We used what we called extended Fan's sub-equation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the Davey-Stewartson (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, first-kind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the Davey-Stewartson (DS) equations including more general soliton solutions, triangular solutions, and double-periodic solutions are constructed by symbolic computation.
1. Introduction
The investigation of the exact travelling wave solutions for nonlinear partial differential
equations plays an important role in the study of nonlinear physical phenomena. These
exact solutions when they exist can help the physicists to well understand the mechanism
of the complicated physical phenomena and dynamically processes modeled by these NLPDEs.
In recent years, large amounts of effort have been directed towards finding exact
solutions. Many powerful method have been proposed, such as Darboux transformation
[1], Hirota bilinear method [2], Lie group method [3], homogeneous balance method [4], tanh method. In this paper, we construct the exact travelling wave solutions for
the Davey-Stewartson (DS) equations for the function 
which are given by [5]
(11)The case 
is called the DSI equation, while 
is the DSII equation. The parameter 
characterizes the focusing or defocusing case. The Davey-Stewartson I and II are
two well-known examples of integrable equations in two-dimensional space, which arise
as higher-dimensional 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 gravity-capillarity
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].
The extended tanh-function method, the modified extended tanh-function method, and
the F-expansion method belong to a class of methods called subequation methods for
which they appear some basic relationships among the complicated NLPDEs under study
and some simple solvable nonlinear ordinary equations. Thus by these subequation methods
we seek for the solutions of the nonlinear partial differential equations in consideration
as a polynomial in one variable satisfying equations (named subequation), for example,
Riccati equation 
, auxiliary ordinary equation 
[7], first kind elliptic equation 
, generalized Riccati equation 
[8], and so on. Fan [9] developed a new algebraic method, belonging to the class of subequation methods,
to seek for more new solutions of nonlinear partial differential equations that can
be expressed as a polynomial in an elementary function which satisfies a more general
subequation than other subequations like Riccati equation, first-kind elliptic equation,
and generalized Riccati equation. Recently Yomba [10] and Soliman and Abdou [11] extended Fan's method to show that the general elliptic equation can be degenerated
in some special conditions to Riccati equation, first-kind elliptic equation, and
generalized Riccati equation. We will consider a general elliptic equation in the
formal will through
(12)In addition, we apply a new compound Riccati equations rational expansion method [12] to the Davey-Stewartson (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 Davey-Stewartson (DS) equations. In Section 4, we briefly describe the new CRERE method. In Section 5, we obtain new complexion solutions of the Davey-Stewartson (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.
For given nonlinear partial differential equations with independent variables 
and dependent variable 
,
(21)where 
is in general a polynomial function of its argument, and the subscripts denote the
partial derivatives. We first consider its travelling wave solutions
(22)where 
are all arbitrary constants. Substituting (2.2) into (2.1), we get
(23)Then 
is expanded into a polynomial in 
as
(24)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.
In some special cases, when 
, and 
, there may exist three parameters 
, and 
such that
(25)Equation (2.5) is satisfied only if the following relations hold:
(26)For example, if the conditions (1.2)–(2.5) are satisfied, the following solutions are obtained [8].
Type 1.
When 
and 
,
(27)where 
are two nonzero real constants and satisfy 
(28)where 
denotes 
.
Type 2.
When 
and 
(2)where 
are two nonzero real constants and satisfy 
(210)where 
denotes 
.
Case 2.
Case 1 includes another special case when 
and 
, and 
. There may exist three parameters 
, and 
such that
(211)Equation (2.2) requires for its existence the following relations:
(212)The following constraint should exist between 
, and 
parameters:
(213)For example, if the conditions (1.2), (2.5), (2.12), and (2.13) are satisfied, the following solutions are obtained.
Type 1.
When 
and
,
(214)where 
are two nonzero real constants and satisfy 
(215)where 
denotes 
. Thus for (2.5), (2.11), the general elliptic equation is reduced to the generalized
Riccati Equation [8].
Case 3.
When 
, the general elliptic equation is reduced to the auxiliary ordinary equation [7]
(216)For example, if the condition (2.16) is satisfied, the following solutions are obtained.
Type 1.
If 
(2.16) has the solution
(217)Type 2.
If 
(2.16) has the solution
(218)Type 3.
If 
(2.16) has the solution
(219)Type 4.
If 
(2.16) has the solution
(220)Type 5.
If 
(2.16) has the solution
(221)Type 6.
If 
(2.16) has the solution
(222)Type 7.
If 
(2.16) has the solution
(223)where 
, and 
are arbitrary constants.
Type 8.
If 
,
(224)When 
, the general elliptic equation is reduced to the elliptic equation
(225)Equation (2.25) includes the Riccati equation
(226)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).
The Jacobi elliptic function degenerates as hyperbolic functions when 
(see Table 3).
Case 5.
When 
, the general elliptic equation is reduced to the following:
(227)For example, if the condition (2.27) holds, the solution is of Weierstrass elliptic doubly periodic type
(228)where 
Case 6.
When 
, and 
the general elliptic equation admits solution
(229)Case 7.
when 
and 
, the general elliptic equation have solution
(230)3. Exact Solutions of the Davey-Stewartson (DS) Equations
Now, we will construct the exact solutions to (DS) equations (1.1). Let us assume the travelling wave solutions of (1.1) in the form
(31)where 
are real functions, and the constants 
are real which can be determined later. Substituting (3.1) into (1.1), we find that
and 
satisfy the following coupled ordinary differential system:
(32)
(33)where "the prime" denotes to 
.
Integrating (3.3) w.r.t. 
and solving for 
we get
(34)where 
is an integration constant. Substituting (3.4) into (3.2), we get
(35)where
(36)Balancing the highest-order derivative term (
) with nonlinear term (
) in (3.5) gives leading 
We thus suppose that (3.5) has the following formal solutions:
(37)where 
are to be determined later; substituting (3.7) along with (1.2) into (3.5) yields
a polynomial equation in 
. Setting to zero their coefficients yields the following set of algebraic equations:
(38)Substituting (3.6) into (3.8) and solving with respect to 
, we obtain the following solutions:
(39)where 
are arbitrary real constants, and 
are real constants so we choose the case 
.
The exact travelling wave solutions of the DSII equations (1.1) are given by
(310)where
(311)and 
are arbitrary constants. We may have many kinds of solutions depending on the special
values for 
Case 1.
If 
, then 
is one of the 
containing 
that are real and 
that are complex. For example, if we select 
then one could write down explicitly the following soliton solutions:
(312)where
(313)Case 2.
If 
, then 
is one of the 
contain 
are real and 
are complex. For example, if we select 
then one may write down explicitly the following soliton solutions:
(314)where
(315)Case 3.
If 
are arbitrary constants, then 
is one of the 
as follows.
Type 1.
If
then the travelling wave solutions are given as follows:
(316)where
(317)Type 2.
If
then the travelling wave solutions are given as follows:
(318)where
(319)Case 4.
If 
are arbitrary constants, then 
is one of the 
containing 
that are real and one 
that is complex. For example, if we select 
, then 
, and the travelling wave solutions are rediscovered:
(320)where
(321)In the limit case, when 
then (3.20) admits the soliton wave solutions
(322)When 
, then (3.20) admits the soliton wave solutions
(323)where
(324)If we select 
, then 
, and the travelling wave solutions are rediscovered:
(325)where
(326)In the limit case when 
then (3.25) admits the soliton wave solutions
(327)When 
, then (3.25) admits the soliton wave solutions
(328)where
(329)Case 5.
If 
are arbitrary constants. The system does not admit solutions of this group.
Case 6.
If 
, the travelling wave solutions of (1.1) are
(330)where
(331)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.
For a given NLPDEs with some physical fields 
in three variables 
,
(41)by using the wave transformation
(42)where 
are constants to be determined later. Then (4.1) is reduced to an ordinary differential
equation
(43)Step 2.
We introduce a solution of (4.3) in terms of finite rational formal expansion in the following forms:
(44)where 
are constants to be determined later and the new variables 
satisfy the Riccati equation.
That is,
(45)where 
are arbitrary constants.
Step 3.
The parameter 
can be found by balancing the highest nonlinear terms and the highest-order 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.
It is well known that the general solutions of the Riccati equation
(46)are the following.
(
)
(47)(
)
(48)(
)
(49)(
)
(410)(
)
(411)(
)
(412)where 
, and 
is arbitrary constant.
5. Application of the New Compound Riccati Equations Rational Expansion Method to the Davey-Stewartson (DS) Equations
By considering the wave transformations,
(51)Equation (1.1) reduces to the following ordinary differential equations:
(52)
(53)where "the prime" denotes to 
.
Integrating (5.3) w.r.t. 
and setting the constant of integration to zero, we obtain
(54)By balancing the highest nonlinear terms and the highest-Order partial derivative terms in (5.2) and (5.4), we suppose that (5.2) and (5.4) have the following solutions:
(55)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 
By using the Maple software to solving the overdetermined algebraic equations, we get the following results:
(56)where 
and 
, 
, 
, 
, 
, 
, 
, 
, 
are arbitrary constants.
So we obtain the following solutions of (1.1).
Family 1
Consider the following
(57)Family 2
Consider the following
(58)where 
and 
are arbitrary constants.
Family 3
Consider the following
(59)Family 4
Consider the following
(510)Family 5
Consider the following
(511)Family 6
Consider the following
(512)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 Davey-Stewartson 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, first-kind elliptic equation, and generalized Riccati equation; many exact travelling wave solutions and new complexion solutions including more general soliton solutions, triangular solutions, double-periodic solutions, hyperbolic function solutions, and trigonometric function solutions are also given.
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