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The general meromorphic solutions of the Petviashvili equation
Advances in Difference Equations volume 2014, Article number: 24 (2014)
In this paper, we employ the complex method to first obtain all meromorphic exact solutions of complex Petviashvili equation, and then find all exact solutions of Petviashvili equation. The idea introduced in this paper can be applied to other non-linear evolution equations. Our results show that the complex method is simpler than other methods. Finally, we give some computer simulations to illustrate our main results.
1 Introduction and main results
In 2006 and 2008, Zhang et al. [1, 2] obtained abundant exact solutions of the Petviashvili equation by using the modified mapping method and the availability of symbolic computation. These solutions include the Jacobi elliptic function solutions, triangular function solutions, and soliton solutions. In this paper, we employ the complex method to obtain first all traveling meromorphic exact solutions of complex Petviashvili equation, and then find all exact solutions of the Petviashvili equation.
In order to state our main result, we need some concepts and some notation. A meromorphic function means that is holomorphic in the complex plane ℂ except for poles. α, b, c, , and are constants, which may be different from each other in different places. We say that a meromorphic function f belongs to the class W if f is an elliptic function, or a rational function of (), or a rational function of z.
The Petviashvili equation [1, 2] is
are two-dimensional Laplace and Jacobian operators, respectively, is the linear zero-dimensional phase velocity of Rossby wave, is the characteristic length about x and y, H is the average thickness of the fluid, and g is the acceleration of gravity; , and is the characteristic value about t.
Substituting the traveling wave transformation
into the Petviashvili equation gives a non-linear ordinary differential equation
and integrating it yields the auxiliary ordinary differential equation
where ω, k, l, are constants.
Our main result is Theorem 1.
Theorem 1 Equation (1) is integrable if and only if , , , and E are arbitrary. Furthermore, the general solutions of the Eq. (1) are of the following form.
The elliptic general solutions are(2)
if , , and E are arbitrary.
The simply periodic solutions are(3)
if , , .
The rational function solutions are(4)
if , .
2 Preliminary lemmas and the complex method
In order to explain our complex method and give the proof of Theorem 1, we need some lemmas and results.
Let , then any meromorphic solution w with at least one pole of the kth order Briot-Bouquet equation,
belongs to W, where are polynomials in w with constant coefficients.
Set , , , . Define
is called the degree of . The differential polynomial is defined as follows:
where are constants, and I is a finite index set. The total degree of is defined by .
We will consider the following complex ordinary differential equations:
where , c are constants, .
Definition 2 Let . Suppose that Eq. (5) has a meromorphic solution w with at least one pole; then we say that Eq. (5) satisfies the weak condition if substituting the Laurent series
into Eq. (5) we can determine p distinct Laurent singular parts in the form below:
Let , . Suppose that the mth order Briot-Bouquet equation
satisfies the weak condition; then all meromorphic solutions w belong to the class W. If for some values of parameters such a solution w exists, then other meromorphic solutions form a one-parametric family , . Furthermore each elliptic solution with a pole at can be written as
where are given by Eq. (6), , and .
Each rational function solution is of the form
with l (≤p) distinct poles of multiplicity q.
Each simply periodic solution is a rational function of (). has l (≤p) distinct poles of multiplicity q, and it is of the form
In order to give the representations of the elliptic solutions, we need some notation and results concerning the elliptic function .
Let , be two given complex numbers such that , let be discrete subset , which is isomorphic to . The discriminant is and
The Weierstrass elliptic function is a meromorphic function with double periods , , and satisfying the equation
where , and .
On changing Eq. (11) to the form
we have , , .
Inversely, given two complex numbers and such that , there exists a Weierstrass elliptic function with double periods , such that the above holds.
The Weierstrass elliptic functions have two successive degeneracies and in addition we have the following.
Degeneracy to simply periodic functions (i.e., rational functions of one exponential ) occurs according to(13)
if one root is double ().
Degeneracy to rational functions of z occurs according to
if one root is triple ().
The addition formula holds according to(14)
By the above lemmas, we can give a new method below, called, say, the complex method, to find exact solutions of some PDEs.
Step 1. Substituting the transform into a given PDE gives a non-linear ordinary differential equation (5) or (7).
Step 2. Substitute Eq. (6) into Eq. (5) or (7) to determine whether the weak condition holds.
Step 3. By the indeterminate relations (8)-(10) we find the elliptic, rational, and simply periodic solutions of Eq. (5) or (7) with pole at , respectively.
Step 4. By Lemmas 1 and 3 we obtain all meromorphic solutions .
Step 5. Substituting the inverse transform into these meromorphic solutions , we get all exact solutions of the originally given PDE.
3 Proof of Theorem 1
Substituting (6) into Eq. (1) we have , , , , , , , , is arbitrary.
Hence, Eq. (1) satisfies the weak condition and is a second-order Briot-Bouquet differential equation. Obviously, Eq. (1) satisfies the dominant condition. So, by Lemma 3, we know that all meromorphic solutions of Eq. (1) belong to W. Now we will give the forms of all meromorphic solutions of Eq. (1).
By Eq. (9), we infer that the indeterminate rational solutions of Eq. (1) with pole at have the form of
Substituting into Eq. (1), we get
Here . Thus for all rational solutions of Eq. (1)
where , .
In order to have simply periodic solutions, set , put into Eq. (1); then
into Eq. (16), we obtain
Here . Substituting into the above relation, and then we get simply periodic solutions of Eq. (1) with pole at :
So all simply periodic solutions of Eq. (1) are obtained by
where , , .
From Eq. (8) of Lemma 3, we have indeterminate relations of the elliptic solutions of Eq. (1) with pole at ,
Putting into Eq. (1), we obtain
Here . Therefore, for all elliptic solutions of Eq. (1)
where . Making use of the addition formula of Lemma 4, we rewrite it in the form
Here, , , , and E are arbitrary.
The complex method is a very important tool in finding the traveling wave exact solutions of non-linear evolution equations such as the Petviashvili equation. In this paper, we employ the complex method to obtain all meromorphic exact solutions of the complex variant Eq. (1); then we find all traveling wave exact solutions of the Petviashvili equation.The idea introduced in this paper can be applied to other non-linear evolution equations. Our result shows that the complex method is simpler than other methods.
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This work is supported by the NSF of China (11271090) and NSF of Guangdong Province (S2012010010121). Also this work was supported by the Visiting Scholar Program of the Chern Institute of Mathematics at Nankai University where the authors worked as visiting scholars. The authors would like to express their hearty thanks to the Chern Institute of Mathematics providing very comfortable research environments to them. The authors finally wish to thank Professor Robert Conte for supplying his useful reprints and suggestions. The authors wish to thank the referees and editors for their very helpful comments and useful suggestions.
The authors declare that they have no competing interests.
ZH and WY carried out the design of the study and performed the analysis. JL participated in its design and coordination. All authors read and approved the final manuscript.
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Huang, Z., Yuan, W. & Lai, J. The general meromorphic solutions of the Petviashvili equation. Adv Differ Equ 2014, 24 (2014). https://doi.org/10.1186/1687-1847-2014-24
- Petviashvili equation
- exact solution
- meromorphic function
- elliptic function