Formal Diagonalisation of Lax-Darboux Schemes
https://doi.org/10.18255/1818-1015-2015-6-795-817
Abstract
We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schro¨dinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model, principal chiral field model as well as with differential-difference integrable systems (including the Toda lattice and differential-difference Heisenberg chain) and integrable partial difference systems. We show that there exists a transformation which formally diagonalises all elements of the Lax-Darboux scheme simultaneously. It provides us with generating functions of local conservation laws for all integrable systems obtained. We discuss the relations between conservation laws for systems belonging to the Lax-Darboux scheme.
About the Author
A. V. Mikhailov
University of Leeds, School of Mathematics (Leeds, UK)
United Kingdom
Professor, University of Leeds, Leeds, LS2 9JT, UK
United Kingdom
Professor, University of Leeds, Leeds, LS2 9JT, UK
References
1. V. B. Matveev, M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics 4, Springer-Verlag, Berlin, 1991.
2. C. Rogers, W. K. Schief, “B¨acklund and Darboux transformations”, Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, 2002.
3. A. I. Bobenko, Yu. B. Suris, “Integrable systems on quad-graphs”, Int. Math. Res. Notices, 11, 573–611.
4. F. Khanizadeh, A. V. Mikhailov, Jing Ping Wang, “Darboux transformations and recursion operators for differential-difference equations”, Theoretical and Mathematical Physics, 177(3) (2013), 1606–1654.
5. A. V. Mikhailov, G. Papamikos, Jing Ping Wang, “Darboux transformation with dihedral reduction group”, Journal of Mathematical Physics, 55(11) (2014), 113507, arXiv: 1402.5660.
6. W. R. Wasow, Asymptotic expansions of solutions of ordinary differential equations, Pure and applied mathematics, Wiley Interscience Publishes, New York, 1965.
7. V. G. Drinfel’d, V. V. Sokolov, “Lie algebras and equations of Korteweg– de Vries type”, Itogi Nauki i Tekhniki, 24, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, 81–180.
8. A. V. Mikhailov, A. B. Shabat, “Conditions for integrability of systems of two equations of the form ut = A(u)uxx + F(u, ux). I”, Teoret. Mat. Fiz., 62(2) (1985), 163–185.
9. A. V. Mikhailov, “Formal diagonalisation of Darboux transformation and conservation laws of integrable PDEs, PD∆Es and P∆Es”, International Workshop “Geometric Structures in Integrable Systems” (October 30 November 02, 2012, M.V. Lomonosov Moscow State University, Moscow), http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=5934.
10. A. V. Mikhailov, “Formal diagonalisation of the Lax-Darboux scheme and conservation laws of integrable partial differential, differential-difference and partial difference”, DIS A follow-up meeting (8–12 July 2013, Isaac Newton Institute for Mathematical Sciences), http://www.newton.ac.uk/programmes/DIS/seminars/2013071114001.html.
11. I. T. Habibullin, M. V. Yangubaeva, “Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems”, Theoretical and Mathematical Physics, 177(3) (2013), 1655–1679.
12. R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Discrete equation on a square lattice with a nonstandard structure of generalized symmetries”, Theoretical and Mathematical Physics, 180(1) (2014), 765–780.
13. V. E. Zakharov, A. B. Shabat, “Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media”, Z.ˇEksper. Teoret. Fiz., 61(1) (1971), 118–134.
14. A. V. Mikhailov, A. B. Shabat, V. V. Sokolov, “The symmetry approach to classification of integrable equations”, Springer Ser. Nonlinear Dynamics, Springer, Berlin, 1991, 115–184.
15. A. V. Mikhailov, editor, “Integrability”, Lecture Notes in Physics, 767 (2009).
16. V. E. Adler, Classification of discrete integrable equations, DSci Thesis, L. D. Landau Institute, 2010.
17. I. Merola, O. Ragnisco, Gui-Zhang Tu, “A novel hierarchy of integrable lattices”, Inverse Problems, 10(6) (1994), 1315–1334.
18. L. A. Takhtadzhyan, V. E. Zakharov, “Equivalence of the nonlinear Schr¨odinger equation and the equation of a Heisenberg ferromagnet”, Theoretical and Mathematical Physics, 38(1) (1979), 26–35.
19. V. E. Zakharov, A. V. Mikhailov, “Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method”, Zh. Eksper. Teoret. Fiz. ` , 74(6) (1978), 1953–1973.
20. A. V. Zhiber, V. V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type”, Uspekhi Mat. Nauk, 56(1(337)) (2001), 63–106.
Review
For citations:
Mikhailov A.V. Formal Diagonalisation of Lax-Darboux Schemes. Modeling and Analysis of Information Systems. 2015;22(6):795-817. https://doi.org/10.18255/1818-1015-2015-6-795-817
Views:
1666
Email this article 