Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation

Abstract

We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity.

1. Introduction

The nonlinear phenomena in the scientific work or engineering fields are more and more attractive to scientists. To depict and analyze such nonlinear phenomena, the nonlinear evolutionary equations are playing an important role and their solitary wave solutions are the main interests of mathematicians and physicists.

To obtain the traveling wave solutions of these nonlinear evolution equations, many methods were attempted, such as the inverse scattering method, Hirota’s bilinear transformation, the tanh-sech method, extended tanh method, sine-cosine method, homogeneous balance method, and exp-function method. With the aid of symbolic computation system, many explicit solutions are easily obtained, and many interesting works deeply promote the research of nonlinear phenomena.

The present work is interested in generalized Benjamin-Bona-Mahony (BBM) equation:𝑢_𝑡+𝑎𝑢_𝑥+(𝑏𝑢^𝑛+𝑐𝑢^2𝑛)𝑢_𝑥+𝑘𝑢_𝑥𝑥𝑥=0. (1.1)

In the above equation, the first term of left side represents the evolution term while parameters 𝑏 and 𝑐 represent the coefficients of dual-power law nonlinearity, 𝑎 and 𝑘 are the coefficients of dispersion terms, 𝑛 is the power law parameter, and variable 𝑢 is the wave profile. In [1], Biswas used the solitary wave ansatz and obtained an exact 1-soliton solution of (1.1). In order to find more exact solutions of some nonlinear evolutionary equations, the Weierstrass elliptic function was introduced. For example, Kuru [2, 3] discussed the BBM-like equation, and Estévez et al. [4] analyzed another type of generalized BBM equations. In [5], Deng et al. also applied the similar method to the study of a nonlinear variant of the PHI-four equation. In this paper, we will apply the method to the generalized Benjamin-Bona-Mahony equation.

The rest of this paper is organized as follows. In Section 2, we first outline the Weierstrass elliptic function method. In Section 3, we give exact expression of some traveling wave solutions of generalized Benjamin-Bona-Mahony (BBM) equation (1.1) by using the Weierstrass elliptic function method. Finally, some conclusions are given in Section 4.

2. Description of the Weierstrass Elliptic Function Method

When we search for the solutions of some evolutionary equations, we will meet the following ordinary differential equation: (𝑑𝜑)𝑑𝜃²=𝑃₄(𝜑)=𝑎₀𝜑⁴+4𝑎₁𝜑³+6𝑎₂𝜑²+4𝑎₃𝜑+𝑎₄. (2.1) In [6], Whittaker and Watson introduced two invariants:𝑔₂=𝑎₀𝑎₄−4𝑎₁𝑎₃+3𝑎²₂,𝑔₃=𝑎₀𝑎₂𝑎₄+2𝑎₁𝑎₂𝑎₃−𝑎³₂−𝑎₀𝑎²₃−𝑎²₁𝑎₄, (2.2) and a discriminant Δ=𝑔³₂−27𝑔²₃; then the solutions of differential (2.1) have the following form: 𝜑(𝜃)=𝜑₀+14𝑃_𝜑(𝜑₀)(℘(𝜃;𝑔₂,𝑔₃1)−𝑃24^𝜑𝜑(𝜑₀))⁻¹, (2.3) where 𝜑₀ is one of the roots of the polynomial 𝑃₄(𝜑), and 𝑃_𝜑(𝜑₀),𝑃_𝜑𝜑(𝜑₀), respectively, denote the first and second derivative of 𝑃₄(𝜑) with respect to 𝜑 at the 𝜑₀. Particularly, if Δ=𝑔³₂−27𝑔²₃=0, then the Weierstrass elliptic function ℘(𝜃;𝑔₂,𝑔₃) satisfies these conditions: ℘(𝜃;12𝑏²,−8𝑏³)=𝑏+3𝑏sinh⁻²(√3𝑏𝜃),℘(𝜃;12𝑏²,8𝑏³)=−𝑏+3𝑏sin⁻²(√3𝑏𝜃). (2.4) Once we get the solution of (2.1) with the form of (2.3), we will get the exact expressions of solutions of many partial differential equations. Suppose that Δ=𝑔³₂−27𝑔²₃=0; it is obvious that if 𝑔₂>0,𝑔₃>0, then there exist period solutions of original evolutionary equation, otherwise if 𝑔₂>0,𝑔₃<0, then there exist solitary solutions.

3. Discussion on Generalized BBM Equation (1.1)

Let 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥−𝑣𝑡, so (1.1) is carried to 𝑏(𝑎−𝑣)𝑢+𝑢𝑛+1^𝑛+1+𝑐𝑢2𝑛+1^2𝑛+1+𝑘𝑢^′′=𝐷, (3.1) where 𝐷 is an integral constant. We introduce the transformation that 𝑢(𝜉)=(𝑘(𝑛+1)𝑏)^1/𝑛𝑊(𝜉), (3.2) so that (3.1) is changed to 𝑊^′′−𝑣−𝑎𝑘𝑊+𝑊^𝑛+1+𝑐𝑘(𝑛+1)²𝑏²𝑊(2𝑛+1)^2𝑛+1=𝐷𝑘(𝑏)𝑘(𝑛+1)^1/𝑛. (3.3) Let 𝐵=(𝑣−𝑎)/𝑘,𝐶=−𝑐𝑘(𝑛+1)²/𝑏²(2𝑛+1),and𝑅₁=(𝐷/𝑘)(𝑏/𝑘(𝑛+1))^1/𝑛; then (3.3) is transformed to the following form: 𝑊^′′=𝐵𝑊−𝑊^𝑛+1+𝐶𝑊^2𝑛+1+𝑅₁. (3.4) By multiplying each side of (3.4) by 𝑊^′, integrating once again, we get that (𝑊^′)²=𝐵𝑊²−2𝑊𝑛+2^𝑛+2+𝐶𝑊𝑛+1^2𝑛+2+2𝑅₁𝑊+𝑅₂, (3.5) where 𝑅₂ is the other integration constant.

Let 𝑊=𝜑^𝑝; then the above equation is changed to (𝑑𝜑)𝑑𝜉²=𝐵𝑝²𝜑²−2(𝑛+2)𝑝²𝜑^𝑛𝑝+2+𝐶(𝑛+1)𝑝²𝜑^2𝑛𝑝+2+2𝑅₁𝑝²𝜑^2−𝑝+𝑅₂𝑝²𝜑^2(1−𝑝). (3.6)

Only if all the power of 𝜑 appearing in the right side of (3.6) are the integers and are between 0 and 4, can we guarantee the integrability of (3.6). Therefore, if 𝑅₁=𝑅₂=0, then 𝑝=±1/𝑛. We choose the only case that if 𝑝=−1/𝑛, then (𝑑𝜑)𝑑𝜉²=𝑛²𝐵𝜑²−2𝑛²𝑛𝑛+2𝜑+²𝐶𝑛+1. (3.7) The polynomial 𝑃(𝜑) has two roots: 𝜑₀=((𝑛+1)±(𝑛+1)[𝐵𝐶(𝑛+2)²+(𝑛+1)])/𝐵(𝑛+1)(𝑛+2). For simplicity, let 𝜆=(𝑛+1)[𝐵𝐶(𝑛+2)²+(𝑛+1)], then 𝜑₀=((𝑛+1)±𝜆)/𝐵(𝑛+1)(𝑛+2)𝑔₂=𝑛⁴𝐵²12,𝑔₃𝑛=−⁶𝐵³216,Δ=0. (3.8) Then at the point 𝜑₀=((𝑛+1)±𝜆)/𝐵(𝑛+1)(𝑛+2), the nonzero solutions of (3.7) are ℘(𝜃;𝑔₂,𝑔₃𝑛)=²𝐵+1124𝑛²𝐵cscℎ²(√𝐵2𝑛𝜃),(if𝐵>0),℘(𝜃;𝑔₂,𝑔₃𝑛)=²𝐵−1124𝑛²𝐵csc²(√−𝐵2𝑛𝜃),(if𝐵<0). (3.9) The solutions of the ordinary differential equation are𝜑(𝜉)=(𝑛+1)±[𝜆−2𝜆cosh²((√𝐵/2)𝑛𝜃)]𝐵(𝑛+1)(𝑛+2),(if𝐵>0),𝜑(𝜉)=(𝑛+1)±[𝜆−2𝜆cos²((√−𝐵/2)𝑛𝜃)]𝐵(𝑛+1)(𝑛+2),(if𝐵<0), (3.10) where 𝐵=(𝑣−𝑎)/𝑘,𝐶=−𝑐𝑘(𝑛+1)²/𝑏²(2𝑛+1),𝑅₁=(𝐷/𝑘)(𝑏/𝑘(𝑛+1))^1/𝑛. Therefore, the solutions of partial differential equation are𝑢(𝑥,𝑡)=(𝑘(𝑛+1)𝑏𝐵(𝑛+1)(𝑛+2)(𝑛+1)±[𝜆−2𝜆cosh²((√)𝐵/2)𝑛𝜃)]^1/𝑛,(if𝐵>0),𝑢(𝑥,𝑡)=(𝑘(𝑛+1)𝑏𝐵(𝑛+1)(𝑛+2)(𝑛+1)±[𝜆−2𝜆cos²((√)−𝐵/2)𝑛𝜃)]^1/𝑛,(if𝐵<0), (3.11) where 𝐵=(𝑣−𝑎)/𝑘,𝐶=−𝑐𝑘(𝑛+1)²/𝑏²(2𝑛+1),and𝑅₁=(𝐷/𝑘)(𝑏/𝑘(𝑛+1))^1/𝑛.

Remark 3.1. We have checked that (3.11) is the solution of (1.1). From the above discussion, we find more solutions than the work of [1].

4. Conclusions

From the above discussion, we find the traveling wave solutions of the generalized BBM equation, which are expressed by the hyperbolic functions and trigonometric functions, without the aid of symbolic computations. In addition, the method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity.

Acknowledgments

The authors would like to express their sincere thanks to referees for the valuable advice. The present work was supported by the National Nature Science Foundation of China (no. 10771088) and Foundation for Younger Personnel of Jiangsu University (no. 1241190006).