Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the DaveyStewartson Equations
 Hassan Zedan^{1, 2}Email author
DOI: 10.1155/2010/915721
© Hassan Zedan. 2010
Received: 13 January 2010
Accepted: 26 March 2010
Published: 6 May 2010
Abstract
We used what we called extended Fan's subequation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the DaveyStewartson (DS) equations. The basic idea of the proposed extended Fan's subequation method is to take fulls advantage of the general elliptic equations, involving five parameters, which have many new solutions and whose degeneracies lead to special subequations involving three parameters like Riccati equation, firstkind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the DaveyStewartson (DS) equations including more general soliton solutions, triangular solutions, and doubleperiodic solutions are constructed by symbolic computation.
1. Introduction
The case is called the DSI equation, while is the DSII equation. The parameter characterizes the focusing or defocusing case. The DaveyStewartson I and II are two wellknown examples of integrable equations in twodimensional space, which arise as higherdimensional generalizations of the nonlinear shrodinger (NLS) equation, from the point of physical view as well as from the study in [6]. Indeed, they appear in many applications, for example, in the description of gravitycapillarity surface wave packets and in the limit of the shallow water.
Davey and Stewartson first derived their model in the context of water waves, just purely physical considerations. In the context, is the amplitude of a surface wave packet, while is the velocity potential of the mean flow interacting with the surface wave [6].
In addition, we apply a new compound Riccati equations rational expansion method [12] to the DaveyStewartson (DS) equations and construct new complexion solutions. The rest of this paper is organized as follows. In Section 2, we simply provide the mathematical framework of Fan's subequation method. In Section 3, we apply the new presented method to the DaveyStewartson (DS) equations. In Section 4, we briefly describe the new CRERE method. In Section 5, we obtain new complexion solutions of the DaveyStewartson (DS) equations. In Section 6 and finally, some conclusions are given.
2. The Extended Fan Subequation Method
In the following we shall outline the main steps of our method.
where are constants to be determined later and satisfies (1.2). In order to determine explicitly, one may take the following steps.
Step 1.
Determine by balancing the linear term of the highest order with the nonlinear term in (2.3).
Step 2.
Substituting (2.4) with (1.2) into (2.3) and collecting all coefficients of = ; = then, setting these coefficients, to zero we get a set of algebraic equations with respect to and
Step 3.
Solve the system of algebraic equations to obtain and . Inserting these results into (2.4), we thus obtained the general form of travelling wave solutions.
Step 4.
By using the results obtained in the above steps, we can derive a series of fundamental solutions to (1.2) depending on the different values chosen for , and [7, 8, 10]. The superscripts , and determine the group of the solution while the subscript determines the rank of the solution. Those solutions are listed as follows.
Case 1.
For example, if the conditions (1.2)–(2.5) are satisfied, the following solutions are obtained [8].
Type 1.
where denotes .
Type 2.
where denotes .
Case 2.
For example, if the conditions (1.2), (2.5), (2.12), and (2.13) are satisfied, the following solutions are obtained.
Type 1.
where denotes . Thus for (2.5), (2.11), the general elliptic equation is reduced to the generalized Riccati Equation [8].
Case 3.
For example, if the condition (2.16) is satisfied, the following solutions are obtained.
Type 1.
Type 2.
Type 3.
Type 4.
Type 5.
Type 6.
Type 7.
where , and are arbitrary constants.
Type 8.
where , and solutions of (2.26) can be deduced from those of (2.25) in the specific case where the modulus of the Jacobi elliptic functions is drived to and .
Case 4.
Assume that the conditions of verification of (2.26) are fulfilled, then the general solutions are just the single solution and the combined nondegenerative Jacobi elliptic functions. The relations between the values of and the corresponding Jacobi elliptic function solution of the NODE (2.25) are given in Table 1.
where the modulus of the Jacobi elliptic function satisfies (
The Jacobi elliptic function degenerates as hyperbolic functions when (see Table 2).
Table 1

























































































Table 2

























Table 3

























Case 5.
where
Case 6.
Case 7.
3. Exact Solutions of the DaveyStewartson (DS) Equations
where "the prime" denotes to .
where are arbitrary real constants, and are real constants so we choose the case .
and are arbitrary constants. We may have many kinds of solutions depending on the special values for
Case 1.
Case 2.
Case 3.
If are arbitrary constants, then is one of the as follows.
Type 1.
Type 2.
Case 4.
Case 5.
If are arbitrary constants. The system does not admit solutions of this group.
Case 6.
Case 7.
If is arbitrary constant. The system does not admit solutions of this group.
4. Summary of the New Compound Riccati Equations Rational Expansion Method
The key steps of our method are as follows.
Step 1.
Step 2.
where are constants to be determined later and the new variables satisfy the Riccati equation.
where are arbitrary constants.
Step 3.
The parameter can be found by balancing the highest nonlinear terms and the highestorder partial derivative terms in (4.1) or (4.3).
Step 4.
Substitute (4.3), (4.4) with (4.5) and then set all coefficients of of the resulting system's numerator to zero to get an overdetermined system of nonlinear algebraic system with respect to
Step 5.
Solving the overdetermined system of nonlinear algebraic equations by use of Maple or Mathematica software, we would end up with the explicit expressions for
Step 6.
are the following.
()
where , and is arbitrary constant.
5. Application of the New Compound Riccati Equations Rational Expansion Method to the DaveyStewartson (DS) Equations
where "the prime" denotes to .
where satisfiy (4.5), with the aid of Mathematica software; substituting (5.5) along with (4.5) into (5.2) and (5.4) yields a set of algebraic equations for setting the coefficients of these terms to zero yields a set of overdetermined algebraic equations with respect to , and
where and , , , , , , , , are arbitrary constants.
So we obtain the following solutions of (1.1).
Family 1
Family 2
where and are arbitrary constants.
Family 3
Family 4
Family 5
Family 6
where − , , and , are arbitrary constants.
6. Conclusion
In this paper, we have used the extended Fan's subequation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions and obtain many explicit solutions for the DaveyStewartson equations.
We deduced a relation between the general elliptic equation involving five parameters and other subequations involving three parameter, like Riccati equation, auxiliary ordinary equation, firstkind elliptic equation, and generalized Riccati equation; many exact travelling wave solutions and new complexion solutions including more general soliton solutions, triangular solutions, doubleperiodic solutions, hyperbolic function solutions, and trigonometric function solutions are also given.
Authors’ Affiliations
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