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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 693076, 13 pages
http://dx.doi.org/10.1155/2013/693076
Research Article

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

1Mathematics Department, Faculty of Science, Damanhour University, Bahira 22514, Egypt
2Mathematics Department, Faculty of Science and Humanity Studies at Al-Quwaiaiah, Shaqra University, Al-Quwaiaiah 11971, Saudi Arabia

Received 8 January 2013; Accepted 10 March 2013

Academic Editor: Elena Litsyn

Copyright © 2013 Shoukry Ibrahim Atia El-Ganaini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [13], the sine-cosine method [46], the extended -function method [7, 8], the modified extended -function method [911], the -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 [1423].

This method was further developed by some other mathematicians [11, 2433]. 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 and and dependent variables and , Applying the transformations and , where , where is an arbitrary constant, converts (1) into a system of ordinary differential equations (ODEs) where the prime denotes the derivatives with respect to the same variable .

Step 2. Using some mathematical operations, the system (2) is converted into a second-order ODE

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 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 .

Theorem 1 (Division Theorem). Suppose that , are polynomials in and is irreducible in . If vanishes at all zero points of , then there exists a polynomial in such that 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 be a field and an algebraic closure of .(1)Every ideal of not containing 1 admits at least one zero in (2)Let ,   be two elements of ; for the set of polynomials of zero at to be identical with the set of polynomials of zero at , it is necessary and sufficient that there exists a -automorphism of such that for .(3)For an ideal of to be maximal, it is necessary and sufficient that there exists an in such that is the set of polynomials of zero at .(4)For a polynomial of to be zero on the set of zeros in of an ideal   of , it is necessary and sufficient that there exists an integer such that .

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] where , , ,   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 defined as , we can convert (6) into ODEs, which read where the prime denotes the derivative with respect to .

Integrating (7), we obtain where   is an arbitrary integration constant.

Substituting into (8) yields Integrating (10), we get where is an arbitrary integration constant.

By introducing new variables and , (11) changes into a system of ODEsAccording to the first integral method, we suppose that and are nontrivial solutions of (12a) and (12b), and is an irreducible polynomial in the complex domain such that where , are polynomials of   and .

Equation (13) is called the first integral to (12a) and (12b). Due to the Division Theorem, there exists a polynomial in the complex domain such that

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

Suppose that , by equating the coefficients of on both sides of (14), we haveSince are polynomials, then from (15a) we have deduced that is constant and . For simplicity, take .

Balancing the degrees of and , we have concluded that only. Suppose that , and , then we find where is an arbitrary integration constant.

Substituting ,    and 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 obtainedSetting (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 and , we have obtained respectively, where is an arbitrary integration constant.

Also, by considering the solution   given by the relations (9), we have obtained respectively, where is an arbitrary integration constant.

Thus, two solutions and   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. -Dimensional Davey-Stewartson System

The -dimensional Davey-Stewartson system [37] reads This equation is completely integrable and used to describe the long-time evolution of a two-dimensional wave packet.

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 ODEsAccording to the first integral method, we suppose that and are nontrivial solutions of (29a) and (29b), and is an irreducible polynomial in the complex domain such that where , are polynomials of   and .

Equation (30) is called the first integral to (29a) and (29b). Due to the Division Theorem, there exists a polynomial in the complex domain such that

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

Case 1. Suppose that , by equating the coefficients of    on both sides of (31), we haveSince are polynomials, then from (32a) it can be deduced that is constant and . For simplicity, take .
Balancing the degrees of and , it can be concluded that only. Suppose that , and , then we find Substituting , , and for (32c) and setting all the coefficients of powers to be zero, then we obtain a system of nonlinear algebraic equations and by solving it, we obtain Using the conditions (34) in (30), we obtain respectively.
Combining (35) with (29a), we obtain the exact solutions to (28), and considering the solution given by the relation (27), thus the exact traveling wave solutions to the -dimensional Davey-Stewartson system (23) were obtained and can be written as respectively, where, is an arbitrary integration constant.

Case 2. Suppose that , by equating the coefficients of on both sides of (31), we haveSince, are polynomials, then from (38a) it can be deduced that is a constant and . For simplicity, we take . Balancing the degrees of   and it can be concluded that only.
In this case, it was assumed that , and , then we find and as follows:where ,  ,  , and are arbitrary constants.
Substituting , , , and 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 gotUsing the conditions (40a) and (40b) in (30), we obtainCombining (41a) and (41b) with (29a) we have obtained the exact solutions to (28), and considering the solution 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 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] When , (46)–(48) reduce to be the well-known Hirota-Satsuma coupled KdV system. We seek traveling wave solutions for (46)–(48) in the form where is an arbitrary constant.

Substituting (49) into (46)-(47) yields an ODE Let where , , , , and are constants [38]. Inserting (53) into (50) and (51) integrating once we know that (50) and (51) give rise to the same equation where is an integration constant. Integrating (54) once again we have where is an integration constant. By means of (53)–(55) we get Integrating (50) once we have where   is an integration constant. Inserting (53) and (56) into (57) gives Let Therefore from (58), we have By introducing new variables and , (60) changes into a system of ODEsAccording to the first integral method, we suppose that and are nontrivial solutions of (61a) and (61b), and is an irreducible polynomial in the complex domain such that where , are polynomials of and .

Equation (62) is called the first integral to (61a) and (61b). Due to the Division Theorem, there exists a polynomial in the complex domain such that

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

Case 1. Suppose that , by equating the coefficients of    on both sides of (63), we haveSince are polynomials, then from (64a) it was deduced that is constant and . For simplicity, take .
Balancing the degrees of and , it was concluded that only. Suppose that , and , then we find where is an arbitrary integration constant.
Substituting ,  , and 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 is an arbitrary constant.

Case 2. Suppose that , by equating the coefficients of on both sides of (63), we haveSince are polynomials, then from (74a) it can be deduced that is a constant and . For simplicity, we take . Balancing the degrees of   and it can be concluded that , only.
In this case, it was assumed that , and , then we find and as follows:where ,  ,  , and are arbitrary constants.
Substituting ,  ,  ,  and 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 gotUsing 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 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 and . The discussion becomes more complicated for and 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 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 -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.

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