Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings

[2011.00690] Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Nonlinear Sciences > Exactly Solvable and Integrable Systems arXiv:2011.00690 (nlin) [Submitted on 2 Nov 2020 ( v1 ), last revised 17 May 2022 (this version, v2)] Title: Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings Authors: David A. Croydon , Makiko Sasada , Satoshi Tsujimoto View a PDF of the paper titled Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings, by David A. Croydon and 2 other authors View PDF HTML (experimental) 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. Subjects: Exactly Solvable and Integrable Systems (nlin.SI) ; Mathematical Physics (math-ph); Probability (math.PR) MSC classes: 37K10 (primary), 35Q53, 37K60 Cite as: arXiv:2011.00690 [nlin.SI] (or arXiv:2011.00690v2 [nlin.SI] for this version) https://doi.org/10.48550/arXiv.2011.00690 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Math Phys Anal Geom 25, 27 (2022) Related DOI : https://doi.org/10.1007/s11040-022-09435-4 Focus to learn more DOI(s) linking to related resources Submission history From: David Croydon [ view email ] [v1] Mon, 2 Nov 2020 02:25:50 UTC (99 KB) [v2] Tue, 17 May 2022 04:46:18 UTC (65 KB) Full-text links: Access Paper: View a PDF of the paper titled Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings, by David A. 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