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Discrete Integrable Systems, Darboux Transformations, and Yang–Baxter Maps

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Symmetries and Integrability of Difference Equations

Part of the book series: CRM Series in Mathematical Physics ((CRM))

Abstract

This chapter is devoted to the integrability of discrete systems and their relation to the theory of Yang–Baxter (YB) maps. Lax pairs play a significant role in the integrability of discrete systems. We introduce the notion of Lax pair by considering the well-celebrated doubly-infinite Toda lattice. In particular, we present solution of the Cauchy initial value problem via the method of the inverse scattering transform, provide a review of scattering theory of Jacobi matrices, and give the Riemann–Hilbert formulation of the inverse scattering transform. On the other hand, the Lax–Darboux scheme constitutes an important tool in the theory of integrable systems, as it relates several concepts of integrability. We explain the role of Darboux and Bäcklund transformations in the theory of integrable systems, and we show how they can be used to construct discrete integrable systems via the Lax–Darboux scheme. Moreover, we give an introduction to the theory of Yang–Baxter maps and we show its relation to discrete integrable systems. Finally, we demonstrate the construction of Yang–Baxter maps via Darboux transformations, using the Nonlinear Schrödinger (NLS) equation as illustrative example.

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Notes

  1. 1.

    We employ two notational conventions in this section. We use bold capital letters to denote a matrix, say A, and use the regular capital type of the same letter to denote its entries: A ij . Whenever it is clear from the context, we use x to denote a sequence \(\{x_{n}\}_{n\in \mathbb{Z}}\).

  2. 2.

    Finite Toda lattice is sometimes called the open Toda lattice.

  3. 3.

    Potential u may depend on a temporal parameter t, namely u = u(x, t).

  4. 4.

    Potential u[1] is a solution of the KdV equation, since it can be readily shown that the pair (y[1], u[1]) also satisfies the temporal part of the Lax pair for KdV.

  5. 5.

    These are orbits corresponding to the fixed points of the fractional linear transformations of the spectral parameter.

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Acknowledgements

We would like to thank the organizers of the summer school Abecedarian of SIDE 12 for the opportunity to participate as lecturers, as well as for the financial support.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Deniz Bilman

  2. Institute of Mathematical Physics & Seismodynamics, Kievskaya 33, 364068, Grozny, Russia

    Sotiris Konstantinou-Rizos

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  1. Deniz Bilman
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  2. Sotiris Konstantinou-Rizos
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Correspondence to Deniz Bilman .

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  1. Mathematics and Physics Departments, and INFN, Sezione Roma Tre, Roma Tre University, Roma, Italy

    Decio Levi

  2. Collège Ahuntsic, Montréal, Québec, Canada

    Raphaël Rebelo

  3. Centre de Recherches Mathématiques and Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada

    Pavel Winternitz

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Bilman, D., Konstantinou-Rizos, S. (2017). Discrete Integrable Systems, Darboux Transformations, and Yang–Baxter Maps. In: Levi, D., Rebelo, R., Winternitz, P. (eds) Symmetries and Integrability of Difference Equations. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-56666-5_5

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