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A Bäcklund transformation and nonlinear superposition formula for the Lotka-Volterra hierarchy

Published online by Cambridge University Press:  17 February 2009

Xing-Biao Hu  and
Johan Springael
Xing-Biao Hu
Affiliation:
Academy of Mathematics and Systems Sciences, State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific Engineering Computing, Academia Sinica, P.O. Box 2719, Beijing 100080, P.R. China; e-mail: hxb@lsec.cc.ac.cn. Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury CT2 7NF, United Kingdom.
Johan Springael
Affiliation:
Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium; e-mail: jspringa@mach.vub.ac.be.
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Abstract

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A hierarchy of bilinear Lotka-Volterra equations with a unified structure is proposed. The bilinear Bäcklund transformation for this hierarchy and the corresponding canonical Lax pair are obtained. Furthermore, the nonlinear superposition formula is proved rigorously.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 121 - 128
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981).CrossRefGoogle Scholar
[2]Ablowitz, M. J., Kaup, D. J., Newell, A. C. and Segur, H., “The inverse scattering transform—Fourier analysis for nonlinear problems”, Stud. Appl. Math. 53 (1974) 249–315.CrossRefGoogle Scholar
[3]Bogoyavlensky, O. I., “Integrable discretizations of the KdV equation”, Phys. Lett. A 134 (1988) 34–38.Google Scholar
[4]Fokas, A. S. and Santini, P. M., “Recursion operators and bi-Hamiltonian structures in multidimensions. II”, Comm. Math. Phys. 116 (1988) 449–474.Google Scholar
[5]Fuchssteiner, B., “Application of hereditary symmetries to nonlinear evolution equations”, Nonlinear Anal. Theory Meth. Appl. 3 (1979) 849–862.CrossRefGoogle Scholar
[6]Hirota, R., “Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett. 27 (1971) 1192–1194.CrossRefGoogle Scholar
[7]Hirota, R., “Nonlinear partial difference equations I. A difference analogue of the Korteweg-de Vries equation”, J. Phys. Soc. Japan 43 (1977) 1424–1433.CrossRefGoogle Scholar
[8]Hirota, R., “Direct methods in soliton theory”, in Solitons (eds. Bullough, R. K. and Caudrey, P. J.), (Springer, Berlin, 1980).Google Scholar
[9]Hirota, R. and Satsuma, J., “N-soliton solutions of nonlinear network equations describing a Volterra system”, J. Phys. Soc. Japan 40 (1976) 891–900.Google Scholar
[10]Hirota, R. and Satsuma, J., “A variety of nonlinear network equations generated from the Bäcklund transformation for the Toda lattice”, Prog. Theor. Phys. Suppl. 59 (1976) 64–100.Google Scholar
[11]Hu, X. B., “Integrable systems and related problems”, Ph. D. Thesis, Computing Center of Academia Sinica, 1990.Google Scholar
[12]Hu, X. B., “Bilinearization of nonlinear integrable evolution equations—Recursion operator approach”, preprint, Computing Center of Academia Sinica, 1991.Google Scholar
[13]Hu, X. B., “Nonlinear superposition formulae for the differential-difference analogue of the KdV equation and two dimensional Toda equation”, J. Phys. A: Math. Gen. 27 (1994) 201–214.Google Scholar
[14]Hu, X. B. and Bullough, R. K., “Bäcklund transformation and nonlinear superposition formula of an extended Lotka-Volterra equation”, J. Phys. A: Math. Gen. 30 (1997) 3635–3641.CrossRefGoogle Scholar
[15]Hu, X. B. and Clarkson, P. A., “Rational solutions of a differential-difference KdV equation, the Toda equation and the discrete KdV equation”, J. Phys. A: Math. Gen. 28 (1995)5009–5016.Google Scholar
[16]Itoh, Y., “On a ruin problem with interaction”, Ann. Inst. Stat. Math. 25 (1973) 635–641.Google Scholar
[17]Itoh, Y., “Integrals of a Lotka-Volterra system of odd number of variables”, Prog. Theor. Phys. 78 (1987) 507–510.Google Scholar
[18]Manakov, S. V., “Complete integrability and stochastization of discrete dynamical systems”, Sov. Phys. JETP 40 (1974) 269–274.Google Scholar
[19]Matsuno, Y., Bilinear Transformation Method (Academic Press, New York, 1984).Google Scholar
[20]Nagai, A. and Satsuma, J., “The Lotka-Volterra equations and the QR algorithm”, J. Phys. Soc. Japan 64 (1995) 3669.Google Scholar
[21]Narita, K., “Soliton solution to extended Volterra equations”, J. Phys. Soc. Japan 51 (1982) 1682.Google Scholar
[22]Newell, A. C., Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985).CrossRefGoogle Scholar
[23]Olver, P. J., “Evolution equations possessing infinitely many symmetries”, J. Math. Phys. 18 (1977) 1212–1215.CrossRefGoogle Scholar
[24]Santini, P. M. and Fokas, A. S., “Recursion operators and bi-Hamiltonian structures in multidimensions. I”, Comm. Math. Phys. 115 (1988) 375–419.Google Scholar
[25]Willox, R., Lambert, F. and Springael, J., “Canonical bilinear systems and soliton resonances”, in Application of Analytic and Geometric Methods to Nonlinear Differential Equations (ed. Clarkson, P. A.), (Kluwer, Dordrecht, 1993) 257–270.Google Scholar
[26]Zakharov, V. E., Musher, S. L. and Rubenchik, A. M., “Nonlinear stage of parametric wave excitation in plasma”, JETP. Lett. 19 (1974) 151–153.Google Scholar
[27]Zhang, H. W., Tu, G. Z., Oevel, W. and Fuchssteiner, B., “Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure”, J. Math. Phys. 32 (1991) 1908–1918.Google Scholar