### Abstract

We use a strong version of the Painlevé property to discover and characterize a new class of n-dimensional Hamiltonian Lotka–Volterra systems, which turn out to be Liouville integrable as well as superintegrable. These systems are in fact Nambu systems, they posses Lax equations and they can be explicitly integrated in terms of elementary functions. We apply our analysis to systems containing only quadratic nonlinearities of the form a_{ij}x_{i}x_{j},i≠j, and require that all variables diverge as t^{−1}. We also require that the leading terms depend on n−2 free parameters. We thus discover a cocycle relation among the coefficients a_{ij} of the equations of motion and by integrating the cocycle equations we show that they are equivalent to the above strong version of the Painlevé property. We also show that these systems remain explicitly solvable even if a linear term b_{i}x_{i} is added to the i-th equation, even though this violates the Painlevé property, as logarithmic singularities are introduced in the Laurent solutions, at the first terms following the leading order pole.

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

Pages (from-to) | 3977-3982 |

Number of pages | 6 |

Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |

Volume | 380 |

Issue number | 47 |

DOIs | |

Publication status | Published - Dec 9 2016 |

### Fingerprint

### Keywords

- Integrable Lotka Volterra systems
- Strong Painlevé property

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Physics Letters, Section A: General, Atomic and Solid State Physics*,

*380*(47), 3977-3982. https://doi.org/10.1016/j.physleta.2016.09.034

**Lotka–Volterra systems satisfying a strong Painlevé property.** / Bountis, Tassos; Vanhaecke, Pol.

Research output: Contribution to journal › Article

*Physics Letters, Section A: General, Atomic and Solid State Physics*, vol. 380, no. 47, pp. 3977-3982. https://doi.org/10.1016/j.physleta.2016.09.034

}

TY - JOUR

T1 - Lotka–Volterra systems satisfying a strong Painlevé property

AU - Bountis, Tassos

AU - Vanhaecke, Pol

PY - 2016/12/9

Y1 - 2016/12/9

N2 - We use a strong version of the Painlevé property to discover and characterize a new class of n-dimensional Hamiltonian Lotka–Volterra systems, which turn out to be Liouville integrable as well as superintegrable. These systems are in fact Nambu systems, they posses Lax equations and they can be explicitly integrated in terms of elementary functions. We apply our analysis to systems containing only quadratic nonlinearities of the form aijxixj,i≠j, and require that all variables diverge as t−1. We also require that the leading terms depend on n−2 free parameters. We thus discover a cocycle relation among the coefficients aij of the equations of motion and by integrating the cocycle equations we show that they are equivalent to the above strong version of the Painlevé property. We also show that these systems remain explicitly solvable even if a linear term bixi is added to the i-th equation, even though this violates the Painlevé property, as logarithmic singularities are introduced in the Laurent solutions, at the first terms following the leading order pole.

AB - We use a strong version of the Painlevé property to discover and characterize a new class of n-dimensional Hamiltonian Lotka–Volterra systems, which turn out to be Liouville integrable as well as superintegrable. These systems are in fact Nambu systems, they posses Lax equations and they can be explicitly integrated in terms of elementary functions. We apply our analysis to systems containing only quadratic nonlinearities of the form aijxixj,i≠j, and require that all variables diverge as t−1. We also require that the leading terms depend on n−2 free parameters. We thus discover a cocycle relation among the coefficients aij of the equations of motion and by integrating the cocycle equations we show that they are equivalent to the above strong version of the Painlevé property. We also show that these systems remain explicitly solvable even if a linear term bixi is added to the i-th equation, even though this violates the Painlevé property, as logarithmic singularities are introduced in the Laurent solutions, at the first terms following the leading order pole.

KW - Integrable Lotka Volterra systems

KW - Strong Painlevé property

UR - http://www.scopus.com/inward/record.url?scp=84992518950&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992518950&partnerID=8YFLogxK

U2 - 10.1016/j.physleta.2016.09.034

DO - 10.1016/j.physleta.2016.09.034

M3 - Article

VL - 380

SP - 3977

EP - 3982

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

SN - 0375-9601

IS - 47

ER -