### Abstract

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 language | English |
---|---|

Pages (from-to) | 51-64 |

Number of pages | 14 |

Journal | Random Structures and Algorithms |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Jan |

### Fingerprint

### Keywords

- Limit distribution
- Random structure
- Soliton
- Wave propagation

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*24*(1), 51-64. https://doi.org/10.1002/rsa.10106

**A Stochastic Model for Solitons.** / Itoh, Yoshiaki; Mahmoud, Hosam M.; Takahashi, Daisuke.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 24, no. 1, pp. 51-64. https://doi.org/10.1002/rsa.10106

}

TY - JOUR

T1 - A Stochastic Model for Solitons

AU - Itoh, Yoshiaki

AU - Mahmoud, Hosam M.

AU - Takahashi, Daisuke

PY - 2004/1

Y1 - 2004/1

N2 - 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].

AB - 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].

KW - Limit distribution

KW - Random structure

KW - Soliton

KW - Wave propagation

UR - http://www.scopus.com/inward/record.url?scp=0348199034&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0348199034&partnerID=8YFLogxK

U2 - 10.1002/rsa.10106

DO - 10.1002/rsa.10106

M3 - Article

VL - 24

SP - 51

EP - 64

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 1

ER -