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Nonlinear self-adjointness and conservation laws of the variable coefficient combined KdV equation with a forced term
- Lihua Zhang^{1}Email author
https://doi.org/10.1186/s13662-015-0455-1
© Zhang; licensee Springer. 2015
Received: 4 February 2015
Accepted: 1 April 2015
Published: 25 July 2015
Abstract
In this paper, nonlinear self-adjointness and conservation laws for the variable coefficient combined KdV equation with a forced term are studied. We discuss its self-adjointness and find that the equation is nonlinearly self-adjoint. At the same time, the formal Lagrangian for the equation is obtained. Having performed Lie symmetry analysis for the equation, we derive several nontrivial conservation laws for the equation by using a general theorem on conservation laws, given by Ibragimov.
Keywords
- variable coefficient combined KdV equation
- forced term
- nonlinear self-adjointness
- conservation laws
- Lie symmetry analysis
MSC
- 35G20
- 35L65
- 58J70
1 Introduction
The notion of conservation laws plays an important role in the study of nonlinear science [1–3]. The existence of a large number of conservation laws of a partial differential equation (system) is a strong indication of its integrability. To search for explicit conservation laws of nonlinear partial differential equations (PDEs), a number of methods have been presented, such as Noether’s theorem [4], the multiplier approach [5, 6], the partial Noether approach [7], and so on [8–10]. Among those, the new conservation theorem given by Ibragimov is one of the most frequently used methods [11–25]. Based on the concept of adjoint equation for a given differential equation [9], Ibragimov gives a general conservation theorem by which conservation laws for the system consisting of the given equation and its adjoint equation can be obtained. In fact, we are only interested in the conservation laws for the given equation. Therefore one has to eliminate the nonlocal variable which is introduced in the adjoint equation. For a self-adjoint nonlinear equation, its adjoint equation is equivalent with the original equation after replacing the nonlocal variable with the dependent variable in the original equation. However, many equations, which have remarkable symmetry properties and physical significance, are not self-adjoint. Thus the nonlocal variables of these equations cannot be eliminated easily. To solve this problem, Ibragimov and Gandarias have extended the concept of self-adjoint equations by introducing the definitions of quasi-self-adjoint equations and weak self-adjoint equations [11–16]. Recently, Ibragimov [17] has introduced the concept of nonlinear self-adjointness, which includes the previous two concepts as particular cases and extends the self-adjointness to the most generalized meaning.
Exact solutions including many kinds of solitary wave-like solutions, quasi-periodical solutions and solitary wave solutions of Eq. (1) have been obtained in [26]. When \(R(t)=0\), \(a(t)\), \(m(t)\), and \(b(t)\) are constants, Eq. (1) becomes the constant coefficient combined KdV equation [27, 28]. If \(a(t)=a\), \(b(t)=b\), \(m(t)=0\), and \(R(t)=f(t)\), Eq. (1) becomes the special case of the forced KdV equation [25]. To the best of our knowledge, Lie symmetries and conservation laws of Eq. (1) when \(a(t)\neq0\), \(b(t)\neq0\), and \(m(t)\neq0\) have not been discussed up to now.
The rest of the paper is organized as follows. In Section 2, we introduce the main notations and theorems used in this paper. In Section 3, we first discuss the nonlinear self-adjointness for the combined KdV equation (1) and get its formal Lagrangian. In Section 4, after performing Lie symmetry analysis, nontrivial conservation laws of Eq. (1) are derived making use of the obtained formal Lagrangian and Lie symmetries. A discussion of the results and our conclusion are given in the last section.
2 Preliminaries
Definition 1
Definition 2
Definition 3
Theorem 1
In the following we recall the ‘new conservation theorem’ given by Ibragimov in [8].
Theorem 2
3 Nonlinear self-adjointness of Eq. (1)
The result obtained here is a special case of [19] when \(f(t,u)=g(t,u)=h(t,u)=0\), \(b(t,u)=-R(t)\), \(r(t,u)=b(t)\), and \(a(t,u)=a(t)u+m(t)u^{2}\), where \(a(t)\), \(m(t)\), \(b(t)\) and \(R(t)\) are the coefficient functions of Eq. (1). We have checked that \(\phi= M_{0} u-M_{0} \int{R(t)\, dt} \) satisfies Eqs. (22)-(26) in [19]. However, conservation laws for this special case is not studied in [19]. If \(a(t)=a\), \(b(t)=b\), \(m(t)=0\), and \(R(t)=f(t)\), the obtained result is the same as that obtained in [25] when the coefficient \(c=0\) in [25].
In summary, we have the following statements.
Theorem 3
Corollary 1
Remark 1
When the formal Lagrangian has the form of (13), the adjoint equation of Eq. (1) expressed by Eq. (8) and Eq. (1) are equivalent.
For simplicity, we take \(M_{0}=1\) in Eq. (13).
4 Lie symmetry analysis and conservation laws of Eq. (1)
In the above equation, \(a=a(t)\), \(b=b(t)\), \(m=m(t)\), and \(R=R(t)\). If \(a(t)=1\), \(m(t)=0\), and \(R(t)=0\), from the equation we can obtain the same result as that obtained in [29] when the coefficient \(a(t)=0\) in [29]. If \(a(t)=1\), \(m(t)=0\), \(b(t)=0\), and \(R(t)=0\), we can also obtain the same result as that obtained in [22] when the function \(a(u)=u\) in [22]. In this paper, we consider symmetries with the coefficients \(a(t)\neq0\), \(b(t)\neq0\), and \(m(t)\neq0\).
Solving Eq. (17) with the aid of Maple, we get the following cases.
Through analysis of self-adjointness, the adjoint equation (8) has become equivalent with Eq. (1). Using the formal Lagrangian and Lie symmetries of Eq. (1), conservation laws for Eq. (1) can be obtained by Theorem 2. According to the classifications of Lie symmetries, the conservation laws for Eq. (1) are as follows.
Remark 2
As Eq. (1) does not depend on x explicitly, \(V_{0}=\frac{\partial }{\partial x}\) is an obvious symmetry for any possible choice of the functions \(a(t)\), \(m(t)\), \(b(t)\), and \(R(t)\). In Case 1, we have checked that the conservation laws corresponding to \(V_{0}\) are trivial. In fact, in the other cases, the conservation laws corresponding to \(V_{0}\) are also trivial, we omit them for simplicity.
Remark 3
The conservation laws corresponding to \(V_{1}\)-\(V_{7}\) are nontrivial. The correctness of them has been checked by Maple software.
5 Conclusion
Conservation laws are used for the development of appropriate numerical methods and for mathematical analysis, in particular, existence, uniqueness and stability analysis. For the variable coefficient combined KdV equation (1) with a forced term, the constructing of conservation laws is not easy because of the arbitrariness of the variable coefficients \(a(t)\), \(b(t)\), \(m(t)\), and the forced term \(R(t)\). Through analysis of the self-adjointness, we show that Eq. (1) possesses nonlinear self-adjointness. This ensures that we can derive conservation laws of Eq. (1) by Theorem 2. After performing a Lie symmetry analysis, seven cases of Lie symmetries are obtained. Making use of the obtained Lie symmetries, nontrivial conservation laws for Eq. (1) are derived. These conservation laws may be useful for the explanation of some practical physical problems.
Declarations
Acknowledgements
The work is supported by the Natural Science Foundation of Shandong Province (ZR2013AQ005).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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