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Bi-infinite Solutions for KdV- and Toda-Type Discrete Integrable Systems Based on Path Encodings

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Abstract

We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical ‘Pitman’s transformation’ of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural ‘carrier process’ for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.

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Acknowledgements

The research was supported by JSPS Grant-in-Aid for Scientific Research (B), 19H01792. The research of DC was also supported by JSPS Grant-in-Aid for Scientific Research (C), 19K03540, and by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. This work was completed while MS was being kindly hosted by the Courant Institute, New York University.

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Authors and Affiliations

  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606–8502, Japan

    David A. Croydon

  2. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153–8914, Japan

    Makiko Sasada

  3. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-ku, Kyoto, 606–8501, Japan

    Satoshi Tsujimoto

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  1. David A. Croydon
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Communicated by Yuji Kodama.

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Croydon, D.A., Sasada, M. & Tsujimoto, S. Bi-infinite Solutions for KdV- and Toda-Type Discrete Integrable Systems Based on Path Encodings. Math Phys Anal Geom 25, 27 (2022). https://doi.org/10.1007/s11040-022-09435-4

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