Abstract and Applied Analysis

Volume 2011, Article ID 935710, 8 pages

http://dx.doi.org/10.1155/2011/935710

## Bäcklund Transformation and New Exact Solutions of the Sharma-Tasso-Olver Equation

^{1}School of Economy and Management, Guangzhou University of Chinese Medicine, Guangzhou 510006, China^{2}Department of Mathematics, Shandong University, Jinan 250100, China^{3}School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received 20 February 2011; Accepted 23 March 2011

Academic Editor: Svatoslav Staněk

Copyright © 2011 Lin Jianming et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Sharma-Tasso-Olver (STO) equation is investigated. The Painlevé analysis is efficiently used for analytic study of this equation. The Bäcklund transformations and some new exact solutions are formally derived.

#### 1. Introduction

Let be a constant. We consider the Sharma-Tasso-Olver (STO) equation

This paper is concerned with the STO equation [1] and [2–17].

Attention has been focused on STO equation (1.1) in [1] and [2–4] and the references therein due to its appearance in scientific applications. In [1], the tanh method, the extended tanh method, and other ansatz involving hyperbolic and exponential functions are efficiently used for the analytic study of this equation. And some new solitons and kinks solutions are formally derived. The proposed schemes are reliable and manageable. In [2], this equation was handled by using the Cole-Hopf transformations method. The simple symmetry reduction procedure is repeatedly used in [3] to obtain exact solutions where soliton fission and fusion were examined. However, in [4], the soliton fission and fusion were examined thoroughly by using Hirota's direct method and the Bäcklund transformations method. In [6], the authors show that symmetry constraints do not always yield exact solutions through analyzing the STO equation. The generalized Kaup-Newell-type hierarchy of nonlinear evolution equations is explicitly related to STO equation from [9]. Using the improved tanh function method in [10], the STO equation with its fission and fusion has some exact solutions. In [11] some exact solution of the STO equation are given by implying a generalized tanh function method for approximating some solutions which have been known. In [12], the method of third-order mode coupling is applied to the STO equation. In [13], the same exact explicit solutions of the STO equation, as this work are derived by using an extension of the homogeneous balance method and the Bäcklund transformation. Recently in [7, 8], the authors have introduced some other discussions about new solution techniques, a multiple exp-function method and linear superposition principle applying to nonlinear equations, which rely on a close observation on relations between linear objects and nonlinear objects.

Many types of travelling waves are of particular interest in solitary wave theory. The solitons, which are localized travelling waves, asymptotically zero at large distances, the periodic solutions, the kink waves which rise or descend from one asymptotic state to another, are well-known types of travelling waves.

The objectives of this work are twofold. First, we seek to establish new solitons and kink solutions of distinct physical structures for the nonlinear equation (1.1). Second, we aim to implement many strategies to achieve our goal, namely, the Painlevé test [18, 19], hyperbolic functions ansatze, and exponential functions ansatze to obtain new exact solutions. In what follows, the Painlevé test will be reviewed briefly.

We give the three steps of the test for a single PDE, in two independent variables, and . Following [18, 20], the Laurent expansion of the solution , should be single valued in the neighborhood of a noncharacteristic, movable and singular manifold , which can be viewed as the surface of the movable poles in the complex plane. In (1.3), is a negative integer and are analytic functions in a neighborhood of .

The Painlevé test have the following steps

*Step 1 (leading order analysis). *Determine the (negative) integer and by balancing the minimal power terms after the substitution of into the given PDE. There may be several branches for , and for each the next two steps must be performed.

*Step 2 (determination of the resonances). *For selected and , calculate the nonnegative integers , called the *resonances*, at which arbitrary functions enter the series (1.3). To do so, substitute (1.3) into (1.1), and normalize all orders of with the minimal power term. Then we have the cycle formula on , through reducing the formula, and the resonance can be obtained.

*Step 3 (verification of the compatibility conditions). *Substituting at which is not equal into the cycle formula, we have nonlinear equation . At resonance levels, should be arbitrary, and then we are deducing a nonlinear equation . If the equation implies , then the compatibility condition is unconditionally satisfied.

An equation for which these three steps can be carried out consistently and unambiguously passes the Painlevé test. Equation (1.3) is called the truncated Painlevé expansion.

#### 2. The Compatibility Conditions and B**ä**cklund Transform

Let where are analytic functions on in the neighborhood of the manifold .

Substituting (2.1) into (1.1), we can get the . Thus, (2.1) becomes Expressing (1.1), we have that Differentiating (2.2), we obtain that Substituting (2.4) into (2.3), we get the cycle formula on Taking in (2.5), we deduce that or .

We will get the Bäcklund transformations, according to the above two cases, respectively.

*Case 1 (). *Substituting into (2.5), we have the following equation on :
By (2.6) we see that , and are arbitrary. Hence are resonances. Moreover the compatibility conditions will be deduced by (2.5) or (2.6).

Setting and , we infer that

It is easy to see that (2.8) does so if (2.7) holds. So is arbitrary.

Taking and letting in (2.6), we have that

By (2.6), noting that , and , we can infer that

Thus, we obtain from the Bäckland transformation of the STO equation that
where is the seed solution of (1.1), the and satisfy the following condition:

*Case 2 (). *Rewrite (2.5) as the following form on
From (2.13) we know that is arbitrary. That is, are resonances.

Taking , , and in (2.13), we get that

It is easy to see that (2.15) implies (2.16). So is arbitrary. Equation (1.1) satisfies Painlevé property.

Taking , and letting in (2.5) or (2.13), we have that

Substituting into the above equation, we deduce the second condition below

By (2.5) or (2.13), noting that , , and , we can easy to have that

Thus, we obtain from the other Bäckland transformation of the STO equation that
where satisfies the following conditions:

#### 3. New Exact Solutions for STO Equation

As it is well known that the Bäcklund transformation is one of the most effective methods for finding exact solutions of nonlinear partial differential equations. By (2.11) and (2.20), choose some seed solutions, and then we can get some new single travelling solitary wave solutions below, respectively.

*Case 1 (for Bäcklund transformation (2.11)). *Taking the seed solution in the Bäcklund transformation (2.11), (2.12) becomes

The arbitrary multiple solitary wave solution of (1.1) can be easily written down. For the sake of simplicity, we only give some new single travelling solitary wave solutions here. Substituting the ansatz which satisfies (3.1) into (2.11), and leads the new exact solutions of (1.1)

Substituting the ansatz into (3.1) will produce the dispersion relation between and ,
and the new exact solutions are that

Taking the seed solution for (1.1) will get

Observing (3.5), we have , and .

Substituting , , and into the Bäcklund transformation (2.11), we obtain the exact solutions for STO equation (1.1)

Taking the seed solution for STO equation (1.1), (2.12) becomes

Substitute the ansatz into (2.11), yielding the solutions for the STO equation (1.1)

*Remark 3.1. *When or , the solution is obtained in [1] by tanh method. In [4], the authors have obtained the . As for the case , the authors of [4, page 238] said that no new meaningful results can be obtained; we could here obtain some meaningful results.

*Case 2 (for Bäcklund transformation (2.20)). *Observe, (2.21), we have ansatz . Substituting into (2.21) and seting the coefficients to be equal to zero, we have the dispersion relation between and ,

Thus, (2.20) yields the new exact solutions for the STO equation (1.1)

Substituting ansatz into (2.21) and balancing the coefficients, we have the new exact solutions for the STO equation (1.1)

#### Acknowledgments

The authors would like to express their hearty thanks to the Chern Institute of Mathematics of Nankai University for supplying the financial supports and wonderful work conditions. The authors wish to thank the referee for his or her very helpful comments and useful suggestions. This work was supported by the NSF of China (10771220), Doctorial Point Fund of National Education, Ministry of China (200810780002).

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