Discrete Dynamics in Nature and Society
Volume 2012, Issue 1 872385
Research Article
Open Access

The Painlevé Tests, Bäcklund Transformation and Bilinear Form for the KdV Equation with a Self-Consistent Source

Yali Shen, 

Corresponding Author

Yali Shen

Department of Mathematics, Yuncheng University, Yuncheng 044000, China ycu.edu.cn

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Fengqin Zhang, 

Fengqin Zhang

Department of Mathematics, Yuncheng University, Yuncheng 044000, China ycu.edu.cn

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Xiaomei Feng, 

Xiaomei Feng

Department of Mathematics, Yuncheng University, Yuncheng 044000, China ycu.edu.cn

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First published: 13 May 2012
https://doi.org/10.1155/2012/872385
Citations: 1
Academic Editor: Beatrice Paternoster

Abstract

The Painlevé property and Bäcklund transformation for the KdV equation with a self-consistent source are presented. By testing the equation, it is shown that the equation has the Painlevé property. In order to further prove its integrality, we give its bilinear form and construct its bilinear Bäcklund transformation by the Hirota′s bilinear operator. And then the soliton solution of the equation is obtained, based on the proposed bilinear form.

1. Introduction

It is well known that some nonlinear partial differential equations such as the soliton equations with self-consistent sources have important physical applications. In recent years, there are many ways for solving the soliton equations that can be used to the soliton equations with self-consistent sources as well. For example, the soliton solutions of some equations such as the KdV, AKNS, and nonlinear schrödinger equation with self-consistent sources are obtained through the inverse scattering method [1, 2]. In [3] a Darboux transformation, positon and negaton solutions to a Schrödinger self-consistent source equation are further constructed. Also, the binary Darboux transformations for the KdV hierarchies with self-consistent sources were presented in [4]. In addition to that, the Hirota bilinear method has been successfully used in the search for exact solutions of continuous and discrete systems, and also in the search for new integrable equations by testing for multisoliton solutions or Bäcklund transformation [5, 6]. Recently a bilinear Bäcklund transformation has been presented for a (3 + 1)-dimensional generalized KP equation. Meanwhile, two classes of exponential and rational traveling wave solutions with arbitrary wave numbers are computed by applying the proposed bilinear Bäcklund transformation (see [7] for details). It is a good reference for solving many high-dimensional soliton equations.

Besides, the Painlevé analysis is a powerful tool for identifying the integrability of a nonlinear system. A partial different equation has the Painlevé property when the solutions of the partial different equation are single-valued about the movable, singularity manifold [8]. The basic thought as follows: if the singularity manifold is determined by ϕ(z1, z2, …, zn) = 0 and u = u(z1, z2, …, zn) is a solution of the partial different equation, then we assume that
(1.1)
where
(1.2)

are analytic functions of (zj) in a neighborhood of the manifold ϕ = 0 and α is an integer. Substitution of (1.1) into the partial different equation determines the values of α and defines the recursion relations for uj,  j ≠ 0,1, 2, ⋯. When the anatz (1.1) is correct, the pde is said to possess the Painlevé property and is conjectured to be integrable [9].

Motivated by the previous works, we focus our attention on the following nonlinear partial differential equations (PDEs) which is expressed by
(1.3)
where λ is an arbitrary constant. In fact, (1.3) is a reduced form of the KdV equation with a source by symmetry constraints [10, 11]. The main purpose of this paper is to demonstrate the connection between the Painlevé property and the Bäcklund transformation for (1.3). Moreover, we get the bilinear Bäcklund transformation and the exact solution for (1.3) by the Hirota bilinear method. Thus we further convince the integrability of the equation.

The paper is organized as follows. In Section 2, we investigate the Painlevé property for (1.3). By testing the equation it is shown that the equation has the Painlevé property. Furthermore, we obtain a Bäcklund transformation of (1.3). In Section 3, using the Hirota’s bilinear operator, we obtain its bilinear form and construct its bilinear Bäcklund transformation. And then its one-soliton solution is obtained. Finally, conclusion is given in Section 4.

2. PainlEvÉ Test

As we know, the basic Painlevé test for ODEs consists of the following steps [12].

Step 1. Identify all possible dominant balances, that is, all singularities of form u ~ u0(z − z0) μ.

Step 2. If all exponents μ are integers, find the resonances where arbitrary constants can appear.

Step 3. If all resonances are integers, check the resonance conditions in each Laurent expansion.

Conclusion. If no obstruction is found in Steps 1–3 for every dominant balances, then the Painlevé test is satisfied.

The above series may be substituted into the PDEs. Now we apply the above steps to (1.3). We will further give all possible solutions with integer resonances but without further analysis of the last cases. The expansions about the singular manifold have the forms:

(2.1)
To find the dominant balances, we are looking for leading order singular behaviour of the form
(2.2)
And the derivatives of (2.2) are given by
(2.3)
Substituting (2.2)-(2.3) into (1.3), we get the following forms
(2.4)
Calculating and simplifying equation (2.4), we get the dominant balances
(2.5)
where . So we complete the first step.

The second step in applying the Painlevé test is to find the resonances. To find the resonances numbers j, we substitute (2.1) into (1.3), and collecting terms of each order of ϕ, we obtain

ϕ−5:

(2.6)
By calculating equation (2.6), we get .

ϕ−4:

(2.7)
From (2.6) and (2.7), we have , , 6b1 = ±(15ϕxx − 17a1), where a0 and b0 are consistent with step one.

ϕ−3:

(2.8)
From (2.7) and (2.8), we get a1 = 3ϕxx, b1 = ∓6ϕxx,

ϕ−2:

(2.9)
From (2.8) and (2.9), we get
(2.10)

ϕj−5:

(2.11)

ϕj−4:

(2.12)
Substituting into (2.11) and (2.12), the coefficients of aj and bj may be rearranged to give
(2.13)
(2.14)
simplifying (2.13)-(2.14), we have
(2.15)
(2.16)
There, it is found that the resonance occurs at j = 4, so the second step is completed.

For the last step, we will check the resonance conditions. So we need to find the orders in the expansion (2.1) where arbitrary constants may appear:

ϕ−1:

(2.17)
From (2.17), we know a4 and b4 are both arbitrary. Thus (1.3) possess the Painlevé property.

We now specialize (2.1) by setting the resonance functions a4 = b4 = 0. Furthermore, we require a3 = b3 = 0, it is easily demonstrated that aj = 0, bj = 0, j ≥ 3 from the recursion relations.

If a2 and b2 satisfy

(2.18)
we obtain a Bäcklund transformation of (1.3):
(2.19)
where we consider the case of , b1 = −6ϕxx moreover, u, φ and satisfy (1.3) and
(2.20)
Many studies [9, 13] show that a new solution can usually be obtained from a given solution of an equation if the so-called Bäcklund transformation for the equation is found. Therefore, it is worth to find the Bäcklund transformation of an equation. In the next section, we will give the bilinear Bäcklund transformation of (1.3).

3. Bilinear Form

As we know, when you want to use Hirota method, the first thing you need to do is to rewrite the equation under consideration as the bilinear form [14]. This can be achieved for (1.3) by the following dependent variable transformation:
(3.1)
Equation (1.3) can be written into bilinear forms
(3.2)
where D is the well-known Hirota bilinear operator
(3.3)

Now we will give the bilinear Bäcklund transformation of (1.3).

Theorem 3.1. Suppose that (f, g) is a solution of (3.2), then (f′, g′), satisfying the following relations:

(3.4)
(3.5)
(3.6)
(3.7)
is another solution of (3.2), where μ and ν are arbitrary constants and μ ≠ 0.

Proof. We consider the following:

(3.8)
We will show that (3.4)–(3.7) imply P1 = 0 and P2 = 0. We first work on the case of P1. We will use various bilinear identities which, for convenience, are presented in the appendix:
(3.9)
Next we come to the second part of the proof:
(3.10)
Thus we have completed the proof of Theorem 3.1.

We will show that our Bäcklund transformation (3.4)–(3.7) supplies us with a Lax representation for (1.3). Suppose
(3.11)
then the variables f, g, f′ and g′ can be eliminated from (3.4)–(3.7). The elimination results in
(3.12)
(3.13)
(3.14)
(3.15)
Therefore, we have the following.

Theorem 3.2. The compatibility condition of (3.12)–(3.15) is (1.3). In fact, using the compatibility conditions Xxxt = Xtxx, one can obtain (1.3) where ν and ω satisfy ν − 2ω = λ.

Finally we will give the soliton solution of the equation (1.3) by the standard perturbation method:
(3.16)

where , c, k are arbirtary constants and λ = k2.

4. Conclusion

In this paper, we investigate the Painlevé property for the KdV equation with a self-consistent source. By tests to the equation, it is shown that only the principal balance of the equation has the Painlevé property. While noninteger resonances are allowed with the weak extension of the Painlevé test [12]. We obtain the two different Bäcklund transformations. And then the soliton solution for (1.3) is given.

Acknowledgments

This work was supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young Foundation of Shanxi Province (no. 2011021001-1), Research Project Supported by Shanxi Scholarship Council of China (2011-093), and the Major Subject Foundation of Shanxi.

    Appendix

    In this appendix, we list the relevant bilinear identities, which can be proved directly. Here a, b, c, and d are arbitrary functions of the independent variables x and t:
    (A.1)
    (A.2)
    (A.3)
    (A.4)
    (A.5)

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