Classification of Single Traveling Wave Solutions to the Generalized Kadomtsev-Petviashvili Equation without Dissipation Terms in p = 2 ()
1. Introduction
In mathematics and physics, the Kadomtsev-Petviashvili (KP) equation is a partial differential equation to describe nonlinear wave motion. It can be used to model water waves of long wavelength with weakly nonlinear restoring forces and frequency dispersion [1]. A number of modified forms of the KP equation have been studied [2-6]. In [1,7], the generalized Kadomtsev-Petviashvili equation without dissipation terms was given by
(1)
where 
are constants, 
,
. Some of modified form of the KP equation can be written in the form of Equation (1).
Many reliable methods are used in the literature to examine the completely integrable nonlinear evolution equations. The Hirota bilinear method, the Bäcklund transformation method, the inverse scattering method, the Painlevé analysis, the simplified Hirotas method established by Hereman et al. [8], and others were effectively used in [1-13]. Liu proposed a complete discrimination system for polynomial method [10-13]. That is, by using of elementary integral method and complete discrimination system for polynomial, the single wave solutions can be classified for some nonlinear differential equations which can be directly reduced to integral forms.
In this paper, we consider the following generalized Kadomtsev-Petviashvili equation without dissipation terms in
:
(2)
where 
are constants,
. By using Liu’s complete discrimination system for polynomial method, the classification of single traveling wave solutions to Equation (2) is obtained.
2. Classification of Solutions to Equation (2)
Take wave transformation
and 
into Equation (1), the following nonlinear ordinary difference equation is given:
(3)
Integrating Equation (3) once with respect to
, and setting the integral constant to zero yields:
(4)
Integrating Equation (4) twice yields
(5)
where 
are arbitrary constants.
Case 2.1.
, we substitute the transformation
into Equation (5) yields
(6)
where
(7)
Let
and 
is the discriminant of the polynomial
. According to the classification of the roots of
, there are three cases to be discussed.
Case 2.1.1.
, when
from Equation (6), we have
(8)
Case 2.1.2.
, when
from Equation (6), we have
(9)
(10)
(11)
When
from Equation (6), we have
(12)
(13)
(14)
where
,
.
Case 2.1.3.
. From Equation (6), we have
(15)
where
.
Case 2.2.
. Substituting the transformation
into Equation (5) yields
(16)
where
(17)
If
we take
; if
we take
. The complete discrimination system for the third order polynomial 
is given as follows:
(18)
According to the classification of the roots of
, there are four cases to be discussed.
Case 2.2.1.
. Then
where 
are real constants, 
, and 
If
, when 
and
, or when 
and
, from Equation (16), we have
(19)
when
, and
, or
, we have
(20)
when
, we have
(21)
If
,when 
and
, or when 
and
, from Equation (16), we have
(22)
when
, and
, or
, and
, we have
(23)
when
, we have
(24)
Case 2.2.2. 
. Then
where 
is a real constant. If
, when
, and
, or
, and
, we have
(25)
If
, when
, and
, or
, and
, we have
(26)
Case 2.2.3.
. Then
where 
are different real constants. If
, when
, or
, we have
(27)
(28)
where
.
If
, when
, and
, we have
(29)
(30)
where
.
Case 2.2.4. 
where 
are all real constants, and
, and
. we have
(31)
where
,
.
Case 2.3.
. The Equation (5) becomes
(32)
where
, and
,
.
The complete discrimination system for the fifth order polynomial 
is given as follows:
(33)
According to the classification of the roots of
, there are seven cases to be discussed.
Case 2.3.1.
, then
and 
are real numbers, 
From Equation (32), we have
(34)
(35)
(36)
(37)
where
.
Case 2.3.2.
,
and 
are real numbers, 
From Equation (32), we have
(38)
(39)
where 
Case 2.3.3.
,
are real numbers, 
From Equation (32), we have
(40)
(41)
where 
Case 2.3.4.
,
.
Respectively, from Equation (32), we have
(42)
where
(43)
where the signs of 
and 
must satisfy
Case 2.3.5.
,
.
Respectively, from Equation (32), we have
(44)
where we renew to queue the orders of
, and
, denote
.When
or 
(other cases can be written similarly, they are omitted), the meaning of every parameter in Equation (44) are given as follows:
(45)
Case 2.3.6.
where we renew to queue the orders of 
and 0, and denote 
When
, or 
(other cases can be written similarly, they are omitted), we have
(46)
(47)
The signs are the same as the ones in Equation (45), furthermore,
(48)
Case 2.3.7.
Now we renew to queue the orders of 
and 0, and denote
, we have
(49)
(50)
where
(51)
where the positive sign and negative sign for 
must satisfy
other signs are the same with the former.
From the description above, using elementary integral method and complete discrimination system for polynomial, we have obtained the solutions of equations (6), (16) and (32) that can be expressed by elementary functions and elliptic functions. What’s more, some solutions are explicit, but some solutions are implicit functions. So we can write concretely the exact traveling wave solutions of Equation (5) in some special cases. They are omitted for simplicity.
3. Conclusion
Using the complete discrimination system for polynomial method, we have obtained the classification of single traveling wave solutions to the generalized KadomtsevPetviashvili equation without dissipation terms in
. With the same method, some of other evolution equations can be dealt with.
4. Acknowledgements
The project is supported by Scientific Research Fund of Education Department of Heilongjiang Province of China under Grant No. 12521049.