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A new method to generate non-autonomous discrete integrable systems via convergence acceleration algorithms

Published online by Cambridge University Press:  07 September 2015

YI HE ,
XING-BIAO HU ,
HON-WAH TAM  and
YING-NAN ZHANG
YI HE
Affiliation:
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, PR China email: heyi@lsec.cc.ac.cn
XING-BIAO HU
Affiliation:
LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, P.O.Box 2719, Beijing 100190, PR Chinahxb@lsec.cc.ac.cn
HON-WAH TAM
Affiliation:
Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong, PR Chinatam@comp.hkbu.edu.hk
YING-NAN ZHANG
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR Chinazhangyingnan@lsec.cc.ac.cn
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Abstract

In this paper, we propose a new algebraic method to construct non-autonomous discrete integrable systems. The method starts from constructing generalizations of convergence acceleration algorithms related to discrete integrable systems. Then the non-autonomous version of the corresponding integrable systems are derived. The molecule solutions of the systems are also obtained. As an example of the application of the method, we propose a generalization of the multistep ϵ-algorithm, and then derive a non-autonomous discrete extended Lotka–Volterra equation. Since the convergence acceleration algorithm from the lattice Boussinesq equation is just a particular case of the multistep ϵ-algorithm, we have therefore arrived at a generalization of this algorithm. Finally, numerical experiments on the new algorithm are presented.

Keywords

multistep ϵ-algorithmG-transformationLotka–Volterra equationLattice Boussinesq equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 2 , April 2016 , pp. 194 - 212
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Aitken, A. C. (1965) Determinants and Matrices, Oliver and Boyd, Edinburgh and London.Google Scholar
[2] Brezinski, C. (1972) Conditions d'application et de convergence de procédés d'extrapolation. Numer. Math. 20 (1), 64–79.Google Scholar
[3] Brezinski, C., He, Y., Hu, X. B., Redivo-Zaglia, M. & Sun, J. Q. (2012) Multistep ϵ-algorithm, Shanks' transformation, and the Lotka–Volterra system by Hirota's method. Math. Comp. 81 (279), 1527–1549.Google Scholar
[4] Brezinski, C., He, Y., Hu, X. B. & Sun, J. Q. (2010) A generalization of the G-transformation and the related algorithms. Appl. Numer. Math. 60 (12), 1221–1230.Google Scholar
[5] Brezinski, C. & Redivo-Zaglia, M. (1991) Extrapolation Methods: Theory and Practice, North-Holland, Amsterdam.Google Scholar
[6] Brualdi, R. A. & Schneider, H. (1983) Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir and Cayley. Linear Algebra Appl. 52/53, 769–791.Google Scholar
[7] Gilson, C. R. (2001) Generalizing of the KP hierarchies: Pfaffian hierarchies. Theor. Math. Phys. 133 (3), 1663–1674.Google Scholar
[8] Gilson, C. R. & Nimmo, J. J. C. (2001) Pfaffianization of the Davey-Stewartson equation. Theor. Math. Phys. 128 (1), 870–882.Google Scholar
[9] Gilson, C. R., Nimmo, J. J. C. & Tsujimoto, S. (2001) Pfaffianization of the discrete KP equation. J. Phys. A: Math. Gen. 34 (48), 10569–10575.Google Scholar
[10] He, Y., Hu, X. B., Sun, J. Q. & Weniger, E. J. (2011) Convergence acceleration algorithm via an equation related to the lattice Boussinesq equation. SIAM J. Sci. Comput. 33 (3), 1234–1245.Google Scholar
[11] He, Y., Hu, X. B. & Tam, H. T. (2009) A q-difference version of the ϵ-algorithm. J. Phys. A: Math. Theor. 42 (9), 095202.Google Scholar
[12] Hirota, R. (1981) Discrete analogue of a generalized Toda equation. J. Phys. Soc. Jpn. 50 (11), 3785–3791.Google Scholar
[13] Hirota, R. (2004) Direct Method in Soliton Theory, Cambridge University Press, Cambridge.Google Scholar
[14] Hirota, R. & Ohta, Y. (1991) Hierarchies of coupled soliton equations. I. J. Phys. Soc. Jpn. 60 (3), 798–809.Google Scholar
[15] Hu, X. B. & Wang, H. Y. (2006) Construction of dKP and BKP equations with self-consistent sources. Inverse Problems 22 (3), 1903–1920.Google Scholar
[16] Hu, X. B. & Wang, H. Y. (2007) New type of Kadomtsev-Petviashvili equation with self-consistent sources and its bilinear Bácklund transformation. Inverse Problems 23 (4), 1433–1444.Google Scholar
[17] Hu, X. B., Zhao, J. X. & Tam, H. W. (2004) Pfaffianization of the two-dimensional Toda lattice. J. Math. Anal. Appl. 296 (1), 256–261.Google Scholar
[18] Iwasaki, M. & Nakamura, Y. (2002) On the convergence of a solution of the discrete Lotka-Volterra system. Inverse Problems 18 (6), 1569–1578.Google Scholar
[19] Iwasaki, M. & Nakamura, Y. (2004) An application of the discrete Lotka-Volterra system with variable step-size to singular value computation. Inverse Problems 20 (2), 553–563.Google Scholar
[20] Minesaki, Y. & Nakamura, Y. (2001) The discrete relativistic Toda molecule equation and a Padé approximation algorithm. Numer. Algorithms 27 (3), 219–235.CrossRefGoogle Scholar
[21] Nagai, A. & Satsuma, J. (1995) Discrete soliton equations and convergence acceleration algorithms. Phys. Lett. A 209 (5–6), 305–312.Google Scholar
[22] Nagai, A., Tokihiro, T. & Satsuma, J. (1998) The Toda molecule equation and the ϵ-algorithm. Math. Comp. 67 (224), 1565–1575.Google Scholar
[23] Nakamura, Y. ed. (2000) Applied Integrable Systems (in Japanese), Syokabo, Tokyo.Google Scholar
[24] Nakamura, Y. (1999) Calculating Laplace transforms in terms of the Toda molecule. SIAM J. Sci. Comput. 20 (1), 306–317.Google Scholar
[25] Ohta, Y., Nimmo, J. J. C. & Gilson, C. R. (2001) A bilinear approach to a Pfaffian self-dual Yang-Mills equation. Glasg. Math. J. 43A, 99–108.Google Scholar
[26] Papageorgiou, V., Grammaticos, B. & Ramani, A. (1993) Integrable lattices and convergence acceleration algorithms. Phys. Letters A 179 (2), 111–115.Google Scholar
[27] Rutishauser, H. (1954) Der Quotienten-Differenzen-Algorithmus. Z. Angew. Math. Pysik. 5 (3), 233–251.Google Scholar
[28] Shanks, D. (1949) An analogy between transient and mathematical sequences and some nonlinear sequence-to-sequence transforms suggested by it. Part I. Memorandum 9994, Naval Ordnance Laboratory, White Oak.Google Scholar
[29] Shanks, D. (1955) Non linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, 1–42.Google Scholar
[30] Sidi, A. (2003) Practical Extrapolation Methods. Theory and Applications, Cambridge University Press, Cambridge.Google Scholar
[31] Sun, J. Q., Chang, X. K., He, Y. & Hu, X. B. (2013) An extended multistep shanks transformation and convergence acceleration algorithm with their convergence and stability analysis. Numer. Math. 125 (4), 785–809.Google Scholar
[32] Symes, W. W. (1981/1982) The QR algorithm and scattering for the nonperiodic Toda lattice. Phys. D 4 (2), 275–280.Google Scholar
[33] Tsujimoto, S., Nakamura, Y. & Iwasaki, M. (2001) The discrete Lotka-Volterra system computes singular values. Inverse Problems 17 (1), 53–58.Google Scholar
[34] Walz, G. (1996) Asymptotics and Extrapolation, Akademie Verlag, Berlin.Google Scholar
[35] Wang, H. Y., Hu, X. B. & Tam, H. W. (2007) A 2+1-dimensional Sasa-Satsuma equation with self-consistent sources. J. Phys. Soc. Jpn. 76 (2), 024007.Google Scholar
[36] Weniger, E. J. (1989) Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comp. Phys. Reports 10 (5–6), 189–371.Google Scholar
[37] Wimp, J. (1981) Sequence Transformations and Their Applications, Academic Press, New York.Google Scholar
[38] Wynn, P. (1956) On a device for computing the e m (S n ) transformation. MTAC 10 (54), 91–96.Google Scholar
[39] Wynn, P. (1956) On a procrustean technique for the numerical transformation of slowly convergent sequences and series. Proc. Cambridge Phil. Soc. 52 (4), 663–671.Google Scholar
[40] Zhao, J. X., Li, C. X. & Hu, X. B. (2004) Pfaffianization of the differential-difference KP equation. J. Phys. Soc. Jpn. 73 (5), 1159–1162.Google Scholar