对称理论在解若干非线性问题中的应用
PDF下载 (291)刘希忠.对称理论在解若干非线性问题中的应用[J].宁波大学学报(理工版),2020,33(5):77-82.DOI:
LIU Xizhong.Application of symmetry theory in solving nonlinear problems[J].Journal of Ningbo University(Natural Science & Engineering Edition),2020,33(5):77-82.DOI:
| Title: | Application of symmetry theory in solving nonlinear problems |
| 作者: | 刘希忠 |
| Author(s): | LIU Xizhong |
| 关键词: | 对称; 摄动理论; 近似约化解; 守恒律; 同伦近似对称 |
| Keywords: | symmetry; perturbation theory; approximate reduction solution; conservation law; approximate homotopy method |
| 分类号: | O411.1 |
| 文献标识码: | A |
| 摘要: | 将摄动理论和对称约化理论结合起来对研究扰动非线性方程具有重要的意义. 本文利用近似对称约化理论研究了扰动mKdV方程, 得到了该方程的各阶近似约化方程和级数约化解. 本文还讨论了同伦近似对称方法在求解不可积系统中的应用以及利用对称和守恒律的关系求解非线性系统的无穷多守恒律等问题. |
| Abstract: | The approximate symmetry method, which combines perturbation theory and Lie symmetry approach, proves very effective in solving perturbed nonlinear systems. In this paper, we investigate the perturbed mKdV equation using the approximate symmetry method to obtain its symmetry reduction equations of different orders and series reduction solutions. Also, we discuss the role of approximate homotopy symmetry method in solving non-integrable systems and how to acquire infinitely many conservation laws using the relationship between symmetry and conservation laws. |
| 参考文献 /References: | [1] Lou S Y, Tang X Y, Lin J. Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method[J]. Journal of Mathematical Physics, 2000, 41(12):8286-8303. [2] Lou S Y, Tang X Y. Conditional similarity reduction approach: Jimbo-Miwa equation[J]. Chinese Physics, 2001, 10(10):897-901. [3] Tang X Y, Lin J, Lou S Y. Conditional similarity solutions of the Boussinesq equation[J]. Communications in Theoretical Physics, 2001, 35(4):399-404. [4] Lou S Y, Tang X Y. Equations of arbitrary order invariant under the Kadomtsev-Petviashvili symmetry group[J]. Journal of Mathematical Physics, 2004, 45(3):1020-1030. [5] Lou S Y, Ma H C. Non-Lie symmetry groups of (2+1)- dimensional nonlinear systems obtained from a simple direct method[J]. Journal of Physics A: Mathematical and General, 2005, 38(7):L129-L137. [6] Ma H C, Lou S Y. Solutions generated from the symmetry group of the (2+1)-dimensional sine-Gordon system[J]. Zeitschrift für Naturforschung A, 2005, 60(4): 229-236. [7] Lou S Y, Ma H C. Finite symmetry transformation groups and exact solutions of Lax integrable systems[J]. Chaos, Solitons & Fractals, 2006, 30(4):804-821. [8] Fushchich W I, Shtelen W M. On approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter[J]. Journal of Physics A: Mathematical and General, 1989, 22(18):L887-L890. [9] Jia M, Wang J Y, Lou S Y. Approximate symmetry reduction to the perturbed one-dimensional nonlinear Schr?dinger equation[J]. Chinese Physics Letters, 2009, 26(2):020201. [10] Yao R X, Jiao X Y, Lou S Y. Approximate symmetries and infinite series symmetry reduction solutions to perturbed Kuramoto-Sivashinsky equation[J]. Communications in Theoretical Physics, 2009, 51(5):785-788. [11] Jiao X Y, Yao R X, Lou S Y. Approximate similarity reduction for singularly perturbed Boussinesq equation via symmetry perturbation and direct method[J]. Journal of Mathematical Physics, 2008, 49(9):093505. [12] 廖世俊. 非线性问题的同伦分析技术[D]. 上海: 上海交通大学, 1992. [13] Hayat T, Sajid M. On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder[J]. Physics Letters A, 2007, 361(4/5):316-322. [14] Liao S J. Comparison between the homotopy analysis method and homotopy perturbation method[J]. Applied Mathematics and Computation, 2005, 169(2):1186-1194. [15] Liao S J. An explicit, totally analytic approximate solution for Blasius viscous flow problems[J]. Inter- national Journal of Non-Linear Mechanics, 1999, 34(4): 759-778. [16] Jiao X Y, Yao R X, Lou S Y. Approximate homotopy direct reduction method: Infinite series reductions to perturbed mKdV equations[J]. Chinese Physics Letters, 2009, 26(4):040202. [17] Jiao X Y, Gao Y, Lou S Y. Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation[J]. Science in China Series G: Physics, Mechanics and Astronomy, 2009, 52(8): 1169-1178. [18] Baikov V A, Gazizov R K, Ibragimov N H. Approximate symmetries of equations with a small parameter[J]. Mathematics of the USSR-Sbornik, 1989, 64:427-441. [19] Baikov V A, Gazizov R K, Ibragimov N H. Approximate Transformation Groups and Deformations of Symmetry Lie Algebras[M]. Boca Raton, Florida: CRC Press, 1995. [20] Euler N, Shul’ga M W, Steeb W H. Approximate symmetries and approximate solutions for a multi- dimensional Landau-Ginzburg equation[J]. Journal of Physics A: Mathematical and General, 1992, 25(18): L1095-L1103. [21] Euler M, Euler N, K?hler A. On the construction of approximate solutions for a multidimensional nonlinear heat equation[J]. Journal of Physics A: Mathematical and General, 1994, 27(6):2083-2092. [22] Euler N, Euler M. Symmetry properties of the approxi- mations of multidimensional generalized van der Pol equations[J]. Journal of Nonlinear Mathematical Physics, 1994, 1(1):41-59. [23] Pakdemirli M, Yürüsoy M, Dolap?? ? T. Comparison of approximate symmetry methods for differential equations [J]. Acta Applicandae Mathematicae, 2004, 80(3):243- 271. [24] Wiltshire R. Two approaches to the calculation of approximate symmetry exemplified using a system of advection-diffusion equations[J]. Journal of Computational and Applied Mathematics, 2006, 197(2):287-301. [25] Noether E. Invariante variations probleme[J]. Nachrichten von der Gesellschaft der Wissenschaften zu G?ttingen, 1918, 1:235-257. [26] Bessel-Hagen E. ?ber die erhaltungss?tze der elektro- dynamik[J]. Mathematische Annalen, 1921, 84(3/4):258- 276. [27] Ibragimov N H. Transformation Groups Applied to Mathematical Physics[M]. Dordrecht, Netherlands: Springer, 1985. [28] Olver P J. Applications of Lie Groups to Differential Equations[M]. New York: Springer, 1986. [29] Bluman G W, Kumei S. Symmetries and Differential Equations[M]. New York: Springer, 1989. [30] Andrzej T. Noether equations and conservation laws[J]. Communications in Mathematical Physics, 1967, 6:248- 261. [31] Anderson I M, Duchamp T E. Variational principles for second-order quasi-linear scalar equations[J]. Journal of Differential Equations, 1984, 51(1):1-47. [32] Kara A, Mahomed F. Relationship between symmetries and conservation laws[J]. International Journal of Theoretical Physics, 2000, 39(1):23-40. [33] Jia M, Gao Y, Lou S Y. Conservation laws of equation family with same Kac-Moody-Virasoro symmetry[J]. Physics Letters A, 2010, 374(15/16):1704-1711. |
| 备注/Memo: | 收稿日期: 2020－05－11. 宁波大学学报（理工版）网址: http://journallg.nbu.edu.cn/ 基金项目: 国家自然科学基金（11405110）. 作者简介: 刘希忠（1980－）, 男, 山东日照人, 副教授, 主要研究方向: 孤立子与可积系统. E-mail: liuxizhong123@163.com 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
