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Register a new userLong-time asymptotics for Toda shock waves in the modulation region
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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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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 apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice for decaying initial data in the soliton region. In addition, we point out how to reduce the problem in the remaining region to the known case without solitons.
We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regions verifies a conjecture from Venakides, Deift, and Oba from 1991.
We consider semigroups ${alpha_t: ; tgeq 0}$ of normal, unital, completely positive maps $alpha_t$ on a von Neumann algebra ${mathcal M}$. The (predual) semigroup $ u_t (rho):= rho circ alpha_t$ on normal states $rho$ of $mathcal M$ leaves invariant the face ${mathcal F}_p:= {rho : ; rho (p)=1}$ supported by the projection $pin {mathcal M}$, if and only if $alpha_t(p)geq p$ (i.e., $p$ is sub-harmonic). We complete the arguments showing that the sub-harmonic projections form a complete lattice. We then consider $r_o$, the smallest projection which is larger than each support of a minimal invariant face; then $r_o$ is subharmonic. In finite dimensional cases $sup alpha_t(r_o)={bf 1}$ and $r_o$ is also the smallest projection $p$ for which $alpha_t(p)to {bf 1}$. If ${ u_t: ; tgeq 0}$ admits a faithful family of normal stationary states then $r_o={bf 1}$ is useless; if not, it helps to reduce the problem of the asymptotic behaviour of the semigroup for large times.
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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