# Poincare-lighthill-kuo method and symbolic computation

## Abstract

This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi-inverse algorithm was proposed by the author, with which the lengthy parts of intermediate expressions are first frozen in the form of symbols till the final stage of seeking perturbation solutions. To discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer-extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of*1* and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head-on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher-order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro-commputers. Finally, it is concluded that with the aid of symbolic computation, the vitality of the PLK method is greatly strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.

## Key words

PLK method perturbation methods symbolic computation intermediate expression swell semi-inverse algorithm## CLC number

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## References

- [1]DAI Shi-qiang. PLK Method [A]. In: CHIEN Wei-zang Ed.
*Singular Perturbation Theory and Its Applications in Mechanics*[M]. Beijing: Science Press, 1981:33–86. (in Chinese)Google Scholar - [2]Poincare H.
*New Methods of Celestial Mechanics*[M]. NASA TTF-450, 1967.Google Scholar - [3]Lighthill M J. A technique for rendering approximate solutions to physical problems uniformly valid [J].
*Phil Mag*, 1949,**40**(5):1179–1120.MATHMathSciNetGoogle Scholar - [4]Kuo Y H. On the flow of an incompressible viscous fluid past a flat plate at moderate Reynolds numbers [J].
*J Math and Phys*, 1953,**32**(1):83–51.MATHGoogle Scholar - [5]Kuo Y H. Viscous flow along a flat plate moving at high supersonic speeds [J].
*J Aero Sci*, 1956,**23**(1):125–136.MATHMathSciNetGoogle Scholar - [6]
- [7]DAI Shi-qiang. On the generalized PLK method and its applications [J].
*Acta Mech Sinica*, 1990,**6**(2):111–118.Google Scholar - [8]DAI Shi-qiang. Generalization of the method of full approximation and its applications [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1991,**12**(3):255–264.MathSciNetCrossRefGoogle Scholar - [9]DAI Shi-qiang, Sigalov G F, Diogenov A V. Approximate analytical solutions for some strongly nonlinear problems [J].
*Scientia Sinica Ser A*, 1991,**34**(7):843–853.Google Scholar - [10]DAI Shi-qiang. Head-on collisions between two interfacial solitary waves [J].
*Acta Mechanica Sinica*, 1983,**15**(6):623–632. (in Chinese)Google Scholar - [11]DAI Shi-qiang. The interaction of two pairs of solitary waves in a two-fluid system [J].
*Scientia Sinica Ser A*, 1984,**27**(5):507–520.Google Scholar - [12]DAI Shi-qiang. The generalized Boussinesq equations and obliquely interacting solitary waves in a stratified fluid [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1984,**5**(4):1469–1478.MathSciNetCrossRefGoogle Scholar - [13]DAI Shi-qiang, ZHANG She-guang. Interaction of convex and concave solitary waves in a stratified fluid [J].
*Kexue Tonbao (Chinese Science Bulletin)*, 1987,**32**(9):589–593.Google Scholar - [14]ZHANG She-guang, DAI Shi-qiang. On chasing collisions between solitary waves of different modes in a stratified fluid [J].
*J Shanghai Univ of Technol*, 1986,**7**(4):375–383. (in Chinese)Google Scholar - [15]ZHANG She-guang, DAI Shi-qiang. Overtaking collisions between solitary waves of same mode in a stratified fluid [J].
*J Comm Appl Math Comput*, 1986,**1**(1):61–69. (in Chinese)Google Scholar - [16]LIU Yu-lu, DAI Shi-qiang. Second-order cnoidal waves at the free surface and interface of a two-fluid system [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1987,**8**(6):497–502.CrossRefGoogle Scholar - [17]ZHU Yong, DAI Shi-qiang. Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1989,**10**(3):213–220.MathSciNetCrossRefGoogle Scholar - [18]DAI Shi-qiang. On the oscillatory interfacial solitary waves [J].
*J Hydrodyn Ser A*, 1992,**7**(1): 1–6. (in Chinese)Google Scholar - [19]ZHU Yong, DAI Shi-qiang. On head-on collision between two gKdV solitary waves in a stratified fluid [J].
*Acta Mechanica Sinica*, 1991,**7**(4):300–308.Google Scholar - [20]ZHU Yong. Head-on collision between two mKdV solitary waves in a two-fluid system [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1992,**13**(5):407–417.MathSciNetGoogle Scholar - [21]DAI Shi-qiang, ZHU Yong. Perturbation solution of gKdV equation and interaction of gKdV solitary waves [A]. In: S Xiao, X Hu, Eds.
*Nonlinear Problems in Engineering and Science*[C]. Beijing, New York: Science Press, 1992.Google Scholar - [22]TANG Ling, DAI Shi-qiang. A class of asymptotic solutions for the KdV-Burgers equation: Monotone shock wave [A]. In: HUANG Qian, PAN Li-zhou Eds.
*Applied Mathematics and Mechanics*(Proc for the Memory of the 80th Anniv of Prof C*HIEN*Wei-zang) [C]. Beijing: Science Press, Chongqing: Chongqing Press, 1993, 400–404. (in Chinese)Google Scholar - [23]ZANG Hong-ming, DAI Shi-qiang. Solution for a nonlinear oscillation equation by computer algebra [J].
*J Shanghai Univ*, 1993,**14**(3):189–197. (in Chinese)Google Scholar - [24]ZANG Hong-ming, Study of some problems in mechanics via computer algebra and perturbation techniques [D]. MSc thesis. Shanghai: Shanghai University of Technology, 1993. (in Chinese)Google Scholar
- [25]WANG Ming-qui, DAI Shi-qiang. Computer-extended perturbation solution to Duffing equation [J].
*J Shanghai Univ of Technol*, 1994,**15**(3):384–389. (in Chinese)Google Scholar - [26]WANG Ming-qi, DAI Shi-qiang. Computer algebra-perturbation solution to a nonlinear wave equation [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1995,**16**(5):429–436.MathSciNetGoogle Scholar - [27]ZANG Hong-ming, DAI Shi-qiang. Higher-order solutions for interfacial solitary waves in a twofluid system [A]. In: Editorial Board of Journal of Hydrodynamics., Ed.
*Proc. 1st Int Conf on Hydrodynamics*[C]. Beijing: China Ocean Press, 1994.Google Scholar - [28]DAI Shi-qiang, ZANG Hong-ming. Investigation of internal solitary waves via computer algebra [J].
*J Nature*, 1995,**17**(3):177–179. (in Chinese)Google Scholar - [29]TIAN Mei. The nineth order computer algebra perturbation solution for Klein-Gordon equation [A]. In: AAI Shi-qiang, LIU Zeng-rong, HANG Qian Eds.
*Modern Mathematics and Mechanics (MMM-VI)*[C]. Suzhou: Soochow Univ Press, 1995. (in Chinese)Google Scholar - [30]CHENG You-liang. The evolution equation for second-order internal solitary waves in stratified fluid of great depth [J].
*J Shanghai Univ*, 1997,**1**(2):130–134. (in Chinese)MathSciNetGoogle Scholar - [31]CHENG You-liang, DAI Shi-qiang. Higher-order solutions for internal solitary waves via symbolic computation [A]. In: H Kim, S H Lee, S J Lee, Eds.
*Proc 3rd Int Conf on Hydrodynamics*[C]. Seoul: Ulam Publishers, 1998.Google Scholar - [32]CHENG You-liang. Theoretical analysis and symbolic computation of internal solitary waves in stratified fluids [D]. Ph D thesis. Shanghai:
*Shanghai University*, 1998. (in Chinese)Google Scholar - [33]DAI Hui-hui, DAI Shi-qiang, HUO Yi. Head-on collision between two solitary waves in a compressible Mooney-Rivlin elastic rod [J].
*Wave Motion*, 2000,**32**(1):93–111.MathSciNetGoogle Scholar - [34]DAI Shi-qiang. Study of problems in nonlinear mechanics via computer algebra [A]. In: DAI Shiqiang, LIU Zeng-rong, HUANG Qian Eds.
*Modern Mathematics and Mechanics (MMM-VI)*[C]. Suzhou: Soochow Univ Press, 1995. (in Chinese)Google Scholar - [35]Calmet J, Van Hulzen J A. Computer algebra applications [A]. In: B Buchberger, G E Collins, R Loos Eds.
*Computer Algebra Symbolic and Algebraic Computation*[M]. Beijing: World Publishing Corporation, 1988.Google Scholar - [36]Beltzer A I B. Engineering analysis via symbolic computation: a breakthrough [J].
*Appl Mech Rev*, 1990,**403**(6):119–127.MathSciNetGoogle Scholar - [37]Heck A.
*Introduction to MAPLE*[M]. New York: Springer-Verlag, 1993.Google Scholar - [38]Rand H R.
*Armbruster D. Perturbation Methods, Bifurcation Theory and Computer Algebra*[M]. New York: Springer-Verlag, 1987.Google Scholar - [39]DAI Shi-qiang, ZANG Hong-ming. A semi-inverse algorithm in application of computer algebra [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1997,**18**(2):113–120.MathSciNetGoogle Scholar - [40]DAI Shi-qiang. Reductive perturbation method and analyses of far fields of nonlinear waves [J].
*Advan in Mech*, 1982**12**(2):2–22. (in Chinese)Google Scholar - [41]DAI Shi-qiang. Solitary waves at the interface of a two-layer fluid [J].
*Applied Mathematics and Mechanics*(*English Edition*), 1982,**3**(6):777–788.CrossRefGoogle Scholar