# Symbolic Computation and the Extended  Hyperbolic Function Method for Constructing Exact Traveling Solutions of Nonlinear PDEs

Authors: Huang Yong; Shang Yadong; Yuan Wenjun
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

On the basis of the computer symbolic system Maple and the extended hyperbolic function method, we develop a more mathematically rigorous and systematic procedure for constructing exact solitary wave solutions and exact periodic traveling wave solutions in triangle form of various nonlinear partial differential equations that are with physical backgrounds. Compared with the existing methods, the proposed method gives new and more general solutions. More importantly, the method provides a straightforward and effective algorithm to obtain abundant explicit and exact particular solutions for large nonlinear mathematical physics equations. We apply the presented method to two variant Boussinesq equations and give a series of exact explicit traveling wave solutions that have some more general forms. So consequently, the efficiency and the generality of the proposed method are demonstrated.


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## Body

## 1. Introduction

The nonlinear phenomena are very important in a variety of scientific fields, especially in fluid dynamics, solid-state physics, hydrodynamics, plasma physics, elastic dynamics, acoustics, chemical physics, and nonlinear optics. Nonlinear evolution partial differential equations are widely used as models to describe complex physical phenomena in various fields of sciences, especially in fluid mechanics, solid-state mechanics, atmospheric physics, chemical reaction-diffusion dynamics, ion acoustics, and nonlinear vibration. The investigation of exact traveling wave solutions to nonlinear evolution partial differential equations plays an important role in the study of nonlinear science. The exact solution, if available, of those nonlinear equations facilitates the verification of numerical solvers and aids in the stability analysis of solutions. It can also provide much physical information and more insights into the physical aspects of the nonlinear physical problem. During the past decades, much effort has been spent on the subject of obtaining the exact analytical solutions to the nonlinear evolution PDEs. Many powerful methods have been proposed such as the inverse scattering transformation method [1], the Bäcklund and Darboux transformation method [2, 3], the Hirota bilinear method [4], the Lie group reduction method [5], the tanh method [6], the tanh-coth method [7], the sine-cosine method [8, 9], the homogeneous balance method [10–12], the Jacobi elliptic function method [13, 14], the extended tanh method [15, 16], the F-expansion method and its extension [17, 18], the Riccati method [19, 20], and extended improved tanh-function method [21, 22]. With the development of symbolic computation, the tanh method, the F-expansion method, the sine-Gordon equation expansion method, and all kinds of auxiliary equation methods attract more and more researchers. In this paper, we present an effective extension to the projective Riccati equation method [19, 20] and extended improved tanh-function method [21, 22] and develop an effective Maple software package “PDESolver” to uniformly construct a series of traveling wave solutions including solitary wave solutions, singular traveling, rational, triangular periodic solitons for general nonlinear evolution equations. Our method can be regarded as an extension of the works by Wazwaz [23, 24] and Soliman [21]. For illustration, we apply the presented method to two variant Boussinesq equations(1.1)Ht+(Hu)x+uxxx=0,ut+Hx+uux=0,(1.2)Ht+ux+(Hu)x-αuxxx=0,ut+Hx+uux-3αuxxt=0,
where H(x,t), u(x,t) are the unknown functions depending on the temporal variable t and the spatial variable x. These two equations were introduced as models for water waves and called variant Boussinesq equations I and II, respectively [25]. Their symmetries, conservation laws, and inverse scattering transformation, and soliton solutions have been investigated by many authors (see [8, 16] and references therein). Many exact solutions have been obtained by many researchers using the sine-cosine method [8], the homogeneous balance method [10], the extended tanh method [16], and the the F-expansion method [17], respectively. We will give a series of new traveling wave solutions for the two equations. Some entirely new exact solitary wave solutions and periodic wave solutions of the equations are obtained.The paper is organized as follows: in Section2, we briefly describe what is the extended hyperbolic function method and how to use it to derive the traveling solutions of nonlinear PDEs. In Section 3, we apply the extended hyperbolic function method to (1.1) and (1.2) and establish many rational form solitary wave, rational form triangular periodic wave solutions. In Section 4, we briefly make a summary to the results that we have obtained.


## 2. The Extended Hyperbolic Function Method

Now we would like to outline the main steps of our method for solving nonlinear PDEs.Consider the coupled Riccati equations(2.1)f′(ξ)=-f(ξ)g(ξ),g′(ξ)=ε-rεf(ξ)-g2(ξ),
where ε=±1 or 0,r is a constant. When ε≠0, we can obtain the following first integral as given:
(2.2)g2(ξ)=ε-2rεf(ξ)+Cf2(ξ).Step 1.
Consider a system ofN nonlinear evolution equations (N≥1) with n independent variables x=(x1,x2,…,xn) and m dependent variables u=(u1,u2,…,um), given by
(2.3)Pi(u,∂u,∂2u,…,∂ru)=0,i=1,2,…,N,
where Pi, i=1,2,…,N, are in general nonlinear functions of their arguments, ∂ju denotes the coordinates with components ∂ju/(∂xi1∂xi2⋯∂xij)=ui1i2⋯ij,ij=1,2,…,n, for j=1,2,…,k, corresponding to all jth-order partial derivatives of u with respect to x.
We seek the following formal traveling wave solutions which are of important physical significance:(2.4)u(x,t)=u(ξ),ξ=k1x1+k2x2+···+knxn+ξ0,(∑i=1n‍ki2≠0),
where ki are constants to be determined later and ξ0 is an arbitrary constant.
Then the nonlinear partial differential equations (2.3) reduce to a nonlinear ordinary differential equations
(2.5)Qi(u,u′,u′′,…uk)=0,i=1,2,…,N,
where ' denotes d/dξ.Step 2.
To seek the exact solutions of nonlinear partial differential equations (2.3), we assume that the solution of the ODEs (2.5) is of the following form.(a)
Whenε=±1 in (2.1), (2.2),
(2.6)ul(x,t)=ul(ξ)=∑i=0ln‍al,i(f(ξ))i+∑j=1ln‍bl,j(f(ξ))j-1g(ξ),l=1,2,…,m,
where the coefficients al,i(i=0,1,2…,ln) and bl,j(j=1,2,…,ln) are constants to be determined latter.(b)
Whenε=0 in (2.1),
(2.7)ul(x,t)=∑i=0ln‍al,i(g(ξ))i,l=1,2,…,m,
where g'(ξ)=-g2(ξ) and the coefficients al,i(i=0,1,2…,ln) are constants to be determined.Step 3.
Balancing the highest-order derivative term and the nonlinear terms in (2.5), we get balance powers ln (usually positive integer). If some one of ln is a fraction or a negative integer, say sn is negative, we make the following transformation:(2.8)us(ξ)=vssn,
then return to determine balance power n again.Step 4.
(a) Whenε≠0, substituting (2.6) along with the conditions (2.1) and (2.2) into (2.5).
(b) Whenε=0, substituting (2.7) along with the condition g'(ξ)=-g2(ξ) into (2.5).
Then eliminating any derivative of(f,g) and any power of g higher than one and setting the coefficients of powers fi(i=0,1,…;) and fjg(j=0,1,…) in the case (a) (setting the coefficients of the different powers g in the case (b)) to zero yield a set of overdetermined nonlinear algebraic equations with respect to the unknown variables ki, i=1,2,…,n, al,i(i=0,1,2…,ln), bl,j(j=1,2,…,ln), r, C. With the aid of Maple, we apply the Wu-elimination method [26] to solve the above overdetermined system of nonlinear algebraic equations, that yields the values of ki, i=1,2,…,n, al,i(i=0,1,2…,ln), bl,j(j=1,2,…,ln), r, C.Step 5.
We know that the coupled Riccati equations (2.1) admit the following special solutions.(a)
Whenε=1,
(2.9)f(ξ)=1acoshξ+bsinhξ+r,g(ξ)=asinhξ+bcoshξacoshξ+bsinhξ+r,
and then g2(ξ)=1-2rf(ξ)+(b2-a2+r2)f2(ξ).(b)
Whenε=-1,
(2.10)f(ξ)=1acosξ+bsinξ+r,g(ξ)=bcosξ-asinξacosξ+bsinξ+r,
and then g2(ξ)=-1+2rf(ξ)+(b2+a2-r2)f2(ξ).(c)
Whenε=0,
(2.11)f(ξ)=Cξ,g(ξ)=1ξ,
where C is constant.
Thus according to (2.4), (2.6), (2.9), (2.10) (or (2.4), (2.7), (2.11)) and the conclusions in Step 4, we can obtain the multiple more general exact explicit traveling wave solutions of nonlinear partial differential equations (2.3).Remark 2.1.
Many types of traveling wave solutions such as tanhξ, coth ξ, sech ξ, cosech ξ, tan ξ, cot ξ, sec ξ, cosec ξ, and 1/ξ can be obtained by considering the different values of a,b in (2.9), (2.10). Our proposed method is a generalization of the tanh method [6], the extended tanh method, the tanh-coth method [7, 15], the extended tanh-function method [27], the extended improved tanh-function method [21], and the recent works of Wazwaz [23, 24].
The proposed method supplies a unified formulation to construct abundant traveling wave solutions to nonlinear evolution partial differential equations of special physical significance. Furthermore, the presented method is readily computerizable by using symbolic software Maple. Based on the extended hyperbolic function method and computer symbolic software, we develop a Maple software package “PDESolver.” Compared with packages RATH, ERATH, AJFM, TRWS, and RAEEM, “PDESolver” is more effective. “PDESolver” can obtain more exact traveling wave solutions.


## 3. Exact Solutions of the Variant Boussinesq Equations

As an example of the use of Maple software package “PDESolver,” we first consider the variant Boussinesq equations (1.1).According to the above method, to seek traveling wave solutions of (1.1), we make the transformation
(3.1)u(x,t)=u(ξ),H(x,t)=H(ξ),ξ=kx+ωt+ξ0,
and thus (1.1) becomes
(3.2)ωH′(ξ)+k(Hu)′(ξ)+k3u′′′(ξ)=0,ωu′(ξ)+kH′(ξ)+ku(ξ)u′(ξ)=0.Firstly we assume that the solutions of (3.1) are of the form
(3.3)H(x,t)=H(ξ)=∑i=0m‍a1,i(f(ξ))i+∑j=1m‍b1,j(f(ξ))j-1g,u(x,t)=u(ξ)=∑i=0n‍a1,i(f(ξ))i+∑j=1n‍b1,j(f(ξ))j-1g,
where a1,i, b1,j, i=0,1,2,…,m, j=1,2,…,m, and a2,i, b2,j, i=0,1,2,…,m, j=1,2,…,m, are constants to be determined latter, f and g satisfy (2.1), (2.2), and ε=±1. We can get the balancing powers m=2, n=1. So we have
(3.4)H(ξ)=a1,0+a1,1f(ξ)+a1,2f2(ξ)+b1,1g(ξ)+b1,2f(ξ)g(ξ),u(ξ)=a2,0+a2,1f(ξ)+b2,1g(ξ),
where a1,0, a1,1, a1,2, b1,1, b1,2, a2,0, a2,1, b2,1 are constants to be determined later.With the aid of Maple, substituting (3.4) along with (2.1) and (2.2) into (3.2), yields a set of algebraic equations for fi(ξ)gj(ξ). Setting the coefficients of these terms fi(ξ)gj(ξ) to zero yields a set of overdetermined nonlinear algebraic equations with respect to a1,0, a1,1, a1,2, b1,1, b1,2, a2,0, a2,1, b2,1, k, ω, and r:
(3.5)-2k(b1,2+a2,1b2,1)=0,-k(b2,12+2a1,2+a2,12)=0,-3k(2k2a2,1+b1,2b2,1+a1,2a2,1)=0,-3k(b1,2a2,1+a1,2b2,1+2k2b2,1)=0,kb2,12εr-ka2,0a2,1-ωa2,1-ka1,1=0,ε(-ka2,1b2,1+ka2,0b2,1r+kb1,1r-kb1,2+ωb2,1r)=0,3kb1,2εr+3ka2,1b2,1εr-kb1,1-ka2,0b2,1-ωb2,1=0,-ka1,1a2,0-kb1,2εb2,1-k3a2,1ε-ka1,0a2,1-ωa1,1+2kb1,1εrb2,1=0,-2kb1,1b2,1-2ka1,1a2,1-2ka1,2a2,0+6k3a2,1εr-2ωa1,2+4kb1,2εrb2,1=0,5kb1,2εra2,1+5ka1,2b2,1εr-2ωb1,2+12k3b2,1εr-2kb1,2a2,0-2kb1,1a2,1-2ka1,1b2,1=0,ε(k3b2,1εr-ωb1,2+ka1,0b2,1r+kb1,1ra2,0+ωb1,1r-kb1,2a2,0-ka1,1b2,1-kb1,1a2,1)=0,3ka1,1b2,1εr-ka1,0b2,1-3k3b2,1ε2r2-kb1,1a2,0-2kb1,2εa2,1-ωb1,1+3ωb1,2εr-4k3b2,1ε+3kb1,2εra2,0+3kb1,1εra2,1-2ka1,2b2,1ε=0.To get a nontrivial solution of (1.1), we assume that k≠0 and a1,12+a1,22+b1,12+b1,22+a2,12+b2,12≠0. By making use of the Maple software package “PDESolver” which is based on the Wu-elimination method [26], we solve (3.5) and get the following nontrivial solutions with the aid of the computer program Maple 12.Setξ=kx+ωt+ξ0, and k(≠0), ω, r are real constants.When   ε=1.Case 1.
r
=
r, a1,0=0, a1,1=k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=k, b1,1=0, b1,2=k2, b2,1=-k. By applying these to (3.4), one gets a solution of (1.1):
(3.6)H1(x,t)=k2racoshξ+bsinhξ+r-k2(acoshξ+bsinhξ+r)2+k2(asinhξ+bcoshξ)(acoshξ+bsinhξ+r)2,u1(x,t)=-ωk+kacoshξ+bsinhξ+r-k(asinhξ+bcoshξ)(acoshξ+bsinhξ+r).Case 2.
r
=
r, a1,0=0, a1,1=k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=k, b1,1=0, b1,2=-k2, b2,1=k. By applying these to (3.4), one gets a solution of (1.1):
(3.7)H2(x,t)=k2racoshξ+bsinhξ+r-k2(acoshξ+bsinhξ+r)2-k2(asinhξ+bcoshξ)(acoshξ+bsinhξ+r)2,u2(x,t)=-ωk+kacoshξ+bsinhξ+r+k(asinhξ+bcoshξ)acoshξ+bsinhξ+r.Case 3.
r
=
r, a1,0=0, a1,1=k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=-k, b1,1=0, b1,2=k2, b2,1=k. By applying these to (3.4), one gets a solution of (1.1):
(3.8)H3(x,t)=k2racoshξ+bsinhξ+r-k2(acoshξ+bsinhξ+r)2+k2(asinhξ+bcoshξ)(acoshξ+bsinhξ+r)2,u3(x,t)=-ωk-kacoshξ+bsinhξ+r+k(asinhξ+bcoshξ)acoshξ+bsinhξ+r.Case 4.
r
=
r, a1,0=0, a1,1=k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=-k, b1,1=0, b1,2=-k2, b2,1=-k. By applying these to (3.4), one gets a solution of (1.1):
(3.9)H4(x,t)=k2racoshξ+bsinhξ+r-k2(acoshξ+bsinhξ+r)2-k2(asinhξ+bcoshξ)(acoshξ+bsinhξ+r)2,u4(x,t)=-ωk-kacoshξ+bsinhξ+r-k(asinhξ+bcoshξ)acoshξ+bsinhξ+r.Case 5.
r
=
(
ω
+
k
a
2,0
)
/
k
2, a1,0=(-k4+ω2+2ωka2,0+k2a2,02)/k2, a1,1=2ω+2ka2,0, a1,2=-2k2, a2,0=a2,0, a2,1=-2k, b1,1=0, b1,2=0, b2,1=0. By applying these to (3.4), one gets a solution of (1.1):
(3.10)H5(x,t)=-k4+(ω+ka2,0)2k2+2ω+2ka2,0(acoshξ+bsinhξ+((ω+ka2,0)/k2))-2k2(acoshξ+bsinhξ+((ω+ka2,0)/k2))2,u5(x,t)=a2,0-2kacoshξ+bsinhξ+((ω+ka2,0)/k2),
where a2,0 is an arbitrary real constant.Case 6.
r
=
-
(
(
ω
+
k
a
2,0
)
/
k
2
), a1,0=(-k4+ω2+2ωka2,0+k2a2,02)/k2, a1,1=-2ω-2ka2,0, a1,2=-2k2, a2,0=a2,0, a2,1=2k, b1,1=0, b1,2=0, b2,1=0. By applying these to (3.4), one gets a solution of (1.1):
(3.11)H6(x,t)=-k4+(ω+ka2,0)2k2-2(ω+ka2,0)acoshξ+bsinhξ-((ω+ka2,0)/k2)-2k2(acoshξ+bsinhξ-((ω+ka2,0)/k2))2,u6(x,t)=a2,0+2kacoshξ+bsinhξ-((ω+ka2,0)/k2).Case 7.
r
=
0, a1,0=0, a1,1=0, a1,2=-2k2, a2,0=-(ω/k), a2,1=0, b1,1=0, b1,2=0, b2,1=2k. By applying these to (3.4), one gets a solution of (1.1):
(3.12)H7(x,t)=-2k2(acoshξ+bsinhξ)2,u7(x,t)=-ωk+2k(asinhξ+bcoshξ)acoshξ+bsinhξ.Case 8.
r
=
0, a1,0=0, a1,1=0, a1,2=-2k2, a2,0=-(ω/k), a2,1=0, b1,1=0, b1,2=0, b2,1=-2k. By applying these to (3.4), one gets a solution of (1.1):
(3.13)H8(x,t)=-2k2(acoshξ+bsinhξ)2,u8(x,t)=-ωk-2k(asinhξ+bcoshξ)acoshξ+bsinhξ.
While takinga=1, b=0 in solutions H7, u7 (3.12) and H8, u8 (3.13), we can obtain the exact soliton solutions obtained in [10] by homogeneous balance method. While taking a=0, b=1 in solutions H7, u7 (3.12) and H8, u8 (3.13), we obtain singular traveling wave solutions of csch2ξ, cothξ type corresponding to the soliton solutions of sech2ξ, tanhξ type. Thus the results obtained in [10] are special cases of this paper. We would like to emphasize that the solutions H5, u5 and H6, u6 are solutions that have an entirely new form and proposed firstly in this paper. It should be pointed out that the solutions H1, u1(3.6) to H4, u4 (3.9) have a more general form than those that appeared in previous literatures. While setting a=1, b=0 (or a=0, b=1, resp.) in some of these solutions, we can obtain all solutions in hyperbolic function form presented in literatures [8, 16]. Since there are some parameters that can take different values, these solutions include abundant new information.
When  ε=-1.Case 9.
r
=
r, a1,0=0, a1,1=-k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=k, b1,1=0, b1,2=k2, b2,1=-k. By applying these to (3.4), one gets a solution of (1.1):
(3.14)H9(x,t)=-k2racosξ+bsinξ+r-k2(acosξ+bsinξ+r)2+k2(-asinξ+bcosξ)(acosξ+bsinξ+r)2,u9(x,t)=-ωk+kacosξ+bsinξ+r-k(-asinξ+bcosξ)(acosξ+bsinξ+r)2.Case 10.
r
=
r, a1,0=0, a1,1=-k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=k, b1,1=0, b1,2=-k2, b2,1=k. By applying these to (3.4), one gets a solution of (1.1):
(3.15)H10(x,t)=-k2racosξ+bsinξ+r-k2(acosξ+bsinξ+r)2-k2(-asinξ+bcosξ)(acosξ+bsinξ+r)2,u10(x,t)=-ωk+kacosξ+bsinξ+r+k(-asinξ+bcosξ)(acosξ+bsinξ+r)2.Case 11.
r
=
r, a1,0=0, a1,1=-k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=-k, b1,1=0, b1,2=k2, b2,1=k. By applying these to (3.4), one gets a solution of (1.1):
(3.16)H11(x,t)=-k2racosξ+bsinξ+r-k2(acosξ+bsinξ+r)2+k2(-asinξ+bcosξ)(acosξ+bsinξ+r)2,u11(x,t)=-ωk-kacosξ+bsinξ+r+k(-asinξ+bcosξ)acosξ+bsinξ+r.Case 12.
r
=
r, a1,0=0, a1,1=-k2r, a1,2=-k2, a2,0=-(ω/k), a2,1=-k, b1,1=0, b1,2=-k2, b2,1=-k. By applying these to (3.4), one gets a solution of (1.1):
(3.17)H12(x,t)=-k2racosξ+bsinξ+r-k2(acosξ+bsinξ+r)2-k2(-asinξ+bcosξ)(acosξ+bsinξ+r)2,u12(x,t)=-ωk-kacosξ+bsinξ+r-k(-asinξ+bcosξ)acosξ+bsinξ+r.Case 13.
r
=
(
k
a
2,0
+
ω
)
/
k
2, a1,0=(k4+k2a2,02+2ka2,0ω+ω2)/k2, a1,1=-2ka2,0-2ω, a1,2=-2k2, a2,0=a2,0, a2,1=2k, b1,1=0, b1,2=0, b2,1=0. By applying these to (3.4), one gets a solution of (1.1):
(3.18)H13(x,t)=k4+(ω+ka2,0)2k2+-2ka2,0-2ωacosξ+bsinξ+((ka2,0+ω)/k2)-2k2(acosξ+bsinξ+((ka2,0+ω)/k2))2,u13(x,t)=a2,0+2kacosξ+bsinξ+((ka2,0+ω)/k2),
where a2,0 is an arbitrary real constant.Case 14.
r
=
-
(
(
k
a
2,0
+
ω
)
/
k
2
), a1,0=(k4+k2a2,02+2ka2,0ω+ω2)/k2, a1,1=2ka2,0+2ω, a1,2=-2k2, a2,0=a2,0, a2,1=-2k, b1,1=0, b1,2=0, b2,1=0. By applying these to (3.4), one gets a solution of (1.1):
(3.19)H14(x,t)=k4+(ω+ka2,0)2k2+2ka2,0+2ωacosξ+bsinξ-((ka2,0+ω)/k2)-2k2(acosξ+bsinξ-((ka2,0+ω)/k2))2,u14(x,t)=a2,0-2kacosξ+bsinξ-((ka2,0+ω)/k2),
where a2,0 is an arbitrary real constant.Case 15.
r
=
0, a1,0=0, a1,1=0, a1,2=-2k2, a2,0=-(ω/k), a2,1=0, b1,1=0, b1,2=0, b2,1=2k. By applying these to (3.4), one gets a solution of (1.1):
(3.20)H15(x,t)=-2k2(acosξ+bsinξ)2,u15(x,t)=-ωk+2k(-asinξ+bcosξ)acosξ+bsinξ.Case 16.
r
=
0, a1,0=0, a1,1=0, a1,2=-2k2, a2,0=-(ω/k), a2,1=0, b1,1=0, b1,2=0, b2,1=-2k. By applying these to (3.4), one gets a solution of (1.1):
(3.21)H16(x,t)=-2k2(acosξ+bsinξ)2,u16(x,t)=-ωk-2k(-asinξ+bcosξ)acosξ+bsinξ.
When  ε=0.
Balancing the highest-order derivative term and the nonlinear terms in (2.5), we get balance powers m=2, n=1. By the method described in Section 2 Step 4(b), we get a set of overdetermined nonlinear algebraic equations with respect to a1,0, a1,1, a1,2, a2,0, a2,1, k, ω, and r:
(3.22)-k(2a1,2+a2,12)=0,-3ka2,1(a1,2+2k2)=0,-ωa1,1-ka1,1a2,0-a2,1ka1,0=0,-2ωa1,2-2ka1,1a2,1-2ka1,2a2,0=0,-a2,1ω-ka1,1-a2,1ka2,0=0.
Solving (3.22), we get the following.Case 17.
a
1,0
=
0, a1,1=0, a1,2=-2k2, a2,0=-(ω/k), a2,1=2k. By applying these to (2.7), one gets a solution of (1.1):
(3.23)H17(x,t)=-2k2(kx+ωt+ξ0)2,u17(x,t)=-ωk+2kkx+ωt+ξ0.Case 18.
a
1,0
=
0, a1,1=0, a1,2=-2k2, a2,0=-(ω/k), a2,1=-2k. By applying these to (2.7), one gets a solution of (1.1):
(3.24)H18(x,t)=-2k2(kx+ωt+ξ0)2,u18(x,t)=-ωk+2kkx+ωt+ξ0.
The periodic wave solutions of triangle function typesH9, u9 (3.14) to H16, u16 (3.21) and the solitary wave solutions of rational function form H17, u17 (3.23), H18, u18 (3.24) have not been appeared in [10]. The periodic wave solutions H13,u13 (3.18) and H14, u14 (3.19) have the entirely new form and proposed firstly in this paper. The periodic wave solutions of triangle function types H9, u9 (3.14) to H12, u12 (3.17) have more general form than those obtained in literatures [8, 16]. While taking a=1, b=0 (or a=0, b=1, resp.) in some of these solutions, we will obtain all the periodic wave solutions in triangle function form that have been obtained in literatures [8, 16, 17].
We next deal with the variant Boussinesq equation (1.2) in a similar way to (1.1).
Using transformation (3.1), we reduce (1.2) to a system of nonlinear ODEs:
(3.25)ωH′(ξ)+ku′+(Hu)′(ξ)-αk3u′′′(ξ)=0,ωu′(ξ)+kH′(ξ)+ku(ξ)u′(ξ)-3αk3u′′′(ξ)=0.
Firstly we assume that the solutions of (3.25) are of the form (3.3) with f and g satisfying (2.1), (2.2), and ε=±1. We can get the balancing powers m=2, n=2. So we have
(3.26)H(ξ)=a1,0+a1,1f(ξ)+a1,2f(ξ)2+b1,1g(ξ)+b1,2f(ξ)g(ξ),u(ξ)=a2,0+a2,1f(ξ)+a2,2f(ξ)2+b2,1g(ξ)+b2,2f(ξ)g(ξ),
where ξ=kx+ωt+ξ0, and a1,0, a1,1, a1,2, a2,0, a2,1, a2,2, b1,1, b1,2, b2,1, b2,2 are constants to be determined later.
Substituting (3.26) with (2.1) and (2.2) into (3.25) and using the Maple yield
(3.27)4kb2,2(18γωk-a2,2)=0,2k(36γωa2,2k-b2,22-a2,22)=0,4k(6γb2,2k2-a1,2b2,2-a2,2b1,2)=0,4k(6γa2,2k2-b2,2b1,2-a1,2a2,2)=0,-3k(30kγωa2,2r-6kγωa2,1+b2,2b2,1-b2,22r+a2,1a2,2)=0,-k(-18kγωb2,1+180kγωb2,2r+3a2,2b2,1+3a2,1b2,2-7a2,2b2,2r)=0,-ka1,1-kb2,1b2,2-ka2,0a2,1+3γωk2a2,1-ωa2,1+kb2,12r=0,-2k2γa2,1+10k2γa2,2r+a1,1a2,2-2b1,2rb2,2+a1,2a2,1+b2,1b1,2+b2,2b1,1=0,-ka1,0a2,1+γk3a2,1-kb1,2b2,1-kb1,1b2,2-ka1,1a2,0-ωa1,1-ka2,1+2kb1,1rb2,1=0,-k(60k2γb2,2r-6k2γb2,1-7b1,2ra2,2+3a1,2b2,1+3a2,1b1,2-7a1,2b2,2r+3a2,2b1,1+3a1,1b2,260k2γb2,2r)=0,-2ka1,2-2ωa2,2-2ka2,0a2,2-18γωk2a2,1r+4kb2,1b2,2r-kb2,22-kb2,12+24γωk2a2,2-ka2,12=0,-ka2,0b2,2+kb1,1r-kb1,2+3γωk2b2,2+ωb2,1r-ka2,1b2,1-ωb2,2-3γωk2b2,1r+ka2,0b2,1r=0,-2kb1,2+60γωk2b2,2+5ka2,1b2,2r-2ka2,1b2,1-2ka2,0b2,2-36γωk2b2,1r+90γωk2b2,2r2-3ka2,2b2,2-2ωb2,2+5ka2,2b2,1r=0,-2ka1,2a2,0-2kb1,1b2,1-2kb1,2b2,2+4kb1,2rb2,1-2ka1,0a2,2+8γk3a2,2-2ωa1,2+4kb1,1rb2,2-6γk3a2,1r-2ka1,1a2,1-2ka2,2=0,-kb1,2a2,0+ka1,0b2,1r-ka1,1b2,1-kb1,1a2,1+kb2,1r+γk3b2,2-ωb1,2+kb1,1ra2,0-kb2,2-ka1,0b2,2+ωb1,1r-γk3b2,1r=0,12γωk2b2,1+3ka2,0b2,2r+3ka2,1b2,1r+9γωk2b2,1r2-2ka2,1b2,2-45γωk2b2,2r-ωb2,1-kb1,1+3kb1,2r-ka2,0b2,1-2ka2,2b2,1+3ωb2,2r=0,5kb1,2ra2,1-2ka1,0b2,2-2kb2,2-3kb1,2a2,2-2ka1,1b2,1-2ωb1,2-12γk3b2,1r+5ka1,1b2,2r+5kb1,1ra2,2+30γk3b2,2r2-2kb1,1a2,1+20γk3b2,2+5ka1,2b2,1r-2kb1,2a2,0-3ka1,2b2,2=0,-ka1,0b2,1-kb1,1a2,0+3kb1,2ra2,0-2ka1,2b2,1-kb2,1-2kb1,2a2,1+3ωb1,2r-2kb1,1a2,2+4γk3b2,1+3kb1,1ra2,1-ωb1,1+3ka1,1b2,1r-15γk3b2,2r+3ka1,0b2,2r-2ka1,1b2,2+3kb2,2r+3γk3b2,1r2=0.
With the aid of the computer program Maple 12, make use of the Maple software package PDESolver developed by the authors, which is based on the Wu-elimination method [26]; solving (3.27), one gets the following nontrivial solutions.
Setξ=kx+ωt+ξ0, and k(≠0), ω, r are real constants.
Forε=1, there are 7 solutions.Case 1.
ε
=
1, r=r, a1,0=(-36ω2+18αω2k2+k2)/36ω2, a1,1=-3k2αr, a1,2=3αk2, a2,0=(-k2+18αω2k2-6ω2)/6ωk, a2,1=-18kαωr, a2,2=18αωk, b1,1=0, b1,2=3αk2, b2,1=0, b2,2=18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.28)H1(x,t)=-36ω2+18αω2k2+k236ω2-3k2αracoshξ+bsinhξ+r+3αk2(1+asinhξ+bcoshξ)(acoshξ+bsinhξ+r)2,u1(x,t)=-k2+18αω2k2-6ω26ωk-18kαωracoshξ+bsinhξ+r+18αωk(1+asinhξ+bcoshξ)(acoshξ+bsinhξ+r)2.Case 2.
ε
=
1,r=r, a1,0=(-36ω2+18αω2k2+k2)/36ω2, a1,1=-3k2αr, a1,2=3αk2, a2,0=(-k2+18γω2k2-6ω2)/6ωk, a2,1=-18kαωr, a2,2=18αωk, b1,1=0, b1,2=-3αk2, b2,1=0, b2,2=-18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.29)H2(x,t)=-36ω2+18αω2k2+k236ω2-3k2αracoshξ+bsinhξ+r-3αk2(asinhξ+bcoshξ-1)(acoshξ+bsinhξ+r)2,u2(x,t)=-k2+18αω2k2-6ω26ωk-18kαωracoshξ+bsinhξ+r-18αωk(asinhξ+bcoshξ-1)(acoshξ+bsinhξ+r)2.Case 3.
ε
=
1, r=0, a1,0=(-36ω2+18αω2k2+k2)/36ω2, a1,1=0, a1,2=3αk2, a2,0=(-k2+18αω2k2-6ω2)/6ωk, a2,1=0, a2,2=18αωk, b1,1=0, b1,2=3αk2, b2,1=0, b2,2=18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.30)H3(x,t)=-36ω2+18αω2k2+k236ω2+3αk2(acoshξ+bsinhξ)2+3αk2(asinhξ+bcoshξ)(acoshξ+bsinhξ)2,u3(x,t)=-k2+18αω2k2-6ω26ωk+18αωk(acoshξ+bsinhξ)2+18αωk(asinhξ+bcoshξ)(acoshξ+bsinhξ)2.Case 4.
ε
=
1, r=0, a1,0=(-36ω2+18αω2k2+k2)/36ω2, a1,1=0, a1,2=3αk2, a2,0=(-k2+18αω2k2-6ω2)/6ωk, a2,1=0, a2,2=18αωk, b1,1=0, b1,2=-3αk2, b2,1=0, b2,2=-18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.31)H4(x,t)=-36ω2+18αω2k2+k236ω2+3αk2(acoshξ+bsinhξ)2-3αk2(asinhξ+bcoshξ)(acoshξ+bsinhξ)2,u4(x,t)=-k2+18αω2k2-6ω26ωk+18αωk(acoshξ+bsinhξ)2-18αωk(asinhξ+bcoshξ)(acoshξ+bsinhξ)2.Case 5.
ε
=
1, r=0, a1,0=(-36ω2+72αω2k2+k2)/36ω2, a1,1=0, a1,2=6αk2, a2,0=(-k2-6ω2+72αω2k2)/6ωk, a2,1=0, a2,2=36αωk, b1,1=0, b1,2=0, b2,1=0, b2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.32)H5(x,t)=-36ω2+72αω2k2+k236ω2+6αk2(acoshξ+bsinhξ)2,u5(x,t)=-k2-6ω2+72αω2k26ωk+36αωk(acoshξ+bsinhξ)2.Case 6.
ε
=
1, r=1, a1,0=(-36ω2+18αω2k2+k2)/36ω2, a1,1=-6αk2, a1,2=6αk2, a2,0=(-k2+18αω2k2-6ω2)/6ωk, a2,1=-36αωk, a2,2=36αωk, b1,1=0, b1,2=0, b2,1=0, b2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.33)H6(x,t)=-36ω2+18αω2k2+k236ω2-6αk2(acoshξ+bsinhξ+1)+6αk2(acoshξ+bsinhξ+1)2,u6(x,t)=-k2+18αω2k2-6ω26ωk-36αωk(acoshξ+bsinhξ+1)+36αωk(acoshξ+bsinhξ+1)2.Case 7.
ε
=
1, r=-1, a1,0=(-36ω2+18αω2k2+k2)/36ω2, a1,1=6αk2, a1,2=6αk2, a2,0=(-k2+18αω2k2-6ω2)/6ωk, a2,1=36αωk, a2,2=36αωk, b1,1=0, b1,2=0, b2,1=0, b2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.34)H7(x,t)=-36ω2+18αω2k2+k236ω2+6αk2(acoshξ+bsinhξ-1)+6αk2(acoshξ+bsinhξ-1)2,u7(x,t)=-k2+18αω2k2-6ω26ωk+36αωk(acoshξ+bsinhξ-1)+36αωk(acoshξ+bsinhξ-1)2.
While takinga=1,b=0 in solutions H5,u5 (3.32), we can get the soliton solutions in sech2ξ,sech2ξ form obtained in [10] by homogeneous balance method. While taking a=1,b=0 in solutions H5,u5 (3.32), we can get the singular traveling soliton-like solutions in csch2ξ,csch2ξ form. It should be emphasized that the solutions H1,u1 (3.28) and H2,u2 (3.29) are entirely new solutions that have not been proposed in previous literatures. We also should point out that H3,u3 (3.30), H4,u4 (3.31), H7,u7 (3.33), H8,u8 (3.34) include all soliton solutions in hyperbolic function form obtained in [8, 16]. Because there are some parameters that can take different values, these solutions possess more new information.
For  ε=-1.Case 8.
ε
=
-
1, r=r, a1,0=-((36ω2+18αω2k2-k2)/36ω2), a1,1=3k2αr, a1,2=3αk2, a2,0=-(k2+18αω2k2+6ω2)/6ωk, a2,1=18kαωr, a2,2=18αωk, b1,1=0, b1,2=3αk2, b2,1=0, b2,2=18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.35)H8(x,t)=-36ω2+18αω2k2-k236ω2+3k2αr(acosξ+bsinξ+r)+3αk2(acosξ+bsinξ+r)2+3αk2(-asinξ+bcosξ)(acosξ+bsinξ+r)2,u8(x,t)=-k2+18αω2k2+6ω26ωk+18kαωr(acosξ+bsinξ+r)+18αωk(acosξ+bsinξ+r)2+18αωk(-asinξ+bcosξ)(acosξ+bsinξ+r)2.Case 9.
ε
=
-
1, r=r, a1,0=-((36ω2+18αω2k2-k2)/36ω2), a1,1=3k2αr, a1,2=3αk2, a2,0=-((k2+18αω2k2+6ω2)/6ωk), a2,1=18kαωr, a2,2=18αωk, b1,1=0, b1,2=-3αk2, b2,1=0, b2,2=-18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.1):
(3.36)H9(x,t)=-36ω2+18αω2k2-k236ω2+3k2αr(acosξ+bsinξ+r)+3αk2(acosξ+bsinξ+r)2-3αk2(-asinξ+bcosξ)(acosξ+bsinξ+r)2,u9(x,t)=-k2+18αω2k2+6ω26ωk+18kαωr(acosξ+bsinξ+r)+18αωk(acosξ+bsinξ+r)2-18αωk(-asinξ+bcosξ)(acosξ+bsinξ+r)2.Case 10.
ε
=
-
1, r=0, a1,0=-((36ω2+18αω2k2-k2)/36ω2), a1,1=0, a1,2=3αk2, a2,0=-((k2+18αω2k2+6ω2)/6ωk), a2,1=0, a2,2=18αωk, b1,1=0, b1,2=3αk2, b2,1=0, b2,2=18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.37)H10(x,t)=-36ω2+18αω2k2-k236ω2+3αk2(acosξ+bsinξ)2+3αk2(-asinξ+bcosξ)(acosξ+bsinξ)2,u10(x,t)=-k2+18αω2k2+6ω26ωk+18αωk(acosξ+bsinξ)2+18αωk(-asinξ+bcosξ)(acosξ+bsinξ)2.Case 11.
ε
=
-
1, r=0, a1,0=-((36ω2+18αω2k2-k2)/36ω2), a1,1=0, a1,2=3αk2, a2,0=-((k2+18αω2k2+6ω2)/6ωk), a2,1=0, a2,2=18αωk, b1,1=0, b1,2=-3αk2, b2,1=0, b2,2=-18αωk. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.38)H11(x,t)=-36ω2+18αω2k2-k236ω2+3αk2(acosξ+bsinξ)2-3αk2(-asinξ+bcosξ)(acosξ+bsinξ)2,u11(x,t)=-k2+18αω2k2+6ω26ωk+18αωk(acosξ+bsinξ)2-18αωk(-asinξ+bcosξ)(acosξ+bsinξ)2.Case 12.
ε
=
-
1, r=0, a1,0=-((36ω2+72αω2k2-k2)/36ω2), a1,1=0, a1,2=6αk2, a2,0=-((k2+72αω2k2+6ω2)/6ωk), a2,1=0, a2,2=36αωk, b1,1=0, b1,2=0, b2,1=0, b2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2):
(3.39)H12(x,t)=-36ω2+72αω2k2-k236ω2+6αk2(acosξ+bsinξ)2,u12(x,t)=-k2+72αω2k2+6ω26ωk+36αωk(acosξ+bsinξ)2.
The periodic wave solutions of triangle functionH8,u8 (3.35) and H9,u9 (3.36) are entirely new periodic wave solutions that have not been obtained in other literatures. The periodic wave solutions of triangle function H10,u10 (3.37) to H12,u12 have more general form than those obtained in literatures [8, 16]. All of the above periodic wave solutions have not been obtained in literature [10].
ForR=0, we can deduce that the solution of (3.25) is of the form
(3.40)H(ξ)=a1,0+a1,1g(ξ)+a1,2g(ξ)2,u(ξ)=a2,0+a2,1g(ξ)+a2,2g(ξ)2,
where ξ=kx+ωt+ξ0 and a1,0, a1,1, a1,2, a2,0, a2,1, a2,2 are constants to be determined later.
Substituting (3.40) along with the condition g'(ξ)=-g2(ξ) into (3.25) yields a set of nonlinear algebraic equations:
(3.41)2ka2,2(36αωk-a2,2)=0,4ka2,2(6αk2-a1,2)=0,3ka2,1(6αωk-a2,2)=0,-ωa2,1-ka2,0a2,1-ka1,1=0,3k(2αa2,1k2-a1,2a2,1-a2,2a1,1)=0,-ωa1,1-ka2,1-ka1,1a2,0-ka1,0a2,1=0,-2ωa2,2-ka2,12-2ka2,0a2,2-2ka1,2=0,-2ωa1,2-2ka2,2-2ka1,1a2,1-2ka1,2a2,0-2ka1,0a2,2=0.
With the aid of the computer program Maple 12, make use of the Maple software package PDESolver by the authors, which is based on the Wu-elimination method [26]; solving (3.41), one gets the following nontrivial solutions.Case 13.
ε
=
0, r=r, a1,0=(-36ω2+k2)/36ω2, a1,1=0, a1,2=6αk2, a2,0=-((k2+6ω2)/6ωk), a2,1=0, a2,2=36αωk. By applying these to (3.40), one gets a solution of the PDEs (1.2):
(3.42)H13(x,t)=-36ω2+k236ω2+6αk2(kx+ωt+ξ0)2,u13(x,t)=-k2+6ω26ωk+36αωk(kx+ωt+ξ0)2.
The solitary wave solutions H13, u13 are in rational function form and have not been obtained in literatures [8, 10, 16].


## 4. Summary and Conclusions

In summary, we propose an effective and a readily computerizable method to uniformly construct a series of traveling wave solutions for nonlinear evolution partial differential equations in mathematical physic, and develop a symbolic computational software package “PDESolver.” As an application of the proposed method, we deal with two types of variant Boussinesq equations. Some more general forms of exact explicit traveling wave solutions for two variant Boussinesq equations are obtained. We not only obtain all known exact solitary wave solutions, periodic wave solutions, and singular traveling wave solutions but also find abundant new exact solitary wave solutions, singular traveling wave solutions, and periodic traveling wave solutions of triangle function. Since there are some parameters that can take different values, these solutions admit abundant new information. Because these two variant Boussinesq equations are integrable, all solitary wave solutions are stable. All known results in [8, 10, 16, 17] are improved and completed. Although the procedure has many advantages than other methods such as the tanh method [6], the extended tanh method, the tanh-coth method [7, 15], the extended tanh-function method [27], the extended improved tanh-function method [21], and the recent method proposed by Wazwaz [23, 24], there exist some limitations in our procedure. We cannot obtain nontraveling wave solution by using the method presented in this paper. Comparing with the Jacobi elliptic function method, the F-expansion method, and the improved F-expansion method, our method failed to obtain elliptic function doubly periodic wave solutions.



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*Source: 716719-2012-10-11.xml*