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

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.

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 equations𝐻_𝑡+(𝐻𝑢)_𝑥+𝑢_𝑥𝑥𝑥𝑢=0,_𝑡+𝐻_𝑥+𝑢𝑢_𝑥𝐻=0,_𝑡+𝑢_𝑥+(𝐻𝑢)_𝑥−𝛼𝑢_𝑥𝑥𝑥𝑢=0,_𝑡+𝐻_𝑥+𝑢𝑢_𝑥−3𝛼𝑢_𝑥𝑥𝑡=0, (1.1)(1.2) 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 Section 2, 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 𝑓^′(𝜉)(𝜉)𝑔(𝜉),𝑔=−𝑓_′(𝜉)=𝜀−𝑟𝜀𝑓(𝜉)−𝑔²(𝜉), (2.1) where 𝜀=±1 or 0,𝑟 is a constant. When 𝜀≠0, we can obtain the following first integral as given: 𝑔²(𝜉)=𝜀−2𝑟𝜀𝑓(𝜉)+𝐶𝑓²(𝜉). (2.2)

Step 1. Consider a system of 𝑁 nonlinear evolution equations (𝑁≥1) with 𝑛 independent variables 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) and 𝑚 dependent variables 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑚), given by 𝑃_𝑖𝑢,𝜕𝑢,𝜕²𝑢,…,𝜕^𝑟𝑢=0,𝑖=1,2,…,𝑁, (2.3) where 𝑃_𝑖, 𝑖=1,2,…,𝑁, are in general nonlinear functions of their arguments, 𝜕^𝑗𝑢 denotes the coordinates with components 𝜕^𝑗𝑢/(𝜕𝑥_𝑖1𝜕𝑥_𝑖2⋯𝜕𝑥_𝑖𝑗)=𝑢_{𝑖1𝑖2⋯𝑖𝑗},𝑖_𝑗=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: 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑘₁𝑥₁+𝑘₂𝑥₂+⋅⋅⋅+𝑘_𝑛𝑥_𝑛+𝜉₀,^𝑛_𝑖=1𝑘²_𝑖≠0, (2.4) where 𝑘_𝑖 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 𝑄_𝑖𝑢,𝑢^′,𝑢^′′,…𝑢^𝑘=0,𝑖=1,2,…,𝑁, (2.5) 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)When 𝜀=±1 in (2.1), (2.2), 𝑢_𝑙(𝑥,𝑡)=𝑢_𝑙(𝜉)=^𝑙𝑛_𝑖=0𝑎_𝑙,𝑖𝑓(𝜉)^𝑖+^𝑙𝑛_𝑗=1𝑏_𝑙,𝑗𝑓(𝜉)^𝑗−1𝑔(𝜉),𝑙=1,2,…,𝑚, (2.6) where the coefficients 𝑎_𝑙,𝑖(𝑖=0,1,2…,𝑙_𝑛) and 𝑏_𝑙,𝑗(𝑗=1,2,…,𝑙_𝑛) are constants to be determined latter.(b)When 𝜀=0 in (2.1), 𝑢_𝑙(𝑥,𝑡)=^𝑙𝑛_𝑖=0𝑎_𝑙,𝑖𝑔(𝜉)^𝑖,𝑙=1,2,…,𝑚, (2.7) where 𝑔^′(𝜉)=−𝑔²(𝜉) and the coefficients 𝑎_𝑙,𝑖(𝑖=0,1,2…,𝑙_𝑛) are constants to be determined.

Step 3. Balancing the highest-order derivative term and the nonlinear terms in (2.5), we get balance powers 𝑙_𝑛 (usually positive integer). If some one of 𝑙_𝑛 is a fraction or a negative integer, say 𝑠_𝑛 is negative, we make the following transformation:𝑢_𝑠(𝜉)=𝑣^𝑠𝑛_𝑠, (2.8) then return to determine balance power 𝑛 again.

Step 4. (a) When 𝜀≠0, substituting (2.6) along with the conditions (2.1) and (2.2) into (2.5).
(b) When 𝜀=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 𝑓_𝑖(𝑖=0,1,…;) and 𝑓_𝑗𝑔(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 𝑘_𝑖, 𝑖=1,2,…,𝑛, 𝑎_𝑙,𝑖(𝑖=0,1,2…,𝑙_𝑛), 𝑏_𝑙,𝑗(𝑗=1,2,…,𝑙_𝑛), 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 𝑘_𝑖, 𝑖=1,2,…,𝑛, 𝑎_𝑙,𝑖(𝑖=0,1,2…,𝑙_𝑛), 𝑏_𝑙,𝑗(𝑗=1,2,…,𝑙_𝑛), 𝑟, 𝐶.

Step 5. We know that the coupled Riccati equations (2.1) admit the following special solutions.(a)When 𝜀=1, 𝑓(𝜉)=1(𝜉)=𝑎cosh𝜉+𝑏sinh𝜉+𝑟,𝑔𝑎sinh𝜉+𝑏cosh𝜉,𝑎cosh𝜉+𝑏sinh𝜉+𝑟 (2.9) and then 𝑔²(𝜉)=1−2𝑟𝑓(𝜉)+(𝑏²−𝑎²+𝑟²)𝑓²(𝜉).(b)When 𝜀=−1, 𝑓(𝜉)=1(𝜉)=𝑎cos𝜉+𝑏sin𝜉+𝑟,𝑔𝑏cos𝜉−𝑎sin𝜉,𝑎cos𝜉+𝑏sin𝜉+𝑟 (2.10) and then 𝑔²(𝜉)=−1+2𝑟𝑓(𝜉)+(𝑏²+𝑎²−𝑟²)𝑓²(𝜉).(c)When 𝜀=0, 𝑓(𝜉)=𝐶𝜉(𝜉)=1,𝑔𝜉, (2.11) 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 tanh 𝜉, 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 𝑢(𝑥,𝑡)=𝑢(𝜉),𝐻(𝑥,𝑡)=𝐻(𝜉),𝜉=𝑘𝑥+𝜔𝑡+𝜉₀, (3.1) and thus (1.1) becomes 𝜔𝐻^′(𝜉)+𝑘(𝐻𝑢)^′(𝜉)+𝑘³𝑢^′′′(𝜉)=0,𝜔𝑢^′(𝜉)+𝑘𝐻^′(𝜉)+𝑘𝑢(𝜉)𝑢^′(𝜉)=0. (3.2)

Firstly we assume that the solutions of (3.1) are of the form 𝐻(𝑥,𝑡)=𝐻(𝜉)=^𝑚_𝑖=0𝑎_1,𝑖𝑓(𝜉)^𝑖+^𝑚_𝑗=1𝑏_1,𝑗𝑓(𝜉)^𝑗−1𝑢()(𝜉)=𝑔,𝑥,𝑡=𝑢^𝑛_𝑖=0𝑎_1,𝑖𝑓(𝜉)^𝑖+^𝑛_𝑗=1𝑏_1,𝑗𝑓(𝜉)^𝑗−1𝑔, (3.3) where 𝑎_1,𝑖, 𝑏_1,𝑗, 𝑖=0,1,2,…,𝑚, 𝑗=1,2,…,𝑚, and 𝑎_2,𝑖, 𝑏_2,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 𝐻(𝜉)=𝑎_1,0+𝑎_1,1𝑓(𝜉)+𝑎_1,2𝑓²(𝜉)+𝑏_1,1𝑔(𝜉)+𝑏_1,2𝑓(𝜉)𝑔(𝜉),𝑢(𝜉)=𝑎_2,0+𝑎_2,1𝑓(𝜉)+𝑏_2,1𝑔(𝜉), (3.4) where 𝑎_1,0, 𝑎_1,1, 𝑎_1,2, 𝑏_1,1, 𝑏_1,2, 𝑎_2,0, 𝑎_2,1, 𝑏_2,1 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 𝑓^𝑖(𝜉)𝑔^𝑗(𝜉). Setting the coefficients of these terms 𝑓^𝑖(𝜉)𝑔^𝑗(𝜉) to zero yields a set of overdetermined nonlinear algebraic equations with respect to 𝑎_1,0, 𝑎_1,1, 𝑎_1,2, 𝑏_1,1, 𝑏_1,2, 𝑎_2,0, 𝑎_2,1, 𝑏_2,1, 𝑘, 𝜔, and 𝑟: −2𝑘𝑏_1,2+𝑎_2,1𝑏_2,1𝑏=0,−𝑘²_2,1+2𝑎_1,2+𝑎²_2,1=0,−3𝑘2𝑘²𝑎_2,1+𝑏_1,2𝑏_2,1+𝑎_1,2𝑎_2,1𝑏=0,−3𝑘_1,2𝑎_2,1+𝑎_1,2𝑏_2,1+2𝑘²𝑏_2,1=0,𝑘𝑏²_2,1𝜀𝑟−𝑘𝑎_2,0𝑎_2,1−𝜔𝑎_2,1−𝑘𝑎_1,1𝜀=0,−𝑘𝑎_2,1𝑏_2,1+𝑘𝑎_2,0𝑏_2,1𝑟+𝑘𝑏_1,1𝑟−𝑘𝑏_1,2+𝜔𝑏_2,1𝑟=0,3𝑘𝑏_1,2𝜀𝑟+3𝑘𝑎_2,1𝑏_2,1𝜀𝑟−𝑘𝑏_1,1−𝑘𝑎_2,0𝑏_2,1−𝜔𝑏_2,1=0,−𝑘𝑎_1,1𝑎_2,0−𝑘𝑏_1,2𝜀𝑏_2,1−𝑘³𝑎_2,1𝜀−𝑘𝑎_1,0𝑎_2,1−𝜔𝑎_1,1+2𝑘𝑏_1,1𝜀𝑟𝑏_2,1=0,−2𝑘𝑏_1,1𝑏_2,1−2𝑘𝑎_1,1𝑎_2,1−2𝑘𝑎_1,2𝑎_2,0+6𝑘³𝑎_2,1𝜀𝑟−2𝜔𝑎_1,2+4𝑘𝑏_1,2𝜀𝑟𝑏_2,1=0,5𝑘𝑏_1,2𝜀𝑟𝑎_2,1+5𝑘𝑎_1,2𝑏_2,1𝜀𝑟−2𝜔𝑏_1,2+12𝑘³𝑏_2,1𝜀𝑟−2𝑘𝑏_1,2𝑎_2,0−2𝑘𝑏_1,1𝑎_2,1−2𝑘𝑎_1,1𝑏_2,1𝜀𝑘=0,₃𝑏_2,1𝜀𝑟−𝜔𝑏_1,2+𝑘𝑎_1,0𝑏_2,1𝑟+𝑘𝑏_1,1𝑟𝑎_2,0+𝜔𝑏_1,1𝑟−𝑘𝑏_1,2𝑎_2,0−𝑘𝑎_1,1𝑏_2,1−𝑘𝑏_1,1𝑎_2,1=0,3𝑘𝑎_1,1𝑏_2,1𝜀𝑟−𝑘𝑎_1,0𝑏_2,1−3𝑘³𝑏_2,1𝜀²𝑟²−𝑘𝑏_1,1𝑎_2,0−2𝑘𝑏_1,2𝜀𝑎_2,1−𝜔𝑏_1,1+3𝜔𝑏_1,2𝜀𝑟−4𝑘³𝑏_2,1𝜀+3𝑘𝑏_1,2𝜀𝑟𝑎_2,0+3𝑘𝑏_1,1𝜀𝑟𝑎_2,1−2𝑘𝑎_1,2𝑏_2,1𝜀=0. (3.5)

To get a nontrivial solution of (1.1), we assume that 𝑘≠0 and 𝑎²_1,1+𝑎²_1,2+𝑏²_1,1+𝑏²_1,2+𝑎²_2,1+𝑏²_2,1≠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.

Set 𝜉=𝑘𝑥+𝜔𝑡+𝜉₀, and 𝑘(≠0), 𝜔, 𝑟 are real constants.

When   𝜀=1.

Case 1. 𝑟=𝑟, 𝑎_1,0=0, 𝑎_1,1=𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=𝑘, 𝑏_1,1=0, 𝑏_1,2=𝑘², 𝑏_2,1=−𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁()=𝑘𝑥,𝑡²𝑟−𝑘𝑎cosh𝜉+𝑏sinh𝜉+𝑟(𝑎cosh𝜉+𝑏²sinh𝜉+𝑟)²+𝑘²(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)²,𝑢₁()𝜔𝑥,𝑡=−𝑘+𝑘−𝑘()𝑎cosh𝜉+𝑏sinh𝜉+𝑟𝑎sinh𝜉+𝑏cosh𝜉().𝑎cosh𝜉+𝑏sinh𝜉+𝑟 (3.6)

Case 2. 𝑟=𝑟, 𝑎_1,0=0, 𝑎_1,1=𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=𝑘, 𝑏_1,1=0, 𝑏_1,2=−𝑘², 𝑏_2,1=𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₂()=𝑘𝑥,𝑡²𝑟−𝑘𝑎cosh𝜉+𝑏sinh𝜉+𝑟(𝑎cosh𝜉+𝑏²sinh𝜉+𝑟)²−𝑘²(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)²,𝑢₂()𝜔𝑥,𝑡=−𝑘+𝑘+𝑘()𝑎cosh𝜉+𝑏sinh𝜉+𝑟𝑎sinh𝜉+𝑏cosh𝜉.𝑎cosh𝜉+𝑏sinh𝜉+𝑟 (3.7)

Case 3. 𝑟=𝑟, 𝑎_1,0=0, 𝑎_1,1=𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=−𝑘, 𝑏_1,1=0, 𝑏_1,2=𝑘², 𝑏_2,1=𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₃()=𝑘𝑥,𝑡²𝑟−𝑘𝑎cosh𝜉+𝑏sinh𝜉+𝑟(𝑎cosh𝜉+𝑏²sinh𝜉+𝑟)²+𝑘²(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)²,𝑢₃()𝜔𝑥,𝑡=−𝑘−𝑘+𝑘()𝑎cosh𝜉+𝑏sinh𝜉+𝑟𝑎sinh𝜉+𝑏cosh𝜉.𝑎cosh𝜉+𝑏sinh𝜉+𝑟 (3.8)

Case 4. 𝑟=𝑟, 𝑎_1,0=0, 𝑎_1,1=𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=−𝑘, 𝑏_1,1=0, 𝑏_1,2=−𝑘², 𝑏_2,1=−𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₄()=𝑘𝑥,𝑡²𝑟−𝑘𝑎cosh𝜉+𝑏sinh𝜉+𝑟(𝑎cosh𝜉+𝑏²sinh𝜉+𝑟)²−𝑘²(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)²,𝑢₄()𝜔𝑥,𝑡=−𝑘−𝑘−𝑘()𝑎cosh𝜉+𝑏sinh𝜉+𝑟𝑎sinh𝜉+𝑏cosh𝜉.𝑎cosh𝜉+𝑏sinh𝜉+𝑟 (3.9)

Case 5. 𝑟=(𝜔+𝑘𝑎_2,0)/𝑘², 𝑎_1,0=(−𝑘⁴+𝜔²+2𝜔𝑘𝑎_2,0+𝑘²𝑎²_2,0)/𝑘², 𝑎_1,1=2𝜔+2𝑘𝑎_2,0, 𝑎_1,2=−2𝑘², 𝑎_2,0=𝑎_2,0, 𝑎_2,1=−2𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0. By applying these to (3.4), one gets a solution of (1.1): 𝐻₅(𝑥,𝑡)=−𝑘⁴+𝜔+𝑘𝑎_2,0²𝑘²+2𝜔+2𝑘𝑎_2,0𝑎cosh𝜉+𝑏sinh𝜉+𝜔+𝑘𝑎_2,0/𝑘²−2𝑘²𝑎cosh𝜉+𝑏sinh𝜉+𝜔+𝑘𝑎_2,0/𝑘²²,𝑢₅(𝑥,𝑡)=𝑎_2,0−2𝑘𝑎cosh𝜉+𝑏sinh𝜉+𝜔+𝑘𝑎_2,0/𝑘², (3.10) where 𝑎_2,0 is an arbitrary real constant.

Case 6. 𝑟=−((𝜔+𝑘𝑎_2,0)/𝑘²), 𝑎_1,0=(−𝑘⁴+𝜔²+2𝜔𝑘𝑎_2,0+𝑘²𝑎²_2,0)/𝑘², 𝑎_1,1=−2𝜔−2𝑘𝑎_2,0, 𝑎_1,2=−2𝑘², 𝑎_2,0=𝑎_2,0, 𝑎_2,1=2𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0. By applying these to (3.4), one gets a solution of (1.1): 𝐻₆(𝑥,𝑡)=−𝑘⁴+𝜔+𝑘𝑎_2,0²𝑘²−2𝜔+𝑘𝑎_2,0𝑎cosh𝜉+𝑏sinh𝜉−𝜔+𝑘𝑎_2,0/𝑘²−2𝑘²𝑎cosh𝜉+𝑏sinh𝜉−𝜔+𝑘𝑎_2,0/𝑘²²,𝑢₆(𝑥,𝑡)=𝑎_2,0+2𝑘𝑎cosh𝜉+𝑏sinh𝜉−𝜔+𝑘𝑎_2,0/𝑘². (3.11)

Case 7. 𝑟=0, 𝑎_1,0=0, 𝑎_1,1=0, 𝑎_1,2=−2𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=0, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=2𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₇(𝑥,𝑡)=−2𝑘²(𝑎cosh𝜉+𝑏sinh𝜉)²,𝑢₇()𝜔𝑥,𝑡=−𝑘+2𝑘(𝑎sinh𝜉+𝑏cosh𝜉).𝑎cosh𝜉+𝑏sinh𝜉 (3.12)

Case 8. 𝑟=0, 𝑎_1,0=0, 𝑎_1,1=0, 𝑎_1,2=−2𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=0, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=−2𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₈(𝑥,𝑡)=−2𝑘²(𝑎cosh𝜉+𝑏sinh𝜉)²,𝑢₈()𝜔𝑥,𝑡=−𝑘−2𝑘(𝑎sinh𝜉+𝑏cosh𝜉).𝑎cosh𝜉+𝑏sinh𝜉 (3.13)
While taking 𝑎=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, 𝑎_1,0=0, 𝑎_1,1=−𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=𝑘, 𝑏_1,1=0, 𝑏_1,2=𝑘², 𝑏_2,1=−𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₉()𝑘𝑥,𝑡=−²𝑟−𝑘𝑎cos𝜉+𝑏sin𝜉+𝑟(𝑎cos𝜉+𝑏²sin𝜉+𝑟)²+𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)²,𝑢₉()𝜔𝑥,𝑡=−𝑘+𝑘−𝑘()𝑎cos𝜉+𝑏sin𝜉+𝑟−𝑎sin𝜉+𝑏cos𝜉(𝑎cos𝜉+𝑏sin𝜉+𝑟)². (3.14)

Case 10. 𝑟=𝑟, 𝑎_1,0=0, 𝑎_1,1=−𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=𝑘, 𝑏_1,1=0, 𝑏_1,2=−𝑘², 𝑏_2,1=𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁₀()𝑘𝑥,𝑡=−²𝑟−𝑘𝑎cos𝜉+𝑏sin𝜉+𝑟(𝑎cos𝜉+𝑏²sin𝜉+𝑟)²−𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)²,𝑢₁₀()𝜔𝑥,𝑡=−𝑘+𝑘+𝑘()𝑎cos𝜉+𝑏sin𝜉+𝑟−𝑎sin𝜉+𝑏cos𝜉(𝑎cos𝜉+𝑏sin𝜉+𝑟)². (3.15)

Case 11. 𝑟=𝑟, 𝑎_1,0=0, 𝑎_1,1=−𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=−𝑘, 𝑏_1,1=0, 𝑏_1,2=𝑘², 𝑏_2,1=𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁₁()𝑘𝑥,𝑡=−²𝑟−𝑘𝑎cos𝜉+𝑏sin𝜉+𝑟(𝑎cos𝜉+𝑏²sin𝜉+𝑟)²+𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)²,𝑢₁₁()𝜔𝑥,𝑡=−𝑘−𝑘+𝑘()𝑎cos𝜉+𝑏sin𝜉+𝑟−𝑎sin𝜉+𝑏cos𝜉.𝑎cos𝜉+𝑏sin𝜉+𝑟 (3.16)

Case 12. 𝑟=𝑟, 𝑎_1,0=0, 𝑎_1,1=−𝑘²𝑟, 𝑎_1,2=−𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=−𝑘, 𝑏_1,1=0, 𝑏_1,2=−𝑘², 𝑏_2,1=−𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁₂()𝑘𝑥,𝑡=−²𝑟−𝑘𝑎cos𝜉+𝑏sin𝜉+𝑟(𝑎cos𝜉+𝑏²sin𝜉+𝑟)²−𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)²,𝑢₁₂()𝜔𝑥,𝑡=−𝑘−𝑘−𝑘()𝑎cos𝜉+𝑏sin𝜉+𝑟−𝑎sin𝜉+𝑏cos𝜉.𝑎cos𝜉+𝑏sin𝜉+𝑟 (3.17)

Case 13. 𝑟=(𝑘𝑎_2,0+𝜔)/𝑘², 𝑎_1,0=(𝑘⁴+𝑘²𝑎²_2,0+2𝑘𝑎_2,0𝜔+𝜔²)/𝑘², 𝑎_1,1=−2𝑘𝑎_2,0−2𝜔, 𝑎_1,2=−2𝑘², 𝑎_2,0=𝑎_2,0, 𝑎_2,1=2𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁₃()=𝑘𝑥,𝑡⁴+𝜔+𝑘𝑎_2,0²𝑘²+−2𝑘𝑎_2,0−2𝜔𝑎cos𝜉+𝑏sin𝜉+𝑘𝑎_2,0+𝜔/𝑘²−2𝑘²𝑎cos𝜉+𝑏sin𝜉+𝑘𝑎_2,0+𝜔/𝑘²²,𝑢₁₃(𝑥,𝑡)=𝑎_2,0+2𝑘𝑎cos𝜉+𝑏sin𝜉+𝑘𝑎_2,0+𝜔/𝑘², (3.18) where 𝑎_2,0 is an arbitrary real constant.

Case 14. 𝑟=−((𝑘𝑎_2,0+𝜔)/𝑘²), 𝑎_1,0=(𝑘⁴+𝑘²𝑎²_2,0+2𝑘𝑎_2,0𝜔+𝜔²)/𝑘², 𝑎_1,1=2𝑘𝑎_2,0+2𝜔, 𝑎_1,2=−2𝑘², 𝑎_2,0=𝑎_2,0, 𝑎_2,1=−2𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁₄()=𝑘𝑥,𝑡⁴+𝜔+𝑘𝑎_2,0²𝑘²+2𝑘𝑎_2,0+2𝜔𝑎cos𝜉+𝑏sin𝜉−𝑘𝑎_2,0+𝜔/𝑘²−2𝑘²𝑎cos𝜉+𝑏sin𝜉−𝑘𝑎_2,0+𝜔/𝑘²²,𝑢₁₄(𝑥,𝑡)=𝑎_2,0−2𝑘𝑎cos𝜉+𝑏sin𝜉−𝑘𝑎_2,0+𝜔/𝑘², (3.19) where 𝑎_2,0 is an arbitrary real constant.

Case 15. 𝑟=0, 𝑎_1,0=0, 𝑎_1,1=0, 𝑎_1,2=−2𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=0, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=2𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁₅(𝑥,𝑡)=−2𝑘²(𝑎cos𝜉+𝑏sin𝜉)²,𝑢₁₅()𝜔𝑥,𝑡=−𝑘+2𝑘(−𝑎sin𝜉+𝑏cos𝜉).𝑎cos𝜉+𝑏sin𝜉 (3.20)

Case 16. 𝑟=0, 𝑎_1,0=0, 𝑎_1,1=0, 𝑎_1,2=−2𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=0, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=−2𝑘. By applying these to (3.4), one gets a solution of (1.1): 𝐻₁₆(𝑥,𝑡)=−2𝑘²(𝑎cos𝜉+𝑏sin𝜉)²,𝑢₁₆()𝜔𝑥,𝑡=−𝑘−2𝑘(−𝑎sin𝜉+𝑏cos𝜉).𝑎cos𝜉+𝑏sin𝜉 (3.21)
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 𝑎_1,0, 𝑎_1,1, 𝑎_1,2, 𝑎_2,0, 𝑎_2,1, 𝑘, 𝜔, and 𝑟: −𝑘2𝑎_1,2+𝑎²_2,1=0,−3𝑘𝑎_2,1𝑎_1,2+2𝑘²=0,−𝜔𝑎_1,1−𝑘𝑎_1,1𝑎_2,0−𝑎_2,1𝑘𝑎_1,0=0,−2𝜔𝑎_1,2−2𝑘𝑎_1,1𝑎_2,1−2𝑘𝑎_1,2𝑎_2,0=0,−𝑎_2,1𝜔−𝑘𝑎_1,1−𝑎_2,1𝑘𝑎_2,0=0. (3.22)
Solving (3.22), we get the following.

Case 17. 𝑎_1,0=0, 𝑎_1,1=0, 𝑎_1,2=−2𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=2𝑘. By applying these to (2.7), one gets a solution of (1.1): 𝐻₁₇(𝑥,𝑡)=−2𝑘²𝑘𝑥+𝜔𝑡+𝜉₀²,𝑢₁₇()𝜔𝑥,𝑡=−𝑘+2𝑘𝑘𝑥+𝜔𝑡+𝜉₀. (3.23)

Case 18. 𝑎_1,0=0, 𝑎_1,1=0, 𝑎_1,2=−2𝑘², 𝑎_2,0=−(𝜔/𝑘), 𝑎_2,1=−2𝑘. By applying these to (2.7), one gets a solution of (1.1): 𝐻₁₈(𝑥,𝑡)=−2𝑘²𝑘𝑥+𝜔𝑡+𝜉₀²,𝑢₁₈()𝜔𝑥,𝑡=−𝑘+2𝑘𝑘𝑥+𝜔𝑡+𝜉₀. (3.24)
The periodic wave solutions of triangle function types 𝐻₉, 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: 𝜔𝐻^′(𝜉)+𝑘𝑢^′+(𝐻𝑢)^′(𝜉)−𝛼𝑘³𝑢^′′′(𝜉)=0,𝜔𝑢^′(𝜉)+𝑘𝐻^′(𝜉)+𝑘𝑢(𝜉)𝑢^′(𝜉)−3𝛼𝑘³𝑢^′′′(𝜉)=0. (3.25)
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 𝐻(𝜉)=𝑎_1,0+𝑎_1,1𝑓(𝜉)+𝑎_1,2𝑓(𝜉)²+𝑏_1,1𝑔(𝜉)+𝑏_1,2𝑓(𝜉)𝑔(𝜉),𝑢(𝜉)=𝑎_2,0+𝑎_2,1𝑓(𝜉)+𝑎_2,2𝑓(𝜉)²+𝑏_2,1𝑔(𝜉)+𝑏_2,2𝑓(𝜉)𝑔(𝜉), (3.26) where 𝜉=𝑘𝑥+𝜔𝑡+𝜉₀, and 𝑎_1,0, 𝑎_1,1, 𝑎_1,2, 𝑎_2,0, 𝑎_2,1, 𝑎_2,2, 𝑏_1,1, 𝑏_1,2, 𝑏_2,1, 𝑏_2,2 are constants to be determined later.
Substituting (3.26) with (2.1) and (2.2) into (3.25) and using the Maple yield 4𝑘𝑏_2,218𝛾𝜔𝑘−𝑎_2,2=0,2𝑘36𝛾𝜔𝑎_2,2𝑘−𝑏²_2,2−𝑎²_2,2=0,4𝑘6𝛾𝑏_2,2𝑘²−𝑎_1,2𝑏_2,2−𝑎_2,2𝑏_1,2=0,4𝑘6𝛾𝑎_2,2𝑘²−𝑏_2,2𝑏_1,2−𝑎_1,2𝑎_2,2=0,−3𝑘30𝑘𝛾𝜔𝑎_2,2𝑟−6𝑘𝛾𝜔𝑎_2,1+𝑏_2,2𝑏_2,1−𝑏²_2,2𝑟+𝑎_2,1𝑎_2,2=0,−𝑘−18𝑘𝛾𝜔𝑏_2,1+180𝑘𝛾𝜔𝑏_2,2𝑟+3𝑎_2,2𝑏_2,1+3𝑎_2,1𝑏_2,2−7𝑎_2,2𝑏_2,2𝑟=0,−𝑘𝑎_1,1−𝑘𝑏_2,1𝑏_2,2−𝑘𝑎_2,0𝑎_2,1+3𝛾𝜔𝑘²𝑎_2,1−𝜔𝑎_2,1+𝑘𝑏²_2,1𝑟=0,−2𝑘²𝛾𝑎_2,1+10𝑘²𝛾𝑎_2,2𝑟+𝑎_1,1𝑎_2,2−2𝑏_1,2𝑟𝑏_2,2+𝑎_1,2𝑎_2,1+𝑏_2,1𝑏_1,2+𝑏_2,2𝑏_1,1=0,−𝑘𝑎_1,0𝑎_2,1+𝛾𝑘³𝑎_2,1−𝑘𝑏_1,2𝑏_2,1−𝑘𝑏_1,1𝑏_2,2−𝑘𝑎_1,1𝑎_2,0−𝜔𝑎_1,1−𝑘𝑎_2,1+2𝑘𝑏_1,1𝑟𝑏_2,1=0,−𝑘60𝑘²𝛾𝑏_2,2𝑟−6𝑘²𝛾𝑏_2,1−7𝑏_1,2𝑟𝑎_2,2+3𝑎_1,2𝑏_2,1+3𝑎_2,1𝑏_1,2−7𝑎_1,2𝑏_2,2𝑟+3𝑎_2,2𝑏_1,1+3𝑎_1,1𝑏_2,2=0,−2𝑘𝑎_1,2−2𝜔𝑎_2,2−2𝑘𝑎_2,0𝑎_2,2−18𝛾𝜔𝑘²𝑎_2,1𝑟+4𝑘𝑏_2,1𝑏_2,2𝑟−𝑘𝑏²_2,2−𝑘𝑏²_2,1+24𝛾𝜔𝑘²𝑎_2,2−𝑘𝑎²_2,1=0,−𝑘𝑎_2,0𝑏_2,2+𝑘𝑏_1,1𝑟−𝑘𝑏_1,2+3𝛾𝜔𝑘²𝑏_2,2+𝜔𝑏_2,1𝑟−𝑘𝑎_2,1𝑏_2,1−𝜔𝑏_2,2−3𝛾𝜔𝑘²𝑏_2,1𝑟+𝑘𝑎_2,0𝑏_2,1𝑟=0,−2𝑘𝑏_1,2+60𝛾𝜔𝑘²𝑏_2,2+5𝑘𝑎_2,1𝑏_2,2𝑟−2𝑘𝑎_2,1𝑏_2,1−2𝑘𝑎_2,0𝑏_2,2−36𝛾𝜔𝑘²𝑏_2,1𝑟+90𝛾𝜔𝑘²𝑏_2,2𝑟²−3𝑘𝑎_2,2𝑏_2,2−2𝜔𝑏_2,2+5𝑘𝑎_2,2𝑏_2,1𝑟=0,−2𝑘𝑎_1,2𝑎_2,0−2𝑘𝑏_1,1𝑏_2,1−2𝑘𝑏_1,2𝑏_2,2+4𝑘𝑏_1,2𝑟𝑏_2,1−2𝑘𝑎_1,0𝑎_2,2+8𝛾𝑘³𝑎_2,2−2𝜔𝑎_1,2+4𝑘𝑏_1,1𝑟𝑏_2,2−6𝛾𝑘³𝑎_2,1𝑟−2𝑘𝑎_1,1𝑎_2,1−2𝑘𝑎_2,2=0,−𝑘𝑏_1,2𝑎_2,0+𝑘𝑎_1,0𝑏_2,1𝑟−𝑘𝑎_1,1𝑏_2,1−𝑘𝑏_1,1𝑎_2,1+𝑘𝑏_2,1𝑟+𝛾𝑘³𝑏_2,2−𝜔𝑏_1,2+𝑘𝑏_1,1𝑟𝑎_2,0−𝑘𝑏_2,2−𝑘𝑎_1,0𝑏_2,2+𝜔𝑏_1,1𝑟−𝛾𝑘³𝑏_2,1𝑟=0,12𝛾𝜔𝑘²𝑏_2,1+3𝑘𝑎_2,0𝑏_2,2𝑟+3𝑘𝑎_2,1𝑏_2,1𝑟+9𝛾𝜔𝑘²𝑏_2,1𝑟²−2𝑘𝑎_2,1𝑏_2,2−45𝛾𝜔𝑘²𝑏_2,2𝑟−𝜔𝑏_2,1−𝑘𝑏_1,1+3𝑘𝑏_1,2𝑟−𝑘𝑎_2,0𝑏_2,1−2𝑘𝑎_2,2𝑏_2,1+3𝜔𝑏_2,2𝑟=0,5𝑘𝑏_1,2𝑟𝑎_2,1−2𝑘𝑎_1,0𝑏_2,2−2𝑘𝑏_2,2−3𝑘𝑏_1,2𝑎_2,2−2𝑘𝑎_1,1𝑏_2,1−2𝜔𝑏_1,2−12𝛾𝑘³𝑏_2,1𝑟+5𝑘𝑎_1,1𝑏_2,2𝑟+5𝑘𝑏_1,1𝑟𝑎_2,2+30𝛾𝑘³𝑏_2,2𝑟²−2𝑘𝑏_1,1𝑎_2,1+20𝛾𝑘³𝑏_2,2+5𝑘𝑎_1,2𝑏_2,1𝑟−2𝑘𝑏_1,2𝑎_2,0−3𝑘𝑎_1,2𝑏_2,2=0,−𝑘𝑎_1,0𝑏_2,1−𝑘𝑏_1,1𝑎_2,0+3𝑘𝑏_1,2𝑟𝑎_2,0−2𝑘𝑎_1,2𝑏_2,1−𝑘𝑏_2,1−2𝑘𝑏_1,2𝑎_2,1+3𝜔𝑏_1,2𝑟−2𝑘𝑏_1,1𝑎_2,2+4𝛾𝑘³𝑏_2,1+3𝑘𝑏_1,1𝑟𝑎_2,1−𝜔𝑏_1,1+3𝑘𝑎_1,1𝑏_2,1𝑟−15𝛾𝑘³𝑏_2,2𝑟+3𝑘𝑎_1,0𝑏_2,2𝑟−2𝑘𝑎_1,1𝑏_2,2+3𝑘𝑏_2,2𝑟+3𝛾𝑘³𝑏_2,1𝑟²=0. (3.27)
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.
Set 𝜉=𝑘𝑥+𝜔𝑡+𝜉₀, and 𝑘(≠0), 𝜔, 𝑟 are real constants.
For 𝜀=1, there are 7 solutions.

Case 1. 𝜀=1, 𝑟=𝑟, 𝑎_1,0=(−36𝜔²+18𝛼𝜔²𝑘²+𝑘²)/36𝜔², 𝑎_1,1=−3𝑘²𝛼𝑟, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=(−𝑘²+18𝛼𝜔²𝑘²−6𝜔²)/6𝜔𝑘, 𝑎_2,1=−18𝑘𝛼𝜔𝑟, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₁(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²+𝑘²36𝜔²−3𝑘²𝛼𝑟+𝑎cosh𝜉+𝑏sinh𝜉+𝑟3𝛼𝑘²(1+𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)²,𝑢₁(𝑥,𝑡)=−𝑘²+18𝛼𝜔²𝑘²−6𝜔²−6𝜔𝑘18𝑘𝛼𝜔𝑟+()𝑎cosh𝜉+𝑏sinh𝜉+𝑟18𝛼𝜔𝑘1+𝑎sinh𝜉+𝑏cosh𝜉(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)². (3.28)

Case 2. 𝜀=1,𝑟=𝑟, 𝑎_1,0=(−36𝜔²+18𝛼𝜔²𝑘²+𝑘²)/36𝜔², 𝑎_1,1=−3𝑘²𝛼𝑟, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=(−𝑘²+18𝛾𝜔²𝑘²−6𝜔²)/6𝜔𝑘, 𝑎_2,1=−18𝑘𝛼𝜔𝑟, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=−3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=−18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₂(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²+𝑘²36𝜔²−3𝑘²𝛼𝑟−𝑎cosh𝜉+𝑏sinh𝜉+𝑟3𝛼𝑘²(𝑎sinh𝜉+𝑏cosh𝜉−1)(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)²,𝑢₂(𝑥,𝑡)=−𝑘²+18𝛼𝜔²𝑘²−6𝜔²−6𝜔𝑘18𝑘𝛼𝜔𝑟−()𝑎cosh𝜉+𝑏sinh𝜉+𝑟18𝛼𝜔𝑘𝑎sinh𝜉+𝑏cosh𝜉−1(𝑎cosh𝜉+𝑏sinh𝜉+𝑟)². (3.29)

Case 3. 𝜀=1, 𝑟=0, 𝑎_1,0=(−36𝜔²+18𝛼𝜔²𝑘²+𝑘²)/36𝜔², 𝑎_1,1=0, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=(−𝑘²+18𝛼𝜔²𝑘²−6𝜔²)/6𝜔𝑘, 𝑎_2,1=0, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₃(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²+𝑘²36𝜔²+3𝛼𝑘²(𝑎cosh𝜉+𝑏sinh𝜉)²+3𝛼𝑘²(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉)²,𝑢₃(𝑥,𝑡)=−𝑘²+18𝛼𝜔²𝑘²−6𝜔²+6𝜔𝑘18𝛼𝜔𝑘(𝑎cosh𝜉+𝑏sinh𝜉)²+18𝛼𝜔𝑘(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉)². (3.30)

Case 4. 𝜀=1, 𝑟=0, 𝑎_1,0=(−36𝜔²+18𝛼𝜔²𝑘²+𝑘²)/36𝜔², 𝑎_1,1=0, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=(−𝑘²+18𝛼𝜔²𝑘²−6𝜔²)/6𝜔𝑘, 𝑎_2,1=0, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=−3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=−18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₄(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²+𝑘²36𝜔²+3𝛼𝑘²(𝑎cosh𝜉+𝑏sinh𝜉)²−3𝛼𝑘²(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉)²,𝑢₄(𝑥,𝑡)=−𝑘²+18𝛼𝜔²𝑘²−6𝜔²+6𝜔𝑘18𝛼𝜔𝑘(𝑎cosh𝜉+𝑏sinh𝜉)²−18𝛼𝜔𝑘(𝑎sinh𝜉+𝑏cosh𝜉)(𝑎cosh𝜉+𝑏sinh𝜉)². (3.31)

Case 5. 𝜀=1, 𝑟=0, 𝑎_1,0=(−36𝜔²+72𝛼𝜔²𝑘²+𝑘²)/36𝜔², 𝑎_1,1=0, 𝑎_1,2=6𝛼𝑘², 𝑎_2,0=(−𝑘²−6𝜔²+72𝛼𝜔²𝑘²)/6𝜔𝑘, 𝑎_2,1=0, 𝑎_2,2=36𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0, 𝑏_2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₅(𝑥,𝑡)=−36𝜔²+72𝛼𝜔²𝑘²+𝑘²36𝜔²+6𝛼𝑘²(𝑎cosh𝜉+𝑏sinh𝜉)²,𝑢₅(𝑥,𝑡)=−𝑘²−6𝜔²+72𝛼𝜔²𝑘²+6𝜔𝑘36𝛼𝜔𝑘(𝑎cosh𝜉+𝑏sinh𝜉)². (3.32)

Case 6. 𝜀=1, 𝑟=1, 𝑎_1,0=(−36𝜔²+18𝛼𝜔²𝑘²+𝑘²)/36𝜔², 𝑎_1,1=−6𝛼𝑘², 𝑎_1,2=6𝛼𝑘², 𝑎_2,0=(−𝑘²+18𝛼𝜔²𝑘²−6𝜔²)/6𝜔𝑘, 𝑎_2,1=−36𝛼𝜔𝑘, 𝑎_2,2=36𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0, 𝑏_2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₆(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²+𝑘²36𝜔²−6𝛼𝑘²()+𝑎cosh𝜉+𝑏sinh𝜉+16𝛼𝑘²(𝑎cosh𝜉+𝑏sinh𝜉+1)²,𝑢₆(𝑥,𝑡)=−𝑘²+18𝛼𝜔²𝑘²−6𝜔²−6𝜔𝑘36𝛼𝜔𝑘()+𝑎cosh𝜉+𝑏sinh𝜉+136𝛼𝜔𝑘(𝑎cosh𝜉+𝑏sinh𝜉+1)². (3.33)

Case 7. 𝜀=1, 𝑟=−1, 𝑎_1,0=(−36𝜔²+18𝛼𝜔²𝑘²+𝑘²)/36𝜔², 𝑎_1,1=6𝛼𝑘², 𝑎_1,2=6𝛼𝑘², 𝑎_2,0=(−𝑘²+18𝛼𝜔²𝑘²−6𝜔²)/6𝜔𝑘, 𝑎_2,1=36𝛼𝜔𝑘, 𝑎_2,2=36𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0, 𝑏_2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₇(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²+𝑘²36𝜔²+6𝛼𝑘²()+𝑎cosh𝜉+𝑏sinh𝜉−16𝛼𝑘²(𝑎cosh𝜉+𝑏sinh𝜉−1)²,𝑢₇(𝑥,𝑡)=−𝑘²+18𝛼𝜔²𝑘²−6𝜔²+6𝜔𝑘36𝛼𝜔𝑘()+𝑎cosh𝜉+𝑏sinh𝜉−136𝛼𝜔𝑘(𝑎cosh𝜉+𝑏sinh𝜉−1)². (3.34)
While taking 𝑎=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, 𝑎_1,0=−((36𝜔²+18𝛼𝜔²𝑘²−𝑘²)/36𝜔²), 𝑎_1,1=3𝑘²𝛼𝑟, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=−(𝑘²+18𝛼𝜔²𝑘²+6𝜔²)/6𝜔𝑘, 𝑎_2,1=18𝑘𝛼𝜔𝑟, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₈(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²−𝑘²36𝜔²+3𝑘²𝛼𝑟()+𝑎cos𝜉+𝑏sin𝜉+𝑟3𝛼𝑘²(𝑎cos𝜉+𝑏sin𝜉+𝑟)²+3𝛼𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)²,𝑢₈()𝑘𝑥,𝑡=−²+18𝛼𝜔²𝑘²+6𝜔²+6𝜔𝑘18𝑘𝛼𝜔𝑟()+𝑎cos𝜉+𝑏sin𝜉+𝑟18𝛼𝜔𝑘(𝑎cos𝜉+𝑏sin𝜉+𝑟)²+18𝛼𝜔𝑘(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)². (3.35)

Case 9. 𝜀=−1, 𝑟=𝑟, 𝑎_1,0=−((36𝜔²+18𝛼𝜔²𝑘²−𝑘²)/36𝜔²), 𝑎_1,1=3𝑘²𝛼𝑟, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=−((𝑘²+18𝛼𝜔²𝑘²+6𝜔²)/6𝜔𝑘), 𝑎_2,1=18𝑘𝛼𝜔𝑟, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=−3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=−18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.1): 𝐻₉(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²−𝑘²36𝜔²+3𝑘²𝛼𝑟()+𝑎cos𝜉+𝑏sin𝜉+𝑟3𝛼𝑘²(𝑎cos𝜉+𝑏sin𝜉+𝑟)²−3𝛼𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)²,𝑢₉()𝑘𝑥,𝑡=−²+18𝛼𝜔²𝑘²+6𝜔²+6𝜔𝑘18𝑘𝛼𝜔𝑟()+𝑎cos𝜉+𝑏sin𝜉+𝑟18𝛼𝜔𝑘(𝑎cos𝜉+𝑏sin𝜉+𝑟)²−18𝛼𝜔𝑘(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉+𝑟)². (3.36)

Case 10. 𝜀=−1, 𝑟=0, 𝑎_1,0=−((36𝜔²+18𝛼𝜔²𝑘²−𝑘²)/36𝜔²), 𝑎_1,1=0, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=−((𝑘²+18𝛼𝜔²𝑘²+6𝜔²)/6𝜔𝑘), 𝑎_2,1=0, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₁₀(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²−𝑘²36𝜔²+3𝛼𝑘²(𝑎cos𝜉+𝑏sin𝜉)²+3𝛼𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉)²,𝑢₁₀()𝑘𝑥,𝑡=−²+18𝛼𝜔²𝑘²+6𝜔²+6𝜔𝑘18𝛼𝜔𝑘(𝑎cos𝜉+𝑏sin𝜉)²+18𝛼𝜔𝑘(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉)². (3.37)

Case 11. 𝜀=−1, 𝑟=0, 𝑎_1,0=−((36𝜔²+18𝛼𝜔²𝑘²−𝑘²)/36𝜔²), 𝑎_1,1=0, 𝑎_1,2=3𝛼𝑘², 𝑎_2,0=−((𝑘²+18𝛼𝜔²𝑘²+6𝜔²)/6𝜔𝑘), 𝑎_2,1=0, 𝑎_2,2=18𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=−3𝛼𝑘², 𝑏_2,1=0, 𝑏_2,2=−18𝛼𝜔𝑘. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₁₁(𝑥,𝑡)=−36𝜔²+18𝛼𝜔²𝑘²−𝑘²36𝜔²+3𝛼𝑘²(𝑎cos𝜉+𝑏sin𝜉)²−3𝛼𝑘²(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉)²,𝑢₁₁()𝑘𝑥,𝑡=−²+18𝛼𝜔²𝑘²+6𝜔²+6𝜔𝑘18𝛼𝜔𝑘(𝑎cos𝜉+𝑏sin𝜉)²−18𝛼𝜔𝑘(−𝑎sin𝜉+𝑏cos𝜉)(𝑎cos𝜉+𝑏sin𝜉)². (3.38)

Case 12. 𝜀=−1, 𝑟=0, 𝑎_1,0=−((36𝜔²+72𝛼𝜔²𝑘²−𝑘²)/36𝜔²), 𝑎_1,1=0, 𝑎_1,2=6𝛼𝑘², 𝑎_2,0=−((𝑘²+72𝛼𝜔²𝑘²+6𝜔²)/6𝜔𝑘), 𝑎_2,1=0, 𝑎_2,2=36𝛼𝜔𝑘, 𝑏_1,1=0, 𝑏_1,2=0, 𝑏_2,1=0, 𝑏_2,2=0. By applying these to (3.26), one gets a solution of the PDEs (1.2): 𝐻₁₂(𝑥,𝑡)=−36𝜔²+72𝛼𝜔²𝑘²−𝑘²36𝜔²+6𝛼𝑘²(𝑎cos𝜉+𝑏sin𝜉)²,𝑢₁₂()𝑘𝑥,𝑡=−²+72𝛼𝜔²𝑘²+6𝜔²+6𝜔𝑘36𝛼𝜔𝑘(𝑎cos𝜉+𝑏sin𝜉)². (3.39)
The periodic wave solutions of triangle function 𝐻₈,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].
For 𝑅=0, we can deduce that the solution of (3.25) is of the form 𝐻(𝜉)=𝑎_1,0+𝑎_1,1𝑔(𝜉)+𝑎_1,2𝑔(𝜉)²,𝑢(𝜉)=𝑎_2,0+𝑎_2,1𝑔(𝜉)+𝑎_2,2𝑔(𝜉)², (3.40) where 𝜉=𝑘𝑥+𝜔𝑡+𝜉₀ and 𝑎_1,0, 𝑎_1,1, 𝑎_1,2, 𝑎_2,0, 𝑎_2,1, 𝑎_2,2 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: 2𝑘𝑎_2,236𝛼𝜔𝑘−𝑎_2,2=0,4𝑘𝑎_2,26𝛼𝑘²−𝑎_1,2=0,3𝑘𝑎_2,16𝛼𝜔𝑘−𝑎_2,2=0,−𝜔𝑎_2,1−𝑘𝑎_2,0𝑎_2,1−𝑘𝑎_1,1=0,3𝑘2𝛼𝑎_2,1𝑘²−𝑎_1,2𝑎_2,1−𝑎_2,2𝑎_1,1=0,−𝜔𝑎_1,1−𝑘𝑎_2,1−𝑘𝑎_1,1𝑎_2,0−𝑘𝑎_1,0𝑎_2,1=0,−2𝜔𝑎_2,2−𝑘𝑎²_2,1−2𝑘𝑎_2,0𝑎_2,2−2𝑘𝑎_1,2=0,−2𝜔𝑎_1,2−2𝑘𝑎_2,2−2𝑘𝑎_1,1𝑎_2,1−2𝑘𝑎_1,2𝑎_2,0−2𝑘𝑎_1,0𝑎_2,2=0. (3.41)
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, 𝑎_1,0=(−36𝜔²+𝑘²)/36𝜔², 𝑎_1,1=0, 𝑎_1,2=6𝛼𝑘², 𝑎_2,0=−((𝑘²+6𝜔²)/6𝜔𝑘), 𝑎_2,1=0, 𝑎_2,2=36𝛼𝜔𝑘. By applying these to (3.40), one gets a solution of the PDEs (1.2): 𝐻₁₃(𝑥,𝑡)=−36𝜔²+𝑘²36𝜔²+6𝛼𝑘²𝑘𝑥+𝜔𝑡+𝜉₀²,𝑢₁₃()𝑘𝑥,𝑡=−²+6𝜔²+6𝜔𝑘36𝛼𝜔𝑘𝑘𝑥+𝜔𝑡+𝜉₀². (3.42) 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.

Acknowledgments

This work is supported by the NSF of China (10771041, 40890153, 11271090), Doctorial Point Fund of National Education Ministry of China (200810780002) and NSF of Guangdong (S2012010010121). It is also supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars. The authors would like to express their hearty thanks to Chern Institute of Mathematics that provided very comfortable research environments to them. The authors would like to thank Professor Wang Mingliang for his helpful suggestions. The authors wish to thank the referee for his or her very helpful comments and useful suggestions.