Abstract
We introduce a canonical Lie algebra in the direct sum of vector fields and 1-1-tensors, the perturbation bundle. This Lie algebra is extended to a full tensor structure and it is related to the Lie algebra obtained by coupling linear systems to nonlinear ones. Using Lie algebra isomorphisms from the original structure to the abstract perturbation bundle, new completely integrable systems are obtained. The formalism of Lax pairs is found to be a special case of the new structure.
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References
F.A. Berezin and A.M. Perelomov, Commun. Math. Phys., 74 (1980) 129–140.
F. Calogero and C. Nucci, J. Math. Phys., 32 (1991) 72–74.
W.L. Chan and Yu-kun Zheng, Inverse Problems, 7 (1991) 63–75.
H.H. Chen and Y. Lee, in “Advances in nonlinear waves” [Ed. L. Debnath], Pitman, Boston-London-Melbourne (1985) 233–239.
Chen Dengyuan and Zhang Hangwei, J. Phys. A: Math. Gen., 24 (1991) 377–383.
Y. Cheng, J. Math. Phys., 32 (1991) 157–162.
Y. Cheng, [Eds. G. Chaohao, L. Yishen, Tu. Guizhang], Research Reports in Nonlinear Physics, Springer-Verlag, Berlin-Heidelberg (1990) 12–22.
Y. Cheng, Physica, 46D (1990) 286–294.
A. Degasperis, “Linear evolution equations associated with isospectral evolutions of differential operators”, University of Rome, preprint 345 (1983)
B. Fuchssteiner, Nonlinear Analysis TMA, 3 (1979)849–862.
B. Fuchssteiner, Progr. Theor. Phys., 65 (1981) 861–876.
B. Fuchssteiner, Progr. Theor.Phys., 70 (1983) 1508–1522.
B. Fuchssteiner and G. Oevel, Reviews in Mathematical Physics, 1 (1990) 415–479.
B. Fuchssteiner, [Eds. S. Carillo and O. Ragnisco], Research reports in Physics — Nonlinear Dynamics,Springer-Verlag, Berlin-Heidelberg-New York (1990) pp114–122.
B. Fuchssteiner, in “Nonlinear Systems in the Applied Sciences” [Eds. C.A. Rogers and W. F. Ames], Mathematics in Science and Engineering, 185 Academic Press, Boston (1991) pp211–256.
B. Fuchssteiner, in “Groups and related Topics, Proceedings of the first Max Born Symposium” [Eds. R. Gielerak, J. Lukierski and Z. Popowicz], Kluwer, Dordrecht (1992) pp165–178.
B. Fuchssteiner, “Integrable Nonlinear Evolution Equations with time-dependent Coefficients”, preprint (1992)
Ma Wen-Xiu, Science in China, 34 (1991) 769–782.
Ma Wen-Xiu, J. Phys. A. Math. Gen., 23 (1990)2707–2716.
Ma Wen-Xiu, “Lax representations and Lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations”, Inst. Math. Fudan University, Shanghai, preprint (1991)
Ma Wen-Xiu, “The algebraic structures of isospectral Lax operators and applications to integrable systems”, Inst. Math. Fudan University, Shanghai, preprint (1991)
Ma Wen-Xiu, “An approach for constructing nonisospectral hierarchies of evolution equations”, Inst. Math. Fudan University, Shanghai, preprint (1991)
E. Nelson, “Tensor Analysis, Princeton University Press, Princeton NJ (1967)
W. Oevel, in “Topics in Soliton Theory and Exactly solvable Nonlinear equations” [Eds. M. Ablowitz, B. Fuchssteiner and M. Kruskal], World Scientific, Singapore (1987) pp108–124.
K.M. Tamizhmani and M. Laksmanan, J. Phys. A: Math. Gen., 16 (1983) 3773–3782.
G. Wilson, “On a so-called Lax equation of Berezin and Perelomov”, Math. Inst. St. Giles Oxford, unpublished note.
H. Zhang, G. Tu, W. Oevel and B. Fuchssteiner, J. Math. Phys., 32 (1991) 1908–1918.
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Fuchssteiner, B. (1993). Coupling of Completely Integrable Systems: The Perturbation Bundle. In: Clarkson, P.A. (eds) Applications of Analytic and Geometric Methods to Nonlinear Differential Equations. NATO ASI Series, vol 413. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2082-1_13
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DOI: https://doi.org/10.1007/978-94-011-2082-1_13
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