thegeneralmeromorphicsolutionsofthe petviashviliequation · thegeneralmeromorphicsolutionsofthe...

Huang et al. Advances in Difference Equations 2014, 2014:24http://www.advancesindifferenceequations.com/content/2014/1/24

RESEARCH Open Access

The general meromorphic solutions of thePetviashvili equationZifeng Huang1,2, Wenjun Yuan1* and Jinchun Lai1

*Correspondence:[email protected] of Mathematics andInformation Science, GuangzhouUniversity, Guangzhou, 510006,ChinaFull list of author information isavailable at the end of the article

AbstractIn this paper, we employ the complex method to first obtain all meromorphic exactsolutions of complex Petviashvili equation, and then find all exact solutions ofPetviashvili equation. The idea introduced in this paper can be applied to othernon-linear evolution equations. Our results show that the complex method is simplerthan other methods. Finally, we give some computer simulations to illustrate ourmain results.MSC: Primary 30D35; secondary 34A05

Keywords: Petviashvili equation; exact solution; meromorphic function; ellipticfunction

1 Introduction andmain resultsIn and , Zhang et al. [, ] obtained abundant exact solutions of the Petviashviliequation by using the modified mapping method and the availability of symbolic compu-tation. These solutions include the Jacobi elliptic function solutions, triangular functionsolutions, and soliton solutions. In this paper, we employ the complex method to obtainfirst all traveling meromorphic exact solutions of complex Petviashvili equation, and thenfind all exact solutions of the Petviashvili equation.In order to state our main result, we need some concepts and some notation. A mero-

morphic function w(z) means that w(z) is holomorphic in the complex plane C except forpoles. α, b, c, ci, and cij are constants, which may be different from each other in differentplaces. We say that a meromorphic function f belongs to the class W if f is an ellipticfunction, or a rational function of eαz (α ∈C), or a rational function of z.The Petviashvili equation [, ] is

∂

∂t(∇φ – φ

)+CR( + φ)

∂φ

∂t+ J

(φ,∇φ

)= ,

where

∇ =∂

∂x+

∂

∂y, J(A,B) =

∂A∂x

∂B∂y

–∂A∂y

∂B∂x

,

CR =βLR√gH

are two-dimensional Laplace and Jacobian operators, respectively, CR is the linear zero-dimensional phase velocity of Rossby wave, LR is the characteristic length about x and y,

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H is the average thickness of the fluid, and g is the acceleration of gravity; β = ∂f∂y , and

f is

the characteristic value about t.Substituting the traveling wave transformation

φ = w(z), z = kx + ly –ωt

into the Petviashvili equation gives a non-linear ordinary differential equation

–ω(k + l

)w′′′ + kCRww′ + (ω + kCR)w′ = ,

and integrating it yields the auxiliary ordinary differential equation

–ω(k + l

)w′′ +

kCRw + (ω + kCR)w + d = , ()

where ω, k, l, CR are constants.Our main result is Theorem .

Theorem Equation () is integrable if and only if dkCR = –ω(k + l)g + (ω+kCk),F = E – gE– g, g, and E are arbitrary. Furthermore, the general solutions of the Eq. ()are of the following form.

(I) The elliptic general solutions are

wd(z) =ω(k + l)

kCR

{–℘(z) +

[℘ ′(z) + F℘(z) – E

]}

–ω(k + l)E

kCR– –

ω

kCR, ()

if dkCR = –ω(k + l)g + (ω + kCk), F = E – gE – g, g and E arearbitrary.

(II) The simply periodic solutions are

ws(z) =ω(k + l)

kCRα coth

α

(z – z) –

ω(k + l)kCR

α – –ω

kCR, ()

if dkCR = –ω(k + l)α + (ω + kCk), α �= , z ∈C.(III) The rational function solutions are

wR(z) =ω(k + l)kCR(z – z)

– –ω

kCR, ()

if dkCR = (ω + kCk), z ∈C.

2 Preliminary lemmas and the complexmethodIn order to explain our complex method and give the proof of Theorem , we need somelemmas and results.



Lemma [, ] Let k ∈ N, then any meromorphic solution w with at least one pole of thekth order Briot-Bouquet equation,

F(w(k),w

)=

n∑i=

Pi(w)(w(k))i = ,

belongs to W , where Pi(w) are polynomials in w with constant coefficients.

Setm ∈N := {, , , . . .}, rj ∈ N =N∪ {}, r = (r, r, . . . , rm), j = , , . . . ,m. Define

Mr[w](z) :=[w(z)

]r[w′(z)]r[w′′(z)

]r · · · [w(m)(z)]rm .

p(r) := r + r + · · · + rm is called the degree of Mr[w]. The differential polynomialP(w,w′, . . . ,w(m)) is defined as follows:

P(w,w′, . . . ,w(m)) :=∑

r∈IarMr[w],

where ar are constants, and I is a finite index set. The total degree of P(w,w′, . . . ,w(m)) isdefined by degP(w,w′, . . . ,w(m)) :=maxr∈I{p(r)}.We will consider the following complex ordinary differential equations:

P(w,w′, . . . ,w(m)) = bwn + c, ()

where b �= , c are constants, n ∈N.

Definition Let p,q ∈ N. Suppose that Eq. () has a meromorphic solution w with atleast one pole; then we say that Eq. () satisfies the weak 〈p,q〉 condition if substituting theLaurent series

w(z) =∞∑

k=–q

ckzk , q > , c–q �= ()

into Eq. () we can determine p distinct Laurent singular parts in the form below:

–∑k=–q

ckzk .

Lemma [–] Let p, l,m,n ∈ N, degP(w,w(m)) < n. Suppose that the mth order Briot-Bouquet equation

P(w(m),w

)= bwn + c ()

satisfies the weak 〈p,q〉 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 w(z – z), z ∈ C. Furthermore each elliptic solution with apole at z = can be written as

w(z) =l–∑i=

q∑j=

(–)jc–ij(j – )!

dj–

dzj–

(

[℘ ′(z) + Bi

℘(z) –Ai

]

–℘(z))

+l–∑i=

c–i

℘ ′(z) + Bi

℘(z) –Ai+

q∑j=

(–)jc–lj(j – )!

dj–

dzj–℘(z) + c, ()

where c–ij are given by Eq. (), Bi = A

i – gAi – g, and∑l

i= c–i = .Each rational function solution w := R(z) is of the form

R(z) =l∑

i=

q∑j=

cij(z – zi)j

+ c, ()

with l (≤ p) distinct poles of multiplicity q.Each simply periodic solution is a rational function R(ξ ) of ξ = eαz (α ∈ C). R(ξ ) has l

(≤ p) distinct poles of multiplicity q, and it is of the form

R(ξ ) =l∑

i=

q∑j=

cij(ξ – ξi)j

+ c. ()

In order to give the representations of the elliptic solutions, we need some notation andresults concerning the elliptic function [].Let ω, ω be two given complex numbers such that Im ω

ω> , let L = L[ω, ω] be dis-

crete subset L[ω, ω] = {ω | ω = nω + mω,n,m ∈ Z}, which is isomorphic to Z×Z.The discriminant is =(c, c) := c – c and

sn = sn(L) :=∑

n≥,n∈N

ωn .

The Weierstrass elliptic function ℘(z) := ℘(z, g, g) is a meromorphic function withdouble periods ω, ω, and satisfying the equation

(℘ ′(z)

) = ℘(z) – g℘(z) – g, ()

where g = s, g = s and (g, g) �= .On changing Eq. () to the form

(℘ ′(z)

) = (℘(z) – e

)(℘(z) – e

)(℘(z) – e

), ()

we have e = ℘(ω), e = ℘(ω), e = ℘(ω +ω).Inversely, given two complex numbers g and g such that (g, g) �= , there exists a

Weierstrass elliptic function ℘(z) with double periods ω, ω such that the above holds.

Lemma [, ] The Weierstrass elliptic functions ℘(z) := ℘(z, g, g) have two successivedegeneracies and in addition we have the following.



(I) Degeneracy to simply periodic functions (i.e., rational functions of one exponentialekz) occurs according to

℘(z, d, –d) = d –

d

coth√dz ()

if one root ej is double ((g, g) = ).(II) Degeneracy to rational functions of z occurs according to

℘(z, , ) =z

if one root ej is triple (g = g = ).(III) The addition formula holds according to

℘(z – z) = –℘(z) –℘(z) +

[℘ ′(z) +℘ ′(z)℘(z) –℘(z)

]

. ()

By the above lemmas, we can give a new method below, called, say, the complex method,to find exact solutions of some PDEs.Step . Substituting the transform T : u(x, y, t) → w(z), (x, y, t) → z into a given PDE

gives a non-linear ordinary differential equation () or ().Step . Substitute Eq. () into Eq. () or () to determine whether the weak 〈p,q〉

condition holds.Step . By the indeterminate relations ()-() we find the elliptic, rational, and simply

periodic solutions w(z) of Eq. () or () with pole at z = , respectively.Step . By Lemmas and we obtain all meromorphic solutions w(z – z).Step . Substituting the inverse transform T– into these meromorphic solutions

w(z – z), we get all exact solutions u(x, t) of the originally given PDE.

3 Proof of Theorem 1Substituting () into Eq. () we have q = , p = , c– = ω(k+l)

kCR, c– = , c = – – ω

kCR,

c = , c = (ω+kCR)–dkCRω(k+l)kCR

, c = , c is arbitrary.Hence, Eq. () satisfies the weak 〈, 〉 condition and is a second-order Briot-Bouquet

differential equation. Obviously, Eq. () satisfies the dominant condition. So, by Lemma ,we know that all meromorphic solutions of Eq. () belong to W . Now we will give theforms of all meromorphic solutions of Eq. ().By Eq. (), we infer that the indeterminate rational solutions of Eq. () with pole at z =

have the form of

R(z) =cz

+cz+ c.

Substituting R(z) into Eq. (), we get

R(z) =ω(k + l)

kCRz– –

ω

kCR.



Here dkCR = (ω + kCR). Thus for all rational solutions of Eq. ()

R(z) =ω(k + l)kCR(z – z)

– –ω

kCR, ()

where dkCR = (ω + kCR), z ∈C.In order to have simply periodic solutions, set ξ = eαz , put w = R(ξ ) into Eq. (); then

–ω(k + l

)[ξR′ + ξ R′′] + (w + kCR)R +

kCRR + d = . ()

Substituting

R(ξ ) =c

(ξ – )+

cξ –

+ c

into Eq. (), we obtain

R(z) =ω(k + l)α

kCR(ξ – )+ω(k + l)α

kCR(ξ – )+

ω(k + l)α

kCR– –

ω

kCR.

Here dkCR = (ω + kCR) –ω(k – l)α. Substituting ξ = eαz into the above relation, andthen we get simply periodic solutions of Eq. () with pole at z = :

ws(z) =ω(k + l)α

kCR(eαz – )+ω(k + l)α

kCR(eαz – )+

ω(k + l)α

kCR– –

ω

kCR

=ω(k + l)αeαz

kCR(eαz – )+

ω(k + l)α

kCR– –

ω

kCR

= ws(z) =ω(k + l)

kCRα coth

α

(z) –

ω(k + l)kCR

α – –ω

kCR.

So all simply periodic solutions of Eq. () are obtained by

ws(z) =ω(k + l)

kCRα coth

α

(z – z) –

ω(k + l)kCR

α – –ω

kCR, ()

where dkCR = –ω(k + l)α + (ω + kCk), α �= , z ∈C.From Eq. () of Lemma , we have indeterminate relations of the elliptic solutions of

Eq. () with pole at z = ,

wd(z) = c–℘(z) + c.

Putting wd(z) into Eq. (), we obtain

wd(z) =ω(k + l)

kCR℘(z) – –

ω

kCR.

Here dkCR = –ω(k + l)g + (ω+ kCk). Therefore, for all elliptic solutions of Eq. ()

wd(z) =ω(k + l)

kCR℘(z – z) – –

ω

kCR,



Figure 1 The solution of the Petviashvili equation corresponding to wR , (a) t = –5, (b) t = 0, (c) t = 5.

where z ∈C. Making use of the addition formula of Lemma , we rewrite it in the form

wd(z) =ω(k + l)

kCR

{–℘(z) +

[℘ ′(z) + F℘(z) – E

]}

–ω(k + l)E

kCR– –

ω

kCR. ()

Here, dkCR = –ω(k + l)g + (ω + kCk), F = E – gE – g, g, and E are arbitrary.

4 Computer simulations for new solutionsIn this section, we give some computer simulations to illustrate our main results. Herewe take the new rational solutions wR(z) and simply periodic solutions ws(z) to furtheranalyze their properties by Figures and .() Take k = l = , ω = CR = –, d = , z = in wR(z).() Take k = l = , ω = CR = –, d = , z = , α = in ws(z).



Figure 2 The solution of the Petviashvili equation corresponding to ws , (a) t = –5, (b) t = 0, (c) t = 5.

5 ConclusionsThe complex method is a very important tool in finding the traveling wave exact solu-tions of non-linear evolution equations such as the Petviashvili equation. In this paper, weemploy the complex method to obtain all meromorphic exact solutions of the complexvariant Eq. (); 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.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsZH 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.

Author details1School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China. 2Key Laboratory ofMathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University,Guangzhou, 510006, China.

AcknowledgementsThis work is supported by the NSF of China (11271090) and NSF of Guangdong Province (S2012010010121). Also thiswork was supported by the Visiting Scholar Program of the Chern Institute of Mathematics at Nankai University wherethe authors worked as visiting scholars. The authors would like to express their hearty thanks to the Chern Institute ofMathematics 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 fortheir very helpful comments and useful suggestions.

Received: 21 September 2013 Accepted: 19 December 2013 Published: 17 Jan 2014

References1. Zhang, L, Zhang, Y, Liu, SK: Two-dimensional Rossby wave: exact solutions to Petviashvili equation. Commun. Theor.

Phys. 45, 414-416 (2006)2. Zhang, L, Zhang, LF, Li, CY, Wang, T, Tan, YK: Some new exact solutions of Jacobian elliptic function of Petviashvili

equation. Commun. Theor. Phys. 49, 1557-1560 (2008)3. Eremenko, A: Meromorphic solutions of equations of Briot-Bouquet type. Teor. Funktsii, Funk. Anal. i Prilozh. 38, 48-56

(1982). English translation: Transl. Am. Math. Soc. 133(2), 15-23 (1986)4. Eremenko, A, Liao, LW, Ng, TW: Meromorphic solutions of higher order Briot-Bouquet differential equations. Math.

Proc. Camb. Philos. Soc. 146(1), 197-206 (2009)5. Kudryashov, NA: Meromorphic solutions of nonlinear ordinary differential equations. Commun. Nonlinear Sci. Numer.

Simul. 15(10), 2778-2790 (2010)6. Lang, S: Elliptic Functions, 2nd edn. Springer, New York (1987)7. Yuan, WJ, Shang, YD, Huang, Y, Wang, H: The representation of meromorphic solutions of certain ordinary differential

equations and its applications. Scientia Sinica Mathematica 43(6), 563-575 (2013). doi:10.1360/012012-1598. Conte, R, Musette, M: Elliptic general analytic solutions. Stud. Appl. Math. 123(1), 63-81 (2009)

10.1186/1687-1847-2014-24Cite this article as: Huang et al.: The general meromorphic solutions of the Petviashvili equation. Advances inDifference Equations 2014, 2014:24


http://dx.doi.org/10.1360/012012-159

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