Geodesic
Soliton scattering in noncommutative spaces
Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 68-88 Cet article a éte moissonné depuis la source Math-Net.Ru

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We discuss exact multisoliton solutions of integrable hierarchies on noncommutative space–times in various dimensions. The solutions are represented by quasideterminants in compact forms. We study soliton scattering processes in the asymptotic region where the configurations can be real-valued. We find that the asymptotic configurations in the soliton scatterings can all be the same as commutative ones, i.e., the configuration of an $N$-soliton solution has $N$ isolated localized lumps of energy, and each solitary wave-packet lump preserves its shape and velocity in the scattering process. The phase shifts are also the same as commutative ones. As new results, we present multisoliton solutions of the noncommutative anti-self-dual Yang–Mills hierarchy and discuss two-soliton scattering in detail.
Mots-clés : soliton
Keywords: integrable system, noncommutative geometry. 
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M. Hamanaka; H. Okabe. Soliton scattering in noncommutative spaces. Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 68-88. http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a3/

[1] P. Etingof, I. Gelfand, V. Retakh, “Factorization of differential operators, quasideterminants, and nonabelian Toda field equations”, Math. Res. Lett., 4:3 (1997), 413–425, arXiv: q-alg/9701008 | DOI | MR

[2] I. M. Gelfand, V. S. Retakh, “Determinanty matrits nad nekommutativnymi koltsami”, Funkts. analiz i ego pril., 25:2 (1991), 13–25 ; “Теория некоммутативных детерминантов и характеристические функции графов”, 26:4 (1992), 1–20 | DOI | MR | Zbl | DOI | MR | Zbl

[3] E. V. Doktorov, S. B. Leble, A Dressing Method in Mathematical Physics, Mathematical Physics Studies, 28, Springer, Dordrecht, 2007 | DOI | MR

[4] P. Etingof, I. Gelfand, V. Retakh, “Nonabelian integrable systems, quasideterminants, and Marchenko lemma”, Math. Res. Lett., 5:1 (1998), 1–12, arXiv: q-alg/9707017 | DOI

[5] C. R. Gilson, M. Hamanaka, J. J. C. Nimmo, “Bäcklund transformations for noncommutative anti-self-dual Yang–Mills equations”, Glasg. Math. J., 51:A (2009), 83–93, arXiv: 0709.2069 | DOI | MR

[6] C. R. Gilson, M. Hamanaka, J. J. C. Nimmo, “Bäcklund transformations and the Atiyah–Ward ansatz for noncommutative anti-self-dual Yang–Mills equations”, Proc. Roy. Soc. Lond. Ser. A, 465:2108 (2009), 2613–2632 | DOI | MR

[7] C. R. Gilson, S. R. Macfarlane, “Dromion solutions of noncommutative Davey–Stewartson equations”, J. Phys. A: Math. Theor., 42:23 (2009), 235202, 20 pp., arXiv: 0901.4918 | DOI | MR

[8] C. R. Gilson, J. J. C. Nimmo, “On a direct approach to quasideterminant solutions of a noncommutative KP equation”, J. Phys. A: Math. Theor., 40:14 (2007), 3839–3850, arXiv: nlin/0701027 | DOI | MR

[9] C. R. Gilson, J. J. C. Nimmo, Y. Ohta, “Quasideterminant solutions of a non-Abelian Hirota–Miwa equation”, J. Phys. A: Math. Theor., 40:42 (2007), 12607–12618, arXiv: nlin/0702020 | DOI

[10] C. R. Gilson, J. J. C. Nimmo, C. M. Sooman, “On a direct approach to quasideterminant solutions of a noncommutative modified KP equation”, J. Phys. A: Math. Theor., 41:8 (2008), 085202, 10 pp., arXiv: 0711.3733 | DOI | MR

[11] B. Haider, M. Hassan, “The $U(N)$ chiral model and exact multi-solitons”, J. Phys. A: Math. Theor., 41:25 (2008), 255202, 17 pp., arXiv: 0912.1984 | DOI | MR

[12] M. Hamanaka, “Notes on exact multi-soliton solutions of noncommutative integrable hierarchies”, JHEP, 02 (2007), 094, 17 pp., arXiv: hep-th/0610006 | DOI | MR

[13] M. Hamanaka, “Noncommutative solitons and quasideterminants”, Phys. Scr., 89:3 (2014), 038006, 11 pp., arXiv: 1101.0005 | DOI

[14] C. X. Li, J. J. C. Nimmo, K. M. Tamizhmani, “On solutions to the non-Abelian Hirota–Miwa equation and its continuum limits”, Proc. Roy. Soc. London Ser. A, 465:2105 (2009), 1441–1451, arXiv: 0809.3833 | DOI | MR

[15] V. Retakh, V. Rubtsov, “Noncommutative Toda chains, Hankel quasideterminants and Painlevé II equation”, J. Phys. A: Math. Theor., 43:50 (2010), 505204, 13 pp., arXiv: 1007.4168 | DOI | MR

[16] M. Siddiq, U. Saleem, M. Hassan, “Darboux transformation and multi-soliton solutions of a noncommutative sine-Gordon system”, Modern Phys. Lett. A, 23:2 (2008), 115–127 | DOI | MR

[17] A. Dimakis, F. Müller-Hoissen, Extension of Moyal-deformed hierarchies of soliton equations, arXiv: nlin.si/0408023

[18] M. Hamanaka, Noncommutative solitons and D-branes, Ph. D. Thesis, University of Tokyo, Tokyo, 2003, arXiv: hep-th/0303256

[19] M. Hamanaka, “Noncommutative solitons and integrable systems”, Noncommutative Geometry and Physics (Keio, February 26 – March 3, 2004), eds. Y. Maeda, N. Tose, N. Miyazaki, S. Watamura, D. Sternheimer, World Sci., Singapore, 2005, 175–198, arXiv: hep-th/0504001 | DOI | MR

[20] M. Hamanaka, K. Toda, “Towards noncommutative integrable equations”, Symmetry in Nonlinear Mathematical Physics (Kiev, Ukraine, 23–29 June, 2003), ed. A. G. Nikitin, Institute of Mathematics of NASU, Kyiv, 2004, 404–411, arXiv: hep-th/0309265 | MR

[21] O. Lechtenfeld, “Noncommutative solitons”, Noncommutative Geometry and Physics 2005, Proceedings of the International Sendai–Beijing Joint Workshop (Sendai, Japan, 1–4 November, 2005; Beijing, China, 7–10 November, 2005), eds. U. Carow-Watamura, S. Watamura, Y. Maeda, H. Moriyoshi, Z. Liu, K. Wu, World Sci., Singapore, 2007, 175–200, arXiv: hep-th/0605034 | DOI | MR

[22] L. Mazzanti, Topics in noncommutative integrable theories and holographic brane-world cosmology, arXiv: 0712.1116

[23] L. Tamassia, Noncommutative supersymmetric/integrable models and string theory, arXiv: hep-th/0506064

[24] A. Dimakis, F. Müller-Hoissen, “Noncommutative Korteweg–de-Vries equation”, Phys. Lett. A, 278:3 (2000), 139–145, arXiv: hep-th/0007074 | DOI

[25] L. Martina, O. K. Pashaev, Burgers' equation in non-commutative space-time, arXiv: hep-th/0302055

[26] L. D. Paniak, Exact noncommutative KP and KdV multi-solitons, arXiv: hep-th/0105185

[27] J. E. Moyal, “Quantum mechanics as a statistical theory”, Proc. Cambridge Phil. Soc., 45:1 (1949), 99–124 ; H. J. Groenewold, “On the principles of elementary quantum mechanics”, Physica, 12:7 (1946), 405–460 | DOI | DOI | MR

[28] M. Hamanaka, K. Toda, “Noncommutative Burgers equation”, J. Phys. A: Math. Theor., 36:48 (2003), 11981–11998, arXiv: hep-th/0301213 | DOI

[29] M. Hamanaka, “Noncommutative Ward's conjecture and integrable systems”, Nucl. Phys. B, 741:3 (2006), 368–389, arXiv: hep-th/0601209 | DOI

[30] K. Toda, “Extensions of soliton equations to non-commutative $(2+1)$ dimensions”, PoS(unesp2002), 038, 12 pp. | DOI | MR

[31] M. Hamanaka, “On reductions of noncommutative anti-self-dual Yang–Mills equations”, Phys. Lett. B, 625:3–4 (2005), 324–332, arXiv: hep-th/0507112 | DOI

[32] M. Hamanaka, K. Toda, “Towards noncommutative integrable systems”, Phys. Lett. A, 316:1–2 (2003), 77–83, arXiv: hep-th/0211148 | DOI

[33] R. S. Ward, “Integrable and solvable systems, and relations among them”, Phil. Trans. Roy. Soc. London Ser. A, 315:1533 (1985), 451–457 | DOI | MR

[34] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149, Cambridge Univ. Press, Cambridge, 1991 | MR

[35] L. J. Mason, N. M. Woodhouse, Integrability, Self-Duality, and Twistor Theory, London Mathematical Society Monographs. New Series, 15, Oxford Univ. Press, Oxford, 1996 | MR

[36] I. Gelfand, S. Gelfand, V. Retakh, R. Wilson, “Quasideterminants”, Adv. Math., 193:1 (2005), 56–141, arXiv: math.QA/0208146 | DOI

[37] O. Babelon, D. Bernard, M. Talon, Introduction to Classical Integrable Systems, Cambridge Univ. Press, Cambridge, 2003 | DOI | MR

[38] M. Błaszak, Multi-Hamiltonian Theory of Dynamical Systems, Springer, Berlin, 1998 | MR

[39] L. A. Dickey, Soliton Equations and Hamiltonian Systems, Advanced Series in Mathematical Physics, 26, World Sci., Singapore, 2003 | DOI | MR

[40] B. A. Kupershmidt, KP or mKP. Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems, Mathematical Surveys and Monographs, 78, AMS, Providence, RI, 2000 | DOI | MR

[41] M. Hamanaka, “Commuting flows and conservation laws for noncommutative Lax hierarchies”, J. Math. Phys., 46:5 (2005), 052701, 13 pp., arXiv: hep-th/0311206 | DOI | MR

[42] S. Carillo, M. Lo Schiavo, C. Schiebold, “Bäcklund transformations and non-abelian nonlinear evolution equations: a novel Bäcklund chart”, SIGMA, 12 (2016), 087, 17 pp., arXiv: 1512.02386 | MR

[43] B. G. Konopelchenko, W. Oevel, “Matrix Sato theory and integrable systems in $2+1$ dimensions”, Nonlinear Evolution Equations and Dynamical Systems, Proceedings of the Workshop NEEDS'91 (Baia Verde, Galipoli, Italy, 19–29 June, 1991), eds. M. Bolti, L. Martina, F. Pempinelli, World Sci., Singapore, 1992, 87–96 | MR

[44] W. X. Ma, “An extended noncommutative KP hierarchy”, J. Math. Phys., 51:7 (2010), 073505, 19 pp. | DOI | MR

[45] P. J. Olver, V. V. Sokolov, “Integrable evolution equations on associative algebras”, Commun. Math. Phys., 193:2 (1998), 245–268 | DOI | MR

[46] J. P. Wang, “On the structure of $(2+1)$–dimensional commutative and noncommutative integrable equations”, J. Math. Phys., 47:11 (2006), 113508, arXiv: nlin/0606036 | DOI

[47] N. Wang, M. Wadati, “Noncommutative KP hierarchy and Hirota triple-product relations”, J. Phys. Soc. Japan, 73:7 (2004), 1689–1698 | DOI | MR

[48] K. Ohkuma, M. Wadati, “The Kadomtsev–Petviashvili equation: the trace method and the soliton resonances”, J. Phys. Soc. Japan, 52:3 (1983), 749–760 | DOI | MR

[49] Y. Kodama, KP Solitons and the Grassmannians, Springer Briefs in Mathematical Physics, 22, Springer, Singapore, 2017 | DOI | MR

[50] I. A. B. Strachan, “A geometry for multidimensional integrable systems”, J. Geom. Phys., 21:3 (1997), 255–278, arXiv: hep-th/9604142 | DOI | MR

[51] M. J. Ablowitz, S. Chakravarty, L. A. Takhtajan, “A selfdual Yang–Mills hierarchy and its reductions to integrable systems in $(1+1)$-dimensions and $(2+1)$-dimensions”, Commun. Math. Phys., 158:2 (1993), 289–314 | DOI | MR

[52] Y. Nakamura, “Transformation group acting on a self-dual Yang–Mills hierarchy”, J. Math. Phys., 29:1 (1988), 244–248 | DOI | MR

[53] K. Suzuki, Explorations of a self-dual Yang–Mills hierarchy, Diploma thesis, Max-Planck-Institut für Dynamik und Selbstorganisation, Göttingen, 2006

[54] K. Takasaki, “A new approacht to the selfdual Yang–Mills equations”, Commun. Math. Phys., 94:1 (1984), 35–59 | DOI

[55] H. J. de Vega, “Non-linear multi-plane wave solutions of self-dual Yang–Mills theory”, Commun. Math. Phys., 116:4 (1988), 659–674 | DOI | MR

[56] D. K. Nimmo, K. R. Dzhilson, Ya. Okhta, “Primeneniya preobrazovanii Darbu k samodualnym uravneniyam Yanga–Millsa”, TMF, 122:2 (2000), 284–293 | DOI | DOI | MR | Zbl