Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T13:38:17.490Z Has data issue: false hasContentIssue false
  • English
  • Français

A NON-COMMUTATIVE SEMI-DISCRETE TODA EQUATION AND ITS QUASI-DETERMINANT SOLUTIONS

Published online by Cambridge University Press:  01 February 2009

C. X. LI and
J. J. C. NIMMO
C. X. LI
Affiliation:
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China e-mail: trisha_li2001@yahoo.com
J. J. C. NIMMO
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK e-mail: j.nimmo@maths.gla.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A non-commutative version of the semi-discrete Toda equation is considered. A Lax pair and its Darboux transformations and binary Darboux transformations are found and they are used to construct two families of quasi-determinant solutions.

Keywords

35Q5846L55

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue A , February 2009, pp. 121 - 127
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Dimakis, A. and Müller-Hoissen, F., An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions, J. Phys. A 38 (2005), 5453–5505.CrossRefGoogle Scholar
2.Etingof, P., Gelfand, I. M. and Retakh, V. S., Nonabelian integrable systems, quasideterminants, and Marchenko lemma, Math. Res. Lett. 5 (1998), 1–12.CrossRefGoogle Scholar
3.Gelfand, I. M., Gelfand, S., Retakh, V. S. and Wilson, R. L., Quasideterminants, Adv. Math. 193 (2005), 56–141.CrossRefGoogle Scholar
4.Gelfand, I. M. and Retakh, V. S., Determinants of matrices over noncommutative rings, Funct. Anal. Prilozhen. 25 (1991) 13–25.Google Scholar
5.Gilson, C. R. and Nimmo, J. J. C., On a direct approach to quasideterminant solutions of a noncommutative KP equation, J. Phys. A 40 (2007), 3839–3850.CrossRefGoogle Scholar
6.Gilson, C. R., Nimmo, J. J. C. and Ohta, Y., Quasideterminant solutions of a non-Abelian Hirota-Miwa equation, J. Phys. A: Math. Theor. 40 (2007), 12607–12617.CrossRefGoogle Scholar
7.Gilson, C. R., Nimmo, J. J. C. and Sooman, C. M., On a direct approach to quasideterminant solutions of a noncommutative modified KP equation, J. Phys. A: Math. Theor. 41 (2008), 085202 (10 pp.).CrossRefGoogle Scholar
8.Hamanaka, M. and Toda, K., Towards noncommutative integrable systems, Phys. Lett. A 316 (2003), 77–83.CrossRefGoogle Scholar
9.Hamanaka, M., Noncommutative solitons and D-branes, PhD Thesis (Nagoya University, Japan). arXiv:hep-th/0303256 (2003).Google Scholar
10.Hirota, R., The direct method in soliton theory (Nagai, A., Nimmo, J. J. C. and Gilson, C. R., Editors) (Cambridge University Press, Cambridge, UK, vol. 155, 2004).CrossRefGoogle Scholar
11.Inoue, R. and Hikami, K., Construction of soliton cellular automaton from the vertex model – the discrete 2D Toda equation and the Bogoyavlensky lattice, J. Phys. A: Math. Gen. 32 (1999), 6853–6868.CrossRefGoogle Scholar
12.Kupershmidt, B. A., KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems. Mathematical Surverys and Monographs (American Mathematical Society, New York, 78, 2000).CrossRefGoogle Scholar
13.Li, C. X. and Hu, X. B., Pfaffianization of the semi-discrete Toda equation, Phys. Lett. A 329 (2004), 193–198.CrossRefGoogle Scholar
14.Li, C. X. and Nimmo, J. J. C., Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation, Proc. R. Soc. A 464 (2008), 951–966.CrossRefGoogle Scholar
15.Mikhailov, A. V., Integrability of a two-dimensional generalization of the Toda chain, JETP Lett. 30 (1979), 443–448.Google Scholar
16.Nimmo, J. J. C., On a non-Abelian Hirota-Miwa equation, J. Phys. A 39 (2006), 5053–5065.CrossRefGoogle Scholar
17.Nimmo, J. J. C. and Willox, R., Darboux transformations for the two-dimensional Toda system, Proc. R. Soc. London Ser. A 453 (1997), 2497–2525.CrossRefGoogle Scholar
18.Paniak, L. D., Exact noncommutative KP and KdV multi-solitons, arXiv:hep-th/0105185 (2001).Google Scholar
19.Sakakibara, M., Factorization methods for noncommutative KP and Toda hierarchy, J. Phys. A 37 (2004), L599–L604.CrossRefGoogle Scholar
20.Wang, H. Y., Integrability of the semi-discrete Toda equation with self-consistent sources, J. Math. Anal. Appl. 330 (2007), 1128–1138.CrossRefGoogle Scholar
21.Wang, N. and Wadati, M., Noncommutative extension of ∂-dressing method, J. Phys. Soc. Jpn. 72 (2003), 1366–1373.CrossRefGoogle Scholar
22.Wang, N. and Wadati, M., Exact Multi-line Soliton Solutions of Noncommutative KP Equation, J. Phys. Soc. Jpn. 72 (2003), 1881–1888.CrossRefGoogle Scholar
23.Wang, N. and Wadati, M., Noncommutative KP hierarchy and Hirota triple-product relations, J. Phys. Soc. Jpn. 73 (2004), 1689–1698.CrossRefGoogle Scholar
24.Zhao, J. X., Commutativity of Pfaffianization and Bäcklund transformation: The semi-discrete Toda equation, Math. Comput. Simul. 74 (2007), 388–396.CrossRefGoogle Scholar