Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T06:22:51.586Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
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 [email protected]
HON-WAH TAM
Affiliation:
Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong, PR [email protected]
YING-NAN ZHANG
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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