The toda rarefaction problem

Percy Deift, Spyridon Kamvissis, Thomas Kriecherbauer, Xin Zhou

Research output: Contribution to journalArticlepeer-review


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problem in which the driving particle now moves away from the lattice at fixed speed, in analogy with a piston being withdrawn, as it were, from a container filled with gas.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 In other words, in analogy with a piston being withdrawn too rapidly from a container filled with gas, cavitation develops.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
Issue number1
StatePublished - Jan 1996

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


Dive into the research topics of 'The toda rarefaction problem'. Together they form a unique fingerprint.

Cite this