Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-19T22:30:33.398Z Has data issue: false hasContentIssue false
  • English
  • Français

EXPLICIT SOLUTION FORMULAS FOR THE MATRIX-KP

Published online by Cambridge University Press:  01 February 2009

CORNELIA SCHIEBOLD
CORNELIA SCHIEBOLD*
Affiliation:
Department of Mathematics, Mid Sweden University, S-851 70 Sundsvall, Sweden e-mail: Cornelia.Schiebold@miun.se
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.

We study a non-commutative version of the Kadomtsev-Petviashvili equations and construct a family of solutions generalizing naturally the soliton to the non-commutative setting. From this we derive explicit solution formulas as well for the scalar as for the matrix-Kadomtsev-Petviashvili equation which still depend on operator parameters.

Keywords

39B4247J2537K1035C05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue A , February 2009 , pp. 147 - 155
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Blohm, H., Solution of nonlinear equations by trace methods. Nonlinearity 13 (2000), 1925–1964.CrossRefGoogle Scholar
2.Calogero, F. and Degasperis, A., New integrable PDEs of boomeronic type, J. Phys. A 39 (2006), 8349–8376.CrossRefGoogle Scholar
3.Carl, B. and Schiebold, C., Nonlinear equations in soliton physics and operator ideals, Nonlinearity 12 (1999), 333–364.Google Scholar
4.Carl, B. and Schiebold, C., Ein direkter Ansatz zur Untersuchung von Solitonengleichungen, Jber. d. Dt. Math.-Verein. 102 (2000), 102–148.Google Scholar
5.Dimakis, A. and Müller-Hoissen, F., With a Cole-Hopff transformation to solutions of the non-commutative KP hierarchy in terms of Wronski matrices, J. Phys. A 40 (2007), F321–F329.CrossRefGoogle Scholar
6.Dimakis, A. and Müller-Hoissen, F., Weakly nonassociative algebras, Riccati and KP hierarchies, Preprint 2007, see arXiv:nlin/0701010v4, 16 pp.Google Scholar
7.Gilson, C. R. and Nimmo, J. J. C., On a direct approach to quasideterminant solutions of the noncommutative KP equation, J. Phys. A 40 (2007), 3839–3850.CrossRefGoogle Scholar
8.Hamanaka, M., Notes on exact multi-soliton solutions of noncommutative integrable hierarchies, J. High Energy Phys. 02 (2007) 094.Google Scholar
9.Kupershmidt, B. A., KP or mKP. Noncommutative Mathematics of Lagarangian, Hamiltonian, and Integrable Systems. Math. Surv. Monogr. 78 (2000), American Mathematical Society, Providence, RI, 600 pp.Google Scholar
10.Sakhnovich, A. L., Matrix Kadomtsev–Petviashvili equation: matrix identities and explicit non-singular solutions, J. Phys. A 36 (2003), 5023–5033.Google Scholar
11.Schiebold, C., From the non-abelian to the scalar two-dimensional Toda lattice, Glasgow Math. J. 47A (2005), 177–189.Google Scholar
12.Schiebold, C., Integrable systems and operator equations, Habilitation Thesis (Friedrich-Schiller Universität Jena 2004).Google Scholar
13.Olver, P. J. and Sokolov, V. V., Integrable evolution equations on associative algebras, Comm. Math. Phys. 193 (1998), 245–268.CrossRefGoogle Scholar
14.Wadati, M. and Wang, N., Exact multi-line soliton solutions of noncommutative KP equation, J. Phys. Soc. Jpn. 72 (2003), 1881–1888.Google Scholar