The general meromorphic solutions of the Petviashvili equation
© Huang et al.; licensee Springer. 2014
Received: 21 September 2013
Accepted: 19 December 2013
Published: 17 January 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.
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.
where ω, k, l, are constants.
Our main result is Theorem 1.
- (I)The elliptic general solutions are(2)
- (II)The simply periodic solutions are(3)
- (III)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.
belongs to W, where are polynomials in w with constant coefficients.
where are constants, and I is a finite index set. The total degree of is defined by .
where , c are constants, .
where are given by Eq. (6), , and .
with l (≤p) distinct poles of multiplicity q.
In order to give the representations of the elliptic solutions, we need some notation and results concerning the elliptic function .
where , and .
we have , , .
Inversely, given two complex numbers and such that , there exists a Weierstrass elliptic function with double periods , such that the above holds.
- (I)Degeneracy to simply periodic functions (i.e., rational functions of one exponential ) occurs according to(13)
- (II)Degeneracy to rational functions of z occurs according to
- (III)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).
where , .
where , , .
Here, , , , and E are arbitrary.
4 Computer simulations for new solutions
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.
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.
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