Premium
The Toda rarefaction problem
Author(s) -
Deift Percy,
Kamvissis Spyridon,
Kriecherbauer Thomas,
Zhou Xin
Publication year - 1996
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199601)49:1<35::aid-cpa2>3.0.co;2-8
Subject(s) - toda lattice , mathematics , exponential function , lattice (music) , factorization , inverse scattering transform , container (type theory) , classical mechanics , mathematical analysis , inverse scattering problem , mechanics , inverse problem , physics , integrable system , mechanical engineering , algorithm , acoustics , engineering
In the Toda shock problem (see [7], [11], [8], and also [3]) one considers a driving particle moving with a fixed velocity 2 a and impinging on a one‐dimensional semi‐infinite lattice of particles, initially equally spaced and at rest, and interacting with exponential forces. In this paper we consider the related Toda rarefaction problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas. We make use of the Riemann‐Hilbert factorization formulation of the related inverse scattering problem. In the case where the speed 2 | a | of the driving particle is sufficiently large (| a | > 1), we show that the particle escapes from the lattice, which then executes a free motion of the type studied, for example, in [5]. In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops. © 1996 John Wiley & Sons, Inc.