Abstract
We propose a new symmetry reduction method for (1+1)-dimensional differential-difference equations (DDEs), namely, the λ-symmetry reduction method of solving ordinary differential equations is generalized to DDEs. Order-reduction processes are a consequence of the invariance of the given DDE under vector fields of the new class. These vector fields satisfy a new prolongation formula. A simple example of order-reduction is provided to illustrate the application.
1 Introduction
Symmetry is closely related to the integrability of the nonlinear evolution equations (NLEEs) in various specific meanings. For example, the existence of infinite Lie-Bäcklund symmetry is a criterion for the integrability of NLEEs, so the study of symmetry of NLEEs is particularly important. The symmetry of the NLEEs is studied systematically by Lie point symmetry theory [1–3]. Although the Lie point symmetry method has relatively mature theories, it also has great limitations [1–10]. When a given NLEE does not allow enough non-trivial Lie point symmetries, this method cannot be applied. Therefore, it is necessary to extend the classical Lie point symmetry concept from various angles [11–20]. For example, if the infinitesimal also depends on the higher derivative, the corresponding Lie-Bäcklund symmetry is obtained [21, 22].
The concept of λ-symmetry proposed by Muriel and Romero [23], aims to show that many of the known order-reduction processes can be explained by the invariance of the equation under some special vector fields that are neither Lie symmetries nor Lie-Bäcklund symmetries. The λ-symmetry reduction method for ordinary differential equations (ODEs) has attracted the attention of more and more scientists [24]. For example, Levi and Rodriguez successfully extended this method to the case of difference equations [25]. Again, the μ-symmetry reduction method is used to deal with partial differential equations (PDEs) [26–30].
For the sake of readability, we will briefly introduce the λ-symmetry reduction method for ODEs in Section 2. Then we extend the λ-symmetry reduction method to the case of (1+1)-dimensional differential-difference equations (DDEs) in Section 3. The last section is devoted to conclusions and discussions.
2 The λ-symmetry reduction method of ODEs
In this section we briefly review the λ-symmetry reduction method of ODEs. For a given mth-order ODEwe can set a vector fieldwhere means the ith-order derivative with respect to the independent variable x. Thus we can construct high-order infinitesimal prolongation vector fieldwhere
Here Dx means the total derivative with respect to x. So the invariance of Eq. 1 needs
Solving this equation, the expressions for X and U can be derived. For complex high-order ODEs or systems, we need to use symbolic computing software to calculate X and U. Theoretically, all of the similarity variables be derived by solving the following characteristic equationand then we can reduce and solve Eq. 1.
The above method is the Lie point symmetry method, also known as the classical symmetry reduction method. In Ref. [23], authors have introduced a new class of symmetries, that strictly includes Lie point symmetries, for which there exists an algorithm that lets us reduce the order of a given ODE. This method is now called the λ-symmetry reduction method. The key step of this generalized method is that the infinitesimal prolongation is modified to the following form
where λ is a smooth function that is determined simultaneously with the coefficients of the infinitesimal generators X and U. Thus the infinitesimal prolongation vector field is modified to
The following theorem that is important for the λ-symmetry reduction method, which is first obtained by Muriel and Romero [23].
(Muriel, Romero [23]).Let us suppose that, for some smooth functionsλ, the vector fieldvis aλ-symmetry of the following ODE
Thenfor some smooth functionsμ. HereAis the vector field of Eq. 9,
Conversely, ifis a vector field such that
for some smooth functionsλ, μ, then the vector fieldis aλ-symmetry of Eq. 9andK = v[λ,(m−1)].
3 The λ-symmetry reduction method of DDEs
In this section, we extend the λ-symmetry reduction method to the case of (1+1)-dimensional DDEs.
Definition 1For the following (1+1)-dimensional DDE with a discrete variablenand a continuous variablex,where, the vector fieldis said to beλ-symmetry for this equation if there exists a differential functionλsuch that themthλ-prolongation of the vector field satisfies.Particularly, for the following (1+1)-dimensional DDEwe can set a vector fieldHere is for ease of writing. So we have Theorem 2.
Let us suppose that, for some differential functionsλ, the vector fieldvis aλ-symmetry of the following DDE
Thenfor some differential functionsμ. HereAis the vector field of Eq. 19,
Conversely, ifis a vector field such thatfor some differential functionsλandμ, then the vector fieldis aλ-symmetry of Eq. 19andK = v[λ,(m−1)].
ProofCompute [v[λ,(m−1)], A] as a function of at each lattice point, withandSince v is a λ-symmetry,Hence, if , Eq. 26 says thatIf we set μ = −A(X(x)) − λX(x), then we can writeTherefore, we conclude that [v[λ,(m−1)], A] = λ ⋅ v[λ,(m−1)] + μ ⋅ A.The vector fielddepends on three lattice points with n − 1, n and n + 1. If we apply both elements of this equation to each coordinate function, we obtainand, for 0 ≤ i ≤ m − 2, the coordinate of K must satisfyHenceThen we apply both elements of [K, A] = λK + μA, to the coordinate function , and , we obtainwhere k = −1, 0, 1. The above equation yieldsCalculatewhen , we obtain, by Eq. 35, thatTherefore v is a λ-symmetry of Eq. 19.In order to reduce the mth-order DDEs to (m − 1)th-order DDEs and first-order DDEs, we can determine invariants for the λ-prolongation of v by deriving invariants of lower order. This can be achieved through the application of the main tools, Theorem 2.
Letvbe a vector field defined onMand letλis a differential function, Ifare such thatthen
Proof 3By Theorem 2, we havewhere μ = −Dx(v(x)) − λv(x). Therefore,
Proposition 1Letvbe aλ-symmetry. Letbe two functionally independent first-order invariants ofv[λ,(m)]. By solving an equation ofand an auxiliary equation, the general solution of the equation can be obtained.With the help of independent first-order invariant, we demonstrate a simple application of λ-symmetry. Considering a (1+1)-dimensional DDEEq. 43 has the fromwhich admits the obvious order reductionLetting X(x) = 0, Un−1(x, un−1) = 1, Un(x, un) = 1, Un+1(x, un+1) = 1 and , we have the following λ-prolongation vector fieldWe can easily prove that the vector field v is the λ-symmetry of Eq. 43. The λ-symmetry generator has two obvious invariants z = x, . Furthermore, the differential invariant . Therefore, Eq. 43 can be reduced to Eq. 45.
4 Conclusion
λ-symmetry reduction method is useful in establishing effective alternative methods analyze ODEs without using Lie point symmetries. At present, there is no programmatic algorithm package to solve λ-symmetry directly. Therefore, it is difficult to determine the general form of λ.
There are many examples of DDEs, without Lie point symmetries, that can be completely integrated. So we have to study the reduction of these DDEs. In this paper, we have extended the λ-symmetry reduction method to the case of (1+1)-dimensional DDEs. We have obtained some theorems Theorem 2, 3 and Proposition 1 which can be used to reduce and solve DDEs in Section 3. By comparison, DDEs can be more complex. Here we have just listed a simple example to illustrate the method. How to combine the integrating factor method and the λ-symmetry reduction method of DDEs to construct more effective examples will be the next work.
Statements
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Author contributions
JL: Methodology, software, formal analysis, writing-original draft. KS: Investigation, formal analysis, writing-original draft. BR: software, formal analysis. YJ: Conceptualization, funding acquisition, resources, supervision, writing-review and editing. All authors contributed to the article and approved the submitted version.
Acknowledgments
The authors thank Prof. S.F. Shen for their helpful discussions.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
λ-symmetry, differential-difference equation, order-reduction, vector field, reduction method
Citation
Lyu J, Shi K, Ren B and Jin Y (2023) New symmetry reduction method for (1+1)-dimensional differential-difference equations. Front. Phys. 11:1237805. doi: 10.3389/fphy.2023.1237805
Received
10 June 2023
Accepted
15 June 2023
Published
03 July 2023
Volume
11 - 2023
Edited by
Xiangpeng Xin, Liaocheng University, China
Reviewed by
Yongshuai Zhang, Zhejiang University of Science and Technology, China
Biao Li, Ningbo University, China
Updates
Copyright
© 2023 Lyu, Shi, Ren and Jin.
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*Correspondence: Bo Ren, renbosemail@163.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.