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

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 𝛼 be a constant. We consider the Sharma-Tasso-Olver (STO) equation𝑢_𝑡+𝛼𝑢³_𝑥+32𝛼𝑢²_𝑥𝑥+𝛼𝑢_𝑥𝑥𝑥=0.(1.1)

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, ℱ(𝑥,𝑡,𝑢(𝑥,𝑡))=0,(1.2) in two independent variables, 𝑥 and 𝑡. Following [18, 20], the Laurent expansion of the solution 𝑢(𝑥,𝑡),𝑢(𝑥,𝑡)=𝜙^𝛼(𝑥,𝑡)^∞_𝑘=0𝑢_𝑘(𝑥,𝑡)𝜙^𝑘(𝑥,𝑡),(1.3) should be single valued in the neighborhood of a noncharacteristic, movable and singular manifold 𝜙(𝑥,𝑡), which can be viewed as the surface of the movable poles in the complex plane. In (1.3), 𝑢₀(𝑥,𝑡)≠0,𝛼 is a negative integer and 𝑢_𝑘(𝑥,𝑡) are analytic functions in a neighborhood of 𝜙(𝑥,𝑡).

The Painlevé test have the following steps

Step 1 (leading order analysis). Determine the (negative) integer 𝛼 and 𝑢₀ by balancing the minimal power terms after the substitution of 𝑢=𝑢₀𝜙^𝛼 into the given PDE. There may be several branches for 𝑢₀, and for each the next two steps must be performed.

Step 2 (determination of the resonances). For selected 𝛼 and 𝑢₀, calculate the nonnegative integers 𝑟, called the resonances, at which arbitrary functions 𝑢_𝑟 enter the series (1.3). To do so, substitute (1.3) into (1.1), and normalize all orders of 𝜙 with the minimal power term. Then we have the cycle formula on 𝑢_𝑗, through reducing the formula, and the resonance 𝑟 can be obtained.

Step 3 (verification of the compatibility conditions). Substituting 𝑗=𝑘 at which 𝑘 is not equal 𝑟 into the cycle formula, we have nonlinear equation 𝑓_𝑘(𝑢_𝑘⋯𝑢₀,𝜙⋯)=0(1<𝑘<𝑟). At resonance levels, 𝑢_𝑟 should be arbitrary, and then we are deducing a nonlinear equation 𝑔(𝑢_𝑟−1⋯𝑢₀,𝜙⋯)=0. If the equation 𝑓 implies 𝑔, 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 Bäcklund Transform

Let 𝑢=𝜙^𝛼∞_𝑗=0𝑢_𝑗𝜙^𝑗,(2.1) where 𝜙=𝜙(𝑥,𝑡),𝑢_𝑗=𝑢_𝑗(𝑥,𝑡) are analytic functions on (𝑥,𝑡) in the neighborhood of the manifold 𝑀={(𝑥,𝑡)∶𝜙(𝑥,𝑡)=0}.

Substituting (2.1) into (1.1), we can get the 𝛼=−1. Thus, (2.1) becomes 𝑢=^∞_𝑗=0𝑢_𝑗𝜙^𝑗−1.(2.2) Expressing (1.1), we have that 𝑢_𝑡+3𝛼𝑢²𝑢_𝑥+3𝛼𝑢^2𝑥+3𝛼𝑢𝑢_𝑥𝑥+𝛼𝑢_𝑥𝑥𝑥=0.(2.3) Differentiating (2.2), we obtain that 𝑢_𝑡=(𝑗−1)^∞_𝑗=0𝑢_𝑗𝜙^𝑗−2𝜙_𝑡+^∞_𝑗=0𝑢_𝑥𝑢_𝑗,𝑡𝜙^𝑗−1,=(𝑗−1)^∞_𝑗=0𝑢_𝑗𝜙^𝑗−2𝜙_𝑥+^∞_𝑗=0𝑢_𝑥𝑥𝑢_𝑗,𝑥𝜙^𝑗−1,=(𝑗−1)^∞_𝑗=0𝑢_𝑗𝜙^𝑗−2𝜙_𝑥𝑥+2(𝑗−1)^∞_𝑗=0𝑢_𝑗,𝑥𝜙^𝑗−2𝜙_𝑥+(𝑗−1)(𝑗−2)^∞_𝑗=0𝑢_𝑗𝜙^𝑗−3𝜙^2𝑥+^∞_𝑗=0𝑢_𝑥𝑥𝑥𝑢_{𝑗,𝑥𝑥}𝜙^𝑗−1,=3(𝑗−1)^∞_𝑗=0𝑢_𝑗,𝑥𝜙^𝑗−2𝜙_𝑥𝑥+(𝑗−1)^∞_𝑗=0𝑢_𝑗𝜙^𝑗−2𝜙_𝑥𝑥𝑥+3(𝑗−1)(𝑗−2)^∞_𝑗=0𝑢_𝑗𝜙^𝑗−3𝜙_𝑥𝜙_𝑥𝑥+3(𝑗−1)^∞_𝑗=0𝑢_{𝑗,𝑥𝑥}𝜙^𝑗−2𝜙_𝑥+^∞_𝑗=0𝑢_{𝑗,𝑥𝑥𝑥}𝜙^𝑗−1+3(𝑗−1)(𝑗−2)^∞_𝑗=0𝑢_𝑗,𝑥𝜙^𝑗−3𝜙^2𝑥+(𝑗−1)(𝑗−2)(𝑗−3)^∞_𝑗=0𝑢_𝑗𝜙^𝑗−4𝜙^3𝑥.(2.4) Substituting (2.4) into (2.3), we get the cycle formula on 𝑢_𝑗(𝑗−3)𝑢_𝑗−2𝜙_𝑡+𝑢_{𝑗−3,𝑡}+3𝛼^𝑗−𝑛_𝑗_𝑚=0_𝑛=0+3𝛼_𝑗𝑢_{𝑗−𝑚−𝑛}𝑢_𝑚(𝑛−1)𝑢_𝑛𝜙_𝑥+𝑢_{𝑛−1,𝑥}_𝑚₌₀(𝑗−𝑚−1)𝑢_𝑗−𝑚+3𝛼_𝑗𝜙_𝑥+𝑢_{𝑗−𝑚−1,𝑥}(𝑚−1)𝑢_𝑚𝜙_𝑥+𝑢_{𝑚−1,𝑥}_𝑚₌₀+𝛼3(𝑗𝑢−_𝑗−𝑚3)𝑢(𝑚−_{𝑗−2,𝑥}2)𝑢_𝑚−1𝜙_𝑥𝑥+2(𝑚−2)𝑢_{𝑚−1,𝑥}𝜙_𝑥+(𝑚−1)(𝑚−2)𝑢_𝑚𝜙^2𝑥+𝑢_{𝑚−2,𝑥𝑥}𝜙_𝑥𝑥+(𝑗−3)𝑢_𝑗−2𝜙_𝑥𝑥𝑥+3(𝑗−2)(𝑗−3)𝑢_𝑗−1𝜙_𝑥𝜙_𝑥𝑥+3(𝑗−2)(𝑗−3)𝑢_{𝑗−1,𝑥}𝜙^2𝑥+𝑢_{𝑗−3,𝑥𝑥𝑥}+3(𝑗−3)𝑢_{𝑗−2,𝑥𝑥}𝜙_𝑥+(𝑗−1)(𝑗−2)(𝑗−3)𝑢_𝑗𝜙^3𝑥=0.(2.5) Taking 𝑗=0 in (2.5), we deduce that 𝑢₀=𝜙_𝑥 or 𝑢₀=2𝜙_𝑥.

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

Case 1 (𝑢₀=𝜙_𝑥). Substituting 𝑢₀=𝜙_𝑥 into (2.5), we have the following equation on 𝑢_𝑗: (𝑗+1)(𝑗−1)(𝑗−3)𝛼𝑢_𝑗𝜙^3𝑥=𝐹_𝑗𝑢_𝑗−1⋯𝑢₀,𝜙_𝑡,𝜙_𝑥,𝜙_𝑥𝑥,𝜙_𝑥𝑥𝑥⋯(𝑗=1,2,…).(2.6) By (2.6) we see that 𝑢₋₁, 𝑢₁ and 𝑢₃ are arbitrary. Hence 𝑗=3 are resonances. Moreover the compatibility conditions will be deduced by (2.5) or (2.6).
Setting 𝑗=2 and 𝑗=3, we infer that 𝜙_𝑡+3𝛼𝑢²¹𝜙_𝑥+𝑢₂𝜙^2𝑥+𝑢_1,𝑥𝜙_𝑥+𝑢₁𝜙_𝑥𝑥+𝛼𝜙_𝑥𝑥𝑥=0,(2.7)𝜕𝜕𝑥𝜙^𝑡+3𝛼𝑢²¹𝜙_𝑥+𝑢₂𝜙^2𝑥+𝑢_1,𝑥𝜙_𝑥+𝑢₁𝜙_𝑥𝑥+𝛼𝜙_𝑥𝑥𝑥=0.(2.8)
It is easy to see that (2.8) does so if (2.7) holds. So 𝑢₃ is arbitrary.
Taking 𝑗=4,𝑢₃=0 and letting 𝑢₂=𝑢₄=0 in (2.6), we have that 𝑢_1,𝑡+3𝛼𝑢²¹𝑢_1,𝑥+3𝛼𝑢^21,𝑥+3𝛼𝑢₁𝑢_1,𝑥𝑥+𝛼𝑢_{1,𝑥𝑥𝑥}=0.(2.9)
By (2.6), noting that 𝑢₂=0, 𝑢₃=0 and 𝑢₄=0, we can infer that 𝑢_𝑗=0(𝑗≥2).(2.10)
Thus, we obtain from the Bäckland transformation of the STO equation that 𝑢=𝜙^𝑥𝜙+𝑢¹,(2.11) where 𝑢₁ is the seed solution of (1.1), the 𝑢₁ and 𝜙 satisfy the following condition: 𝜙_𝑡+3𝛼𝑢²¹𝜙_𝑥+𝑢_1,𝑥𝜙_𝑥+𝑢₁𝜙_𝑥𝑥+𝛼𝜙_𝑥𝑥𝑥=0.(2.12)

Case 2 (𝑢₀=2𝜙_𝑥). Rewrite (2.5) as the following form on 𝑢_𝑗(𝑗+1)(𝑗+2)(𝑗−3)𝛼𝑢_𝑗𝜙^3𝑥=𝐺_𝑗𝑢_𝑗−1⋯𝑢₀,𝜙_𝑡,𝜙_𝑥,𝜙_𝑥𝑥,𝜙_𝑥𝑥𝑥⋯,(𝑗=1,2,…).(2.13) From (2.13) we know that 𝑢₃ is arbitrary. That is, 𝑗=3 are resonances.
Taking 𝑗=1, 𝑗=2, and 𝑗=3 in (2.13), we get that 𝑢₁=−𝜙^𝑥𝑥𝜙_𝑥𝜙,(2.14)_𝑡𝜕+6𝛼𝑢₂𝜙^2𝑥−3𝛼𝑢₁𝜙_𝑥𝑥−2𝛼𝜙_𝑥𝑥𝑥=0,(2.15)𝜕𝑥𝜙^𝑡+6𝛼𝑢₂𝜙^2𝑥−3𝛼𝑢₁𝜙_𝑥𝑥−2𝛼𝜙_𝑥𝑥𝑥=0.(2.16)
It is easy to see that (2.15) implies (2.16). So 𝑢₃ is arbitrary. Equation (1.1) satisfies Painlevé property.
Taking 𝑗=4, 𝑢₃=0 and letting 𝑢₂=𝑢₄=0 in (2.5) or (2.13), we have that 𝑢_1,𝑡+3𝛼𝑢²¹𝑢_1,𝑥+3𝛼𝑢^21,𝑥+3𝛼𝑢₁𝑢_1,𝑥𝑥+𝛼𝑢_{1,𝑥𝑥𝑥}=0.(2.17)
Substituting 𝑢₁=−𝜙_𝑥𝑥/𝜙_𝑥 into the above equation, we deduce the second condition below −𝜙^3𝑥𝜙_𝑥𝑥𝑡+𝜙^2𝑥𝜙_𝑥𝑥𝜙_𝑥𝑡+18𝛼𝜙^4𝑥𝑥−30𝛼𝜙_𝑥𝜙^2𝑥𝑥𝜙_𝑥𝑥𝑥+7𝛼𝜙^2𝑥𝜙_𝑥𝑥𝜙_{𝑥𝑥𝑥𝑥}+6𝛼𝜙^2𝑥𝜙^{2𝑥𝑥𝑥}−𝛼𝜙^3𝑥𝜙_{𝑥𝑥𝑥𝑥𝑥}=0(2.18)
By (2.5) or (2.13), noting that 𝑢₂=0, 𝑢₃=0, and 𝑢₄=0, we can easy to have that 𝑢_𝑗=0(𝑗≥2).(2.19)
Thus, we obtain from the other Bäckland transformation of the STO equation that 𝑢=2𝜙^𝑥𝜙−𝜙𝜙^𝑥𝑥_𝑥,(2.20) where 𝜙(𝑥,𝑡) satisfies the following conditions: −𝜙_3𝑥𝜙_𝑥𝜙_𝑡+3𝛼𝜙^2𝑥𝑥−2𝛼𝜙_𝑥𝜙_𝑥𝑥𝑥=0,𝜙_𝑥𝑥𝑡+𝜙^2𝑥𝜙_𝑥𝑥𝜙_𝑥𝑡+18𝛼𝜙^4𝑥𝑥−30𝛼𝜙_𝑥𝜙^2𝑥𝑥𝜙_𝑥𝑥𝑥+7𝛼𝜙^2𝑥𝜙_𝑥𝑥𝜙_{𝑥𝑥𝑥𝑥}+6𝛼𝜙^2𝑥𝜙^{2𝑥𝑥𝑥}−𝛼𝜙^3𝑥𝜙_{𝑥𝑥𝑥𝑥𝑥}=0.(2.21)

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 𝑢₁=0 in the Bäcklund transformation (2.11), (2.12) becomes 𝜙_𝑡+𝛼𝜙_𝑥𝑥𝑥=0.(3.1)
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 𝜙₁=𝑘₁𝑥²+𝑘₂𝑥+𝑟 which satisfies (3.1) into (2.11), and leads the new exact solutions of (1.1) 𝑢₁(𝑥,𝑡)=2𝑘¹𝑥+𝑘₂𝑘₁𝑥²+𝑘₂𝑥+𝑟.(3.2)
Substituting the ansatz 𝜙₂=𝑎cosh(𝑘𝑥+𝜔𝑡+𝑟)+𝑏sinh(𝑘𝑥+𝜔𝑡+𝑟)+𝑐 into (3.1) will produce the dispersion relation between 𝜔 and 𝑘, 𝜔+𝛼𝑘³=0,(3.3) and the new exact solutions are that 𝑢₂(𝑥,𝑡)=𝑎𝑘sinh𝑘𝑥−𝛼𝑘³𝑡+𝑟+𝑏𝑘cosh𝑘𝑥−𝛼𝑘³𝑡+𝑟𝑎cosh𝑘𝑥−𝛼𝑘³𝑡+𝑟+𝑏sinh𝑘𝑥−𝛼𝑘³𝑡+𝑟+𝑐.(3.4)
Taking the seed solution 𝑢₁=𝑐 for (1.1) will get 𝜙_𝑡+3𝛼𝑐²𝜙_𝑥+𝑐𝜙_𝑥𝑥+𝛼𝜙_𝑥𝑥𝑥=0.(3.5)
Observing (3.5), we have 𝜙₃=𝑘𝑥−3𝛼𝑐²𝑘𝑡+𝑟, 𝜙₄=𝑎+𝑒^{𝑘𝑥−(3𝛼𝑐2𝑘+3𝛼𝑐𝑘2+𝛼𝑘3)𝑡+𝑟} and 𝜙₅=sinh[𝑘𝑥−(3𝛼𝑐²𝑘+3𝛼𝑐𝑘²+𝛼𝑘³)𝑡+𝑟]+cosh[𝑘𝑥−(3𝛼𝑐²𝑘+3𝛼𝑐𝑘²+𝛼𝑘³)𝑡+𝑟]+𝑏.
Substituting 𝜙₃, 𝜙₄, 𝜙₅ and 𝑢₁=𝑐 into the Bäcklund transformation (2.11), we obtain the exact solutions for STO equation (1.1) 𝑢₃(𝑥,𝑡)=𝑘𝑢𝑘𝑥−3𝛼𝑐²𝑘𝑡+𝑟+𝑐,₄(𝑥,𝑡)=𝑘𝑒^{𝑘𝑥−(3𝛼𝑐2𝑘+3𝛼𝑐𝑘2+𝛼𝑘3)𝑡+𝑟}𝑎+𝑒^{𝑘𝑥−(3𝛼𝑐2𝑘+3𝛼𝑐𝑘2+𝛼𝑘3)𝑡+𝑟}𝑢+𝑐,₅(𝑥,𝑡)=𝑘sinh𝑘𝑥−3𝛼𝑐²𝑘+3𝛼𝑐𝑘²+𝛼𝑘³𝑡+𝑟+𝑘cosh𝑘𝑥−3𝛼𝑐²𝑘+3𝛼𝑐𝑘²+𝛼𝑘³𝑡+𝑟sinh𝑘𝑥−3𝛼𝑐²𝑘+3𝛼𝑐𝑘²+𝛼𝑘³𝑡+𝑟+cosh𝑘𝑥−3𝛼𝑐²𝑘+3𝛼𝑐𝑘²+𝛼𝑘³𝑡+𝑟+𝑏+(𝑐3..6)
Taking the seed solution 𝑢₁=1/𝑥 for STO equation (1.1), (2.12) becomes 𝜙_𝑡+3𝛼𝜙^𝑥𝑥𝑥+𝛼𝜙^𝑥𝑥𝑥=0.(3.7)
Substitute the ansatz 𝜙₆=𝑘₁𝑥³+𝑘₃𝑥−24𝛼𝑘₁𝑡+𝑏 into (2.11), yielding the solutions for the STO equation (1.1) 𝑢₆(𝑥,𝑡)=3𝑘¹𝑥²+𝑘₃𝑘₁𝑥³+𝑘₃𝑥−24𝛼𝑘₁𝑡+𝑏+𝑥1.(3.8)

Remark 3.1. When 𝑎=0 or 𝑏=0, the solution 𝑢₂(𝑥,𝑡) is obtained in [1] by tanh method. In [4], the authors have obtained the 𝑢₄(𝑥,𝑡). As for the case 𝑢₀=2𝜙_𝑥, 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 𝜙₁=sinh(𝑘𝑥+𝜔𝑡+𝑟)+cosh(𝑘𝑥+𝜔𝑡+𝑟). Substituting 𝜙₁ into (2.21) and seting the coefficients to be equal to zero, we have the dispersion relation between 𝜔 and 𝑘, 𝜔=−𝛼𝑘³.(3.9)
Thus, (2.20) yields the new exact solutions for the STO equation (1.1) 𝑢₇=2𝑘sinh𝑘𝑥−𝛼𝑘³𝑡+𝑟+𝑘cosh𝑘𝑥−𝛼𝑘³𝑡+𝑟sinh𝑘𝑥−𝛼𝑘³𝑡+𝑟+cosh𝑘𝑥−𝛼𝑘³𝑡+𝑟+𝑏−𝑘.(3.10)
Substituting ansatz 𝜙₂=𝑒^{𝑘𝑥+𝜔𝑡+𝑟}+𝑏 into (2.21) and balancing the coefficients, we have the new exact solutions for the STO equation (1.1) 𝑢₇(𝑥,𝑡)=2𝑘𝑒^{𝑘𝑥−𝛼𝑘3𝑡+𝑟}𝑒^{𝑘𝑥−𝛼𝑘3𝑡+𝑟}+𝑏−𝑘.(3.11)

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).