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Register a new userA finite Toda representation of the box-ball system with box capacity
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Authors
Kazuki Maeda
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A connection between the finite ultradiscrete Toda lattice and the box-ball system is extended to the case where each box has own capacity and a carrier has a capacity parameter depending on time. In order to consider this connection, new carrier rules size limit for solitons and recovery of balls, and a concept expansion map are introduced. A particular solution to the extended system of a special case is also presented.
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The Box-Ball System, shortly BBS, was introduced by Takahashi and Satsuma as a discrete counterpart of the KdV equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size $k$ solitons in each $k$-slot. The dynamics of the components is linear: the $k$-th component moves rigidly at speed $k$. Let $zeta$ be a translation invariant family of independent random vectors under a summability condition and $eta$ the ball configuration with components $zeta$. We show that the law of $eta$ is translation invariant and invariant for the BBS. This recipe allows us to construct a big family of invariant measures, including product measures and stationary Markov chains with ball density less than $frac12$. We also show that starting BBS with an ergodic measure, the position of a tagged $k$-soliton at time $t$, divided by $t$ converges as $ttoinfty$ to an effective speed $v_k$. The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws.
A nonautonomous version of the ultradiscrete hungry Toda lattice with a finite lattice boundary condition is derived by applying reduction and ultradiscretization to a nonautonomous two-dimensional discrete Toda lattice. It is shown that the derived ultradiscrete system has a direct connection to the box-ball system with many kinds of balls and finite carrier capacity. Particular solutions to the ultradiscrete system are constructed by using the theory of some sort of discrete biorthogonal polynomials.
We analyze the non-relativistic problem of a quantum particle that bounces back and forth between two moving walls. We recast this problem into the equivalent one of a quantum particle in a fixed box whose dynamics is governed by an appropriate time-dependent Schroedinger operator.
In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including: - Time discretization based on the notion of Backlund transformation; - Symplectic realizations of multi-Hamiltonian structures; - Interrelations between discrete 1D systems and lattice 2D systems; - Multi-dimensional consistency as integrability of discrete systems; - Interrelations between integrable systems of quad-equations and integrable systems of Laplace type; - Pluri-Lagrangian structure as integrability of discrete variational systems. All these concepts are illustrated by the discrete time Toda lattices and their relativistic analogs.
We show that Toda shock waves are asymptotically close to a modulated finite gap solution in the region separating the soliton and the elliptic wave regions. We previously derived formulas for the leading terms of the asymptotic expansion of these shock waves in all principal regions and conjectured that in the modulation region the next term is of order $O(t^{-1})$. In the present paper we prove this fact and investigate how resonances and eigenvalues influence the leading asymptotic behaviour. Our main contribution is the solution of the local parametrix Riemann-Hilbert problems and a rigorous justification of the analysis. In particular, this involves the construction of a proper singular matrix model solution.
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