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An Exact Discretization of a Lax Equation for Shock Clustering and Burgers Turbulence I: Dynamical Aspects and Exact Solvability

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Abstract

We consider a finite dimensional system which arises as an exact discretization of the Lax equation for shock clustering, which describes the evolution of the generator of a Markov process with a finite number of states. The spectral curve of the system is a nodal curve which is fully reducible. In this work, we will show that the flow is conjugate to a straight line motion, and we will also show that the equation is exactly solvable. En route, we will establish a dictionary between an open, dense set of the lower triangular infinitesimal generator matrices and an associated set of algebro-geometric data.

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  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Luen-Chau Li

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  1. Luen-Chau Li
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Correspondence to Luen-Chau Li.

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Communicated by P. Deift

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Li, LC. An Exact Discretization of a Lax Equation for Shock Clustering and Burgers Turbulence I: Dynamical Aspects and Exact Solvability. Commun. Math. Phys. 361, 415–466 (2018). https://doi.org/10.1007/s00220-018-3179-8

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  • DOI: https://doi.org/10.1007/s00220-018-3179-8

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