigate three NPDEs by using the first integral method.
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  defined as  = ( − ) + , we can convert (6) into ODEs, which read where the prime denotes the derivative with respect to .Integrating (7), we obtain where  1 is an arbitrary integration constant.Substituting  into (8) yields Integrating (10), we get where  2 is an arbitrary integration constant.By introducing new variables  = V() and  = V  (), (11) changes into a system of ODEs According to the first integral method, we suppose that () and () are nontrivial solutions of (12a) and (12b), and (, ) = ∑  =0   ()  is an irreducible polynomial in the complex domain [, ] such that where   (), ( = 0, 1, 2, . . ., ) are polynomials of  and   () ̸ = 0. Equation ( 13) is called the first integral to (12a) and (12b).Due to the Division Theorem, there exists a polynomial ℎ() + () in the complex domain [, ] such that Here, we have considered one case only, assuming that  = 1 in (13).
Balancing the degrees of ℎ() and  0 (), we have concluded that deg(ℎ()) = 1 only.Suppose that ℎ() =  + , and  ̸ = 0, then we find  0 () where  is an arbitrary integration constant.Substituting  0 (),  1 () and ℎ() for (15c) and setting all the coefficients of powers  to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we have obtained Setting (17a) and (17b) in ( 13) leads to Combining ( 18) with (12a), a first-order ordinary differential equation is derived, then by solving this derived equation and considering  = V() and V(, ) = V(), we have obtained ) respectively, where  0 is an arbitrary integration constant.Also, by considering the solution  given by the relations (9), we have obtained respectively, where  0 is an arbitrary integration constant.Thus, two solutions ( 1 , V 1 ) and ( 2 , V 2 ) 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.
Using the wave variables where , , , , , and  are real constants, converts (23) into the ODE Integrating (26) in the system and neglecting constants of integration, we have found Substituting ( 27) into (25) of the system and integrating we find By introducing new variables  = () and  =   (), (28) changes into a system of ODEs According to the first integral method, we suppose that () and () are nontrivial solutions of (29a) and (29b), and where   (), ( = 0, 1, 2, . . ., ) are polynomials of  and   () ̸ = 0. Equation ( 30) is called the first integral to (29a) and (29b).Due to the Division Theorem, there exists a polynomial ℎ() + () in the complex domain [, ] such that Here, we have considered two different cases, assuming that  = 1 and  = 2 in (30).
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 respectively, where,  0 is an arbitrary integration constant.
In this case, it was assumed that ℎ() =  + , and  ̸ = 0, then we find  1 () and  0 () as follows: where , , , and  are arbitrary constants.Substituting  0 (),  1 (),  2 (), and ℎ() for (38d) and setting all the coefficients of powers  to be zero, a system of nonlinear algebraic equations was obtained and by solving it, we got Using the conditions (40a) and (40b) in (30), we obtain 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 respectively, where  0 is an arbitrary integration constant.
Balancing the degrees of ℎ() and  0 (), it was concluded that deg(ℎ()) = 1 only.Suppose that ℎ() =  + , and  ̸ = 0, then we find where  is an arbitrary integration constant.Substituting  0 (),  1 (), and ℎ() for (64c) and setting all the coefficients of powers  to be zero, then we have obtained a system of nonlinear algebraic equations and by solving it, we obtain Using the conditions (66) in (62), we obtain 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 respectively, where  0 is an arbitrary constant.
In this case, it was assumed that ℎ() =  + , and  ̸ = 0, then we find  1 () and  0 () as follows: where , , , and  are arbitrary constants.Substituting  0 (),  1 (),  2 (), and ℎ() for (74d) and setting all the coefficients of powers  to be zero, a system of nonlinear algebraic equations was obtained and by solving it, we got Using the conditions (76a) in (62), we obtain 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 Similarly, as for the case of (76b) from (62) we obtain 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 respectively, where  0 is an arbitrary integration constant.
Remark 3. We note that our results were based on the assumptions  = 1 and  = 2.The discussion becomes more complicated for  = 3 and  = 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  ≥ 5 because an algebraic equation with its degree greater than or equal to 5 is generally not solvable.

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.