### Abstract

In this paper, conserved quantities of the discrete hungry Lotka- Volterra (dhLV) system are derived. Our approach is based on the Lax representation of the dhLV system, which expresses the time evolution of the dhLV system as a similarity transformation on a certain square matrix. Thus, coefficients of the characteristic polynomial of this matrix constitute conserved quantities of the dhLV system. These coefficients are calculated explicitly through a recurrence relation among the characteristic polynomials of its leading principal submatrices. The conserved quantities of the discrete hungry Toda (dhToda) equation is also derived with the help of the Bäcklund transformation between the dhLV system and the dhToda equation.

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

Pages (from-to) | 889-899 |

Number of pages | 11 |

Journal | Discrete and Continuous Dynamical Systems - Series S |

Volume | 8 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 Oct 1 |

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### Keywords

- Characteristic polynomial
- Conserved quantities
- Discrete hungry Lotka-Volterra system
- Discrete hungry Toda equation
- Leading principal submatrix

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete and Continuous Dynamical Systems - Series S*,

*8*(5), 889-899. https://doi.org/10.3934/dcdss.2015.8.889

**Conserved quantities of the integrable discrete hungry systems.** / Kakizaki, Sonomi; Fukuda, Akiko; Yamamoto, Yusaku; Iwasaki, Masashi; Ishiwata, Emiko; Nakamura, Yoshimasa.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems - Series S*, vol. 8, no. 5, pp. 889-899. https://doi.org/10.3934/dcdss.2015.8.889

}

TY - JOUR

T1 - Conserved quantities of the integrable discrete hungry systems

AU - Kakizaki, Sonomi

AU - Fukuda, Akiko

AU - Yamamoto, Yusaku

AU - Iwasaki, Masashi

AU - Ishiwata, Emiko

AU - Nakamura, Yoshimasa

PY - 2015/10/1

Y1 - 2015/10/1

N2 - In this paper, conserved quantities of the discrete hungry Lotka- Volterra (dhLV) system are derived. Our approach is based on the Lax representation of the dhLV system, which expresses the time evolution of the dhLV system as a similarity transformation on a certain square matrix. Thus, coefficients of the characteristic polynomial of this matrix constitute conserved quantities of the dhLV system. These coefficients are calculated explicitly through a recurrence relation among the characteristic polynomials of its leading principal submatrices. The conserved quantities of the discrete hungry Toda (dhToda) equation is also derived with the help of the Bäcklund transformation between the dhLV system and the dhToda equation.

AB - In this paper, conserved quantities of the discrete hungry Lotka- Volterra (dhLV) system are derived. Our approach is based on the Lax representation of the dhLV system, which expresses the time evolution of the dhLV system as a similarity transformation on a certain square matrix. Thus, coefficients of the characteristic polynomial of this matrix constitute conserved quantities of the dhLV system. These coefficients are calculated explicitly through a recurrence relation among the characteristic polynomials of its leading principal submatrices. The conserved quantities of the discrete hungry Toda (dhToda) equation is also derived with the help of the Bäcklund transformation between the dhLV system and the dhToda equation.

KW - Characteristic polynomial

KW - Conserved quantities

KW - Discrete hungry Lotka-Volterra system

KW - Discrete hungry Toda equation

KW - Leading principal submatrix

UR - http://www.scopus.com/inward/record.url?scp=84936761869&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84936761869&partnerID=8YFLogxK

U2 - 10.3934/dcdss.2015.8.889

DO - 10.3934/dcdss.2015.8.889

M3 - Article

VL - 8

SP - 889

EP - 899

JO - Discrete and Continuous Dynamical Systems - Series S

JF - Discrete and Continuous Dynamical Systems - Series S

SN - 1937-1632

IS - 5

ER -