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WEAKLY NON-ASSOCIATIVE ALGEBRAS AND THE KADOMTSEV–PETVIASHVILI HIERARCHY

Published online by Cambridge University Press:  01 February 2009

ARISTOPHANES DIMAKIS  and
FOLKERT MÜLLER-HOISSEN
ARISTOPHANES DIMAKIS
Affiliation:
Department of Financial and Management Engineering, University of the Aegean, 31 Fostini Str., GR-82100 Chios, Greece e-mail: dimakis@aegean.gr
FOLKERT MÜLLER-HOISSEN
Affiliation:
Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, D-37073 Göttingen, Germany e-mail: folkert.mueller-hoissen@ds.mpg.de
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Abstract

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On any ‘weakly non-associative’ algebra there is a universal family of compatible ordinary differential equations (provided that differentiability with respect to parameters can be defined), any solution of which yields a solution of the Kadomtsev–Petviashvili (KP) hierarchy with dependent variable in an associative sub-algebra, the middle nucleus.

Keywords

37K10 17Axx
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue A , February 2009 , pp. 49 - 57
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Bogdanov, L. V. and Konopelchenko, B. G., Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies, J. Math. Phys. 39 (1998), 4701–4728.Google Scholar
2.Dickey, L. A., Soliton equations and Hamiltonian systems (World Scientific, Singapore, 2003).Google Scholar
3.Dimakis, A. and Müller-Hoissen, F., A new approach to deformation equations of noncommutative KP hierarchies, J. Phys. A: Math. Theor. 40 (2007), 7573–7596.Google Scholar
4.Dimakis, A. and Müller-Hoissen, F., Functional representations of integrable hierarchies, J. Phys. A: Math. Gen. 39 (2006), 9169–9186.Google Scholar
5.Dimakis, A. and Müller-Hoissen, F., Nonassociativity and integrable hierarchies, nlin.SI/0601001.Google Scholar
6.Dimakis, A. and Müller-Hoissen, F., Weakly nonassociative algebras, Riccati and KP hierarchies, nlin.SI/0701010, in Generalized Lie Theory in Mathematics, Physics and Beyond (Silvestrov, S., Paal, E., Abramov, V. and Stolin, A., Editors) (Springer, 2008) 9–27.Google Scholar
7.Dimakis, A. and Müller-Hoissen, F., With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices, J. Phys. A: Math. Theor. 40 (2007), F321–F329.CrossRefGoogle Scholar
8.Golubchik, I. Z., Sokolov, V. V. and Svinolupov, S. I., A new class of nonassociative algebras and a generalized factorization method, ESI preprint 53 (1993). ftp://ftp.esi.ac.at/pub/Preprints/esi053.pdf.Google Scholar
9.Kinyon, M. K. and Sagle, A. A., Quadratic dynamical systems and algebras, J. Diff. Equations 117 (1995), 67–126.Google Scholar
10.Kupershmidt, B. A., Mathematical surveys and monographs: KP or mKP, vol. 78 of (American Math. Society, Providence, RI, USA, 2000).Google Scholar
11.Kuz'min, E. N. and Shestakov, I. P, Non-associative structures, in Algebra VI (Kostrikin, A. I. and Shafarevich, I. R., Editors) (Springer, Berlin, 1995) 197–280.Google Scholar
12.Pflugfelder, H. O., Quasigroups and loops, vol. 7 (Heldermann, Berlin, 1990).Google Scholar
13.Sato, M. and Sato, Y., Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, in Nonlinear partial differential equations in applied science, vol. 5 (Fujita, H., Lax, P. D. and Strang, G., Editors) (North-Holland, Amsterdam, 1982) 259–271.Google Scholar