A Stochastic Model for Solitons

Yoshiaki Itoh, Hosam M. Mahmoud*, Daisuke Takahashi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


The soliton physics for the propagation of waves is represented by a stochastic model in which the particles of the wave can jump ahead according to some probability distribution. We demonstrate the presence of a steady state (stationary distribution) for the wavelength. It is shown that the stationary distribution is a convolution of geometric random variables. Approximations to the stationary distribution are investigated for a large number of particles. The model is rich and includes Gaussian cases as limit distribution for the wavelength (when suitably normalized). A sufficient Lindeberg-like condition identifies a class of solitons with normal behavior. Our general model includes, among many other reasonable alternatives, an exponential aging soliton, of which the uniform soliton is one special subcase (with Gumbel's stationary distribution). With the proper interpretation, our model also includes the deterministic model proposed in Takahashi and Satsuma [A soliton cellular automaton, J Phys Soc Japan 59 (1990), 3514-3519].

Original languageEnglish
Pages (from-to)51-64
Number of pages14
JournalRandom Structures and Algorithms
Issue number1
Publication statusPublished - 2004 Jan


  • Limit distribution
  • Random structure
  • Soliton
  • Wave propagation

ASJC Scopus subject areas

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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