# Some New Traveling Wave Exact Solutions of the (2+1)-Dimensional Boiti-Leon-Pempinelli Equations

Authors: Jian-ming Qi; Fu Zhang; Wen-jun Yuan; Zi-feng Huang
Journal: The Scientific World Journal
 (2014)

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## Abstract

We employ the complex method to obtain all meromorphic exact solutions of complex (2+1)-dimensional Boiti-Leon-Pempinelli equations (BLP system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations.
Our results show that all rational and simply periodic traveling wave exact solutions of the equations (BLP) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutionsur,2(z) and simply periodic solutions us,2–6(z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.


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## Body

## 1. Introduction

Boiti et al. [1] introduced the Boiti-Leon-Pempinelli equations (BLP system of equations)
(1)uty=(u2-ux)xy+2vxxx,(2)vt=vxx+2uvx.A considerable research work has been invested in [2–5] to study the BLP system (1) and (2). The integrability of this system was studied in [1] by using the Sine-Gordon and the Sinh-Gordon equations. Other works have been conducted by using other methods such as Jacobi elliptic methods and balance methods [3–5].For finding exact solutions of the BLP system, many authors applied the tanh-coth method and the Exp-function method to derive them [6–17] and [8–22], respectively.In 2010, Wazwaz and Mehanna [17] used the tanh-coth method and Exp-function method to the BLP equations to derive many new varieties of travelling wave solutions with distinct physical structures. Substituting the traveling wave transformation
(3)u(x,y,t)=u(z),v(x,y,t)=v(z),z=μ(x+y-ct)
into (1) and (2), one carries out the system of nonlinear ordinary differential equations as follows:
(4)-cu′′=(u2)′′-μu′′′+2μv′′′,(5)-cv′=μv′′+2uv′,
where c is the wave velocity and μ is a nonzero constant (see [17, 23]).Afterwards we integrated (1) twice with respect to z and considered the constants of integration to be zero and obtained
(6)v′=12u′-u2+cu2μ.Furthermore substituting (6) into (5) yields
(7)μ2u′′-2u3-3cu2-c2u=0.In 2011, Kudryashov [23] got the general solutions of (7) via Jacobi elliptic functions and analyzed the application of the tanh-coth method for finding exact solutions of (7) and showed that all the solutions which are presented by Wazwaz and Mehanna can be reduced to a single one and so on.In this paper, we employ the complex method which was introduced by Yuan et al. [24–26] to obtain the general solutions and some new solutions of (7). In order to state our results, we need some concepts and notations.A meromorphic functionw(z) means that w(z) is holomorphic in the complex plane ℂ except for poles. ℘(z;g2,g3) is the Weierstrass elliptic function with invariants g2 and g3. We say that a meromorphic function f belongs to the class W if f is an elliptic function, or a rational function of eαz,α∈ℂ, or a rational function of z.Our main result is the following theorem.Theorem 1.
All meromorphic solutionsu of (7) belong to the class W. Furthermore, (7) has the following three forms of solutions.
(I) The elliptic general solutions(8)wd(z)=±μ2×(((-℘+A)(4℘A2+4℘2A((-℘+A)(4℘A2+4℘2A+4B℘3+12AB℘2-3Bg2℘-ABg2)-1+2℘′B-℘g2-Ag2))×(((12A2-g2)℘+4A3-3Ag2)℘′((-℘+A)(4℘A2+4℘2A+4B℘3+12AB℘2-3Bg2℘-ABg2)-1),
where g3=0, B2=4A3-g2A, and g2 and A are arbitrary constants.
(II) The simply periodic solutions, whereξ=eαz are obtained, for z0∈ℂ,
(9)us,1(z)=ceαz-1=c2(coth⁡α2(z-z0)-1),
where μ=±c/α;
(10)us,2(z)=-c2(1+coth⁡α2(z-z0)),
where μ=±c/α;
(11)us,3(z)=ccoth⁡α(z-z0)-c,
where μ=±c/2α;
(12)us,4(z)=-ccoth⁡α(z-z0),
where μ=±c/2α;
(13)us,5(z)=2ci2Sinhα(z-z0)-c2,
where μ=±2ci/2α;
(14)us,6(z)=-2ci2Sinhα(z-z0)-c2,
where μ=±2ci/2α.
(III) All rational function solutions are of the following two distinct forms. For anyz0∈ℂ,
(15)ur,1(z)=±μz-z0,ur,2(z)=±μz-z0±μz-z0-z1,
where z0∈ℂ,z1≠0,c=0.


## 2. Preliminary Lemmas and the Complex Method

In order to give complex method and the proof of Theorem1, we need some notations and results.Setm∈ℕ:={1,2,3,…}, rj∈ℕ0=ℕ∪{0}, r=(r0,r1,…,rm), j=0,1,…,m. We define a differential monomial denoted by
(16)Mr[w](z):=[w(z)]r0[w′(z)]r1×[w′′(z)]r2⋯[w(m)(z)]rm.p(r):=r0+2r1+⋯+(m+1)rm and deg⁡(M) are called the weight and degree of Mr[w], respectively.A differential polynomialP(w,w′,…,w(m)) is defined as follows:
(17)P(w,w′,…,w(m)):=∑r∈I‍arMr[w],
where ar are constants and I is a finite index set. The total weight and degree of P(w,w′,…,w(m)) are defined by W(P):=max⁡r∈I{p(r)} and deg⁡(P):=max⁡r∈I{deg⁡(Mr)}, respectively.We will consider the following complex ordinary differential equations:(18)P(w,w′,…,w(m))=bwn+c,
where b≠0,c are constants, n∈ℕ.Letp,q∈ℕ. Suppose that (18) has a meromorphic solution w with at least one pole; we say that (18) satisfies weak 〈p,q〉 condition if, substituting Laurent series
(19)w(z)=∑k=-q∞‍ckzk,q>0,c-q≠0
into (18), we can determine p distinct Laurent singular parts below
(20)∑k=-q-1‍ckzk.Lemma 2.
Letp,l,m,n∈ℕ, deg⁡P(w,w(m))<n. Suppose that an m order Briot-Bouquet equation
(21)P(w(m),w)=bwn+c
satisfies weak 〈p,q〉 condition, then whose all meromorphic solutions w belong to the class W. If for some values of parameters such solution w exists, then other meromorphic solutions form a one parametric family w(z-z0),z0∈ℂ. Furthermore each elliptic solution with pole at z=0 can be written as
(22)w(z)=∑i=1l-1‍∑j=2qi(-1)jc-ij(j-1)!dj-2dzj-2‍×(14[℘′(z)+Bi℘(z)-Ai]2-℘(z))+∑i=1l-1‍c-i12℘′(z)+Bi℘(z)-Ai+∑j=2ql‍(-1)jc-lj(j-1)!dj-2dzj-2℘(z)+c0,
where c-ij are given by (19), Bi2=4Ai3-g2Ai-g3, and ∑i=1lc-i1=0.
Each rational function solutionw:=R(z) is of the form
(23)R(z)=∑i=1l‍∑j=1q‍cij(z-zi)j+c0,
with l(≤p) distinct poles of multiplicity q.
Each simply periodic solution is a rational functionR(ξ) of ξ=eαz(α∈ℂ). R(ξ) has l(≤p) distinct poles of multiplicity q and is of the form
(24)R(ξ)=∑i=1l‍∑j=1q‍cij(ξ-ξi)j+c0.In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [27].Letω1,ω2 be two given complex numbers such that Im⁡ω1/ω2>0,L=L[2ω1,2ω2] be discrete subset L[2ω1,2ω2]={ω∣ω=2nω1+2mω2,n,m∈ℤ}, which is isomorphic to ℤ×ℤ. The discriminant Δ=Δ(c1,c2):=c13-27c22 and
(25)sn=sn(L):=∑ω∈L∖{0}‍1ωn.Weierstrass elliptic function℘(z):=℘(z,g2,g3) is a meromorphic function with double periods 2ω1,2ω2 and satisfying the equation
(26)(℘′(z))2=4℘(z)3-g2℘(z)-g3,
where g2=60s4,g3=140s6 and Δ(g2,g3)≠0.If we change (24) to the form
(27)(℘′(z))2=4(℘(z)-e1)(℘(z)-e2)(℘(z)-e2),
we have e1=℘(ω1),e2=℘(ω2),ande3=℘(ω1+ω2).Inversely, given two complex numbersg2 and g3 such that Δ(g2,g3)≠0, then there exists double periods 2ω1,2ω2 Weierstrass elliptic function ℘(z) such that the above results hold.Lemma 3 (see [27, 28]).
Weierstrass elliptic functions℘(z):=℘(z,g2,g3) have two successive degeneracies and addition formula.
(I) Degeneracy to simply periodic functions (i.e., rational functions of one exponentialekz) according to
(28)℘(z,3d2,-d3)=2d-3d2coth⁡23d2z,
if one root ej is double (Δ(g2,g3)=0).
(II) Degeneracy to rational functions ofz according to
(29)℘(z,0,0)=1z2
if one root ej is triple (g2=g3=0).
(III) Addition formula(30)℘(z-z0)=-℘(z)-℘(z0)+14[℘′(z)+℘′(z0)℘(z)-℘(z0)]2.By the above lemma and results, we can give a new method below, saycomplex method, to find exact solutions of some PDEs.Step 1.
Substituting the transformT:u(x,t)→w(z),(x,t)→z into a given PDE gives a nonlinear ordinary differential equation (18) or (21).Step 2.
Substitute (19) into (18) or (21) to determine that weak 〈p,q〉 condition holds.Step 3.
By determinant relation (22)–(24) we find the elliptic, rational, and simply periodic solutions u(z) of (18) or (21) with pole at z=0, respectively.Step 4.
By Lemmas2 and 3 we obtain all the meromorphic solutions w(z-z0).Step 5.
Substituting the inverse transformT-1 into these meromorphic solutions w(z-z0), then we get all exact solutions u(x,t) of the original given PDE.


## 3. Proof of Theorem1

Substituting (19) into (7) we have q=1. p=2, c-1=±μ. Hence, (7) satisfies weak 〈2,1〉 condition and is a 2nd order Briot-Bouquet differential equation. Obviously, (7) satisfies the dominant condition. So, by Lemma 2, we know that all meromorphic solutions of (7) belong to W. Now we will give the forms of all meromorphic solutions of (7).By (22), we infer the indeterminant rational solutions of (7) with pole at z=0 that
(31)ur(z)=c11z+c12z-z1+c10.
Substituting ur(z) into (7), we get two distinct forms. One of them is
(32)ur0,1(z)=±μz,
where c=0. The other is
(33)ur0,2(z)=±μz±μz-z1,
where c=0.Thus all rational solutions of (7) are
(34)ur,1(z)=±μz-z0,ur,2(z)=±μz-z0±μz-z0-z1,
where z0∈ℂ,z1≠0,c=0.In order to have simply periodic solutions, setξ=exp⁡(αz), put u=u(ξ) into (7), and then
(35)μ2α2(ξ2u′′+ξu′)-2u3-3cu2-c2u=0.
Substituting
(36)u2(ξ)=c2ξ-1+c1ξ-ξ1+c0
into (7), we obtain the indeterminant simply periodic solutions of (35) with pole at ξ=1 that
(37)us1,1(ξ)=cξ-1,
where μ=±c/α;
(38)us1,2(ξ)=-cξ-1-c,
where μ=±c/α;
(39)us1,3(ξ)=c2(ξ-1)+-c2(ξ+1)=cξ+ξ-1ξ-ξ-1-c,
where μ=±c/2α;
(40)us1,4(ξ)=-c2(ξ-1)+c2(ξ+1)-c=-cξ+ξ-1ξ-ξ-1,
where μ=±c/2α;
(41)us1,5(ξ)=2ci2(ξ-1)+2ci2(ξ+1)-c2=2ciξ-ξ-1-c2,
where μ=±2ci/2α;
(42)us1,6(ξ)=-2ci2(ξ-1)-2ci2(ξ+1)-c2=-2ciξ-ξ-1-c2,
where μ=±2ci/2α.Substituteξ=eαz into the above six relations, and then we get all simply periodic solutions of (7) with pole at z=0(43)us0,1(z)=c2(coth⁡α2z-1),
where μ=±c/α;
(44)us0,2(z)=-c2(1+coth⁡α2z),
where μ=±c/α;
(45)us0,3(z)=ccoth⁡αz-c,
where μ=±c/2α;
(46)us0,4(z)=-ccoth⁡αz,
where μ=±c/2α;
(47)us0,5(z)=2ci2Sinhαz-c2,
where μ=±2ci/2α;
(48)us0,6(z)=-2ci2Sinhαz-c2,
where μ=±2ci/2α.So all simply periodic solutions of (7) are obtained, for z0∈ℂ, by
(49)us,1(z)=c2(coth⁡α2(z-z0)-1),
where μ=±c/α;
(50)us,2(z)=-c2(1+coth⁡α2(z-z0)),
where μ=±c/α;
(51)us,3(z)=ccoth⁡α(z-z0)-c,
where μ=±c/2α;
(52)us,4(z)=-ccoth⁡α(z-z0),
where μ=±c/2α;
(53)us,5(z)=2ci2sinh⁡α(z-z0)-c2,
where μ=±2ci/2α;
(54)us,6(z)=-2ci2Sinhα(z-z0)-c2,
where μ=±2ci/2α.From (21) in Lemma 2, we have indeterminant relations of elliptic solutions of (7) with pole at z=0(55)ud0(z)=c-12℘′(z)+F℘(z)-E+c30,
where F2=4E3-g2E-g3. Applying conclusion II of Lemma 2 to ud0(z) and noting the results of rational solutions obtained above, we deduce that c30=0, E=F=0, and g3=0. Then we get that
(56)ud0(z)=±μ2℘′(z)℘(z).
Therefore, all elliptic function solutions of (7)
(57)ud0(z)=±μ2℘′(z-z0)℘(z-z0).
Here z0∈ℂ,g3=0. Making use of the addition of Lemma 3, we rewrite it to the form
(58)wd(z)=±μ2×(((-℘+A)(4℘A2+4℘2A((12A2-g2)℘+4A3-3Ag2)℘′+4B℘3+12AB℘2-3Bg2℘-ABg2)-1+2℘′B-℘g2-Ag2))×(((12A2-g2)℘+4A3-3Ag2)℘′((12A2-g2)℘+4A3-3Ag2)℘′+4B℘3+12AB℘2-3Bg2℘-ABg2)-1).
Here g3=0,  B2=4A3-g2A,  g2, and A are arbitrary constants.This completes the proof of Theorem1.


## 4. Computer Simulations for New Solutions

In this section, we give some computer simulations to illustrate our main results. Here we take the new rational solutionsur,2(z) and simply periodic solutions us,2–6(z) to further analyze their properties by Figures 1, 2, 3, 4, 5, and 6.The solution of BLP corresponding tour,2: take c=1, μ=1, z0=0, z1=1; (a) t=-1/100, (b) t=0, and (c) t=1/100.
(a)
(b)
(c)The solution of BLP corresponding tous,2: take c=1, α=1, z0=0; (a) t=-1/115, (b) t=0, and (c) t=1/115.
(a)
(b)
(c)The solution of BLP corresponding tous,3: take c=1, α=1, z0=0; (a) t=-1/132, (b) t=0, and (c) t=1/132.
(a)
(b)
(c)The solution of BLP corresponding tous,4: take c=1, α=1, z0=0; (a) t=-1/144, (b) t=0, and (c) t=1/144.
(a)
(b)
(c)The solution of BLP corresponding tous,5: take c=1, α=1, z0=0; (a) t=-1/125, (b) t=1/10000, and (c) t=1/125.
(a)
(b)
(c)The solution of BLP corresponding tous,6: take c=1, α=1, z0=0; (a) t=-1/123, (b) t=1/10000, and (c) t=1/123.
(a)
(b)
(c)


## 5. Conclusions

Complex method is a very important tool in finding the exact solutions of nonlinear evolution equations, and the (2+1)-dimensional Boiti-Leon-Pempinelli equation is classic and simplest case of the nonlinear reaction-diffusion equation. In this paper, we employ the complex method to obtain the general meromorphic solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli equation, which improves the corresponding result obtained by Kudryashov [23] and Wazwaz and Mehanna [17]. Our results show that all rational and simply periodic traveling wave exact solutions of the equations (BLP) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions ur,2(z) and simply periodic solutions us,2–6(z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.



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