Generalized Toda flows
HTML articles powered by AMS MathViewer
- by Darren C. Ong and Christian Remling PDF
- Trans. Amer. Math. Soc. 371 (2019), 5069-5081 Request permission
Abstract:
The classical hierarchy of Toda flows can be thought of as an action of the (abelian) group of polynomials on Jacobi matrices. We present a generalization of this to the larger groups of $C^2$ and entire functions, and in this second case, we also introduce associated cocycles and in fact give center stage to this object.References
- Ilia Binder, David Damanik, Milivoje Lukic, and Tom VandenBoom, Almost periodicity in time of solutions of the Toda lattice, C. R. Math. Acad. Sci. Soc. R. Can. 40 (2018), no. 1, 1–28 (English, with English and French summaries). MR 3751318
- L. A. Dickey, Soliton equations and Hamiltonian systems, 2nd ed., Advanced Series in Mathematical Physics, vol. 26, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1964513, DOI 10.1142/5108
- Fritz Gesztesy, Helge Holden, Johanna Michor, and Gerald Teschl, Soliton equations and their algebro-geometric solutions. Vol. II, Cambridge Studies in Advanced Mathematics, vol. 114, Cambridge University Press, Cambridge, 2008. $(1+1)$-dimensional discrete models. MR 2446594, DOI 10.1017/CBO9780511543203
- Gert K. Pedersen, Operator differentiable functions, Publ. Res. Inst. Math. Sci. 36 (2000), no. 1, 139–157. MR 1749015, DOI 10.2977/prims/1195143229
- Christian Remling, Generalized reflection coefficients, Comm. Math. Phys. 337 (2015), no. 2, 1011–1026. MR 3339168, DOI 10.1007/s00220-015-2341-9
- C. Remling, Toda maps, cocycles, and canonical systems, J. Spectr. Theory (to appear).
- Alexei Rybkin, On the evolution of a reflection coefficient under the Korteweg-de Vries flow, J. Math. Phys. 49 (2008), no. 7, 072701, 15. MR 2432031, DOI 10.1063/1.2951897
- Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. MR 533824
- Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536, DOI 10.1090/surv/072
- V. Vinnikov and P. Yuditskii, Functional models for almost periodic Jacobi matrices and the Toda hierarchy, Mat. Fiz. Anal. Geom. 9 (2002), no. 2, 206–219. MR 1964065
Additional Information
- Darren C. Ong
- Affiliation: Department of Mathematics, Xiamen University Malaysia, Jalan Sunsuria, Bandar Sunsuria, 43900 Sepang, Selangor Darul Ehsan, Malaysia
- MR Author ID: 845285
- Email: darrenong@xmu.edu.my
- Christian Remling
- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
- MR Author ID: 364973
- Email: christian.remling@ou.edu
- Received by editor(s): February 4, 2018
- Received by editor(s) in revised form: July 15, 2018
- Published electronically: November 13, 2018
- Additional Notes: The first author was supported by a Xiamen University Malaysia Research Fund (Grant No. XMUMRF/2018-C1/IMAT/0001).
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5069-5081
- MSC (2010): Primary 34L40, 37K10, 47B36, 81Q10
- DOI: https://doi.org/10.1090/tran/7695
- MathSciNet review: 3934478