The toda rarefaction problem

Percy Deift, Spyridon Kamvissis, Thomas Kriecherbauer, Xin Zhou

Research output: Contribution to journalArticle

Abstract

In the Toda shock problem (see [7], [11], [8], and also [3]) one considers a driving particle moving with a fixed velocity 2a 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problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas.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 In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops.

Original languageEnglish (US)
Pages (from-to)35-83
Number of pages49
JournalCommunications on Pure and Applied Mathematics
Volume49
Issue number1
DOIs
StatePublished - Jan 1996

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ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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