### Abstract

Consider the discrete equation y _{n+1} + y _{n-1} = a_{n} + b_{n}y_{n} + c_{n}y^{2}_{n}/1-y^{2}_{n} where the right side is of degree two in y_{n} and where the coefficients a_{n}, b_{n} and c_{n} are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, and the height of y_{n} dominates the height of the coefficient functions a_{n}, b_{n} and c_{n}. We show that if the logarithmic height of y_{n} grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dP_{II} or its autonomous version or y_{n} is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.

Original language | English |
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Pages (from-to) | 2379-2396 |

Number of pages | 18 |

Journal | Nonlinearity |

Volume | 28 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 1 2015 |

### Fingerprint

### Keywords

- Diophantine integrability
- discrete integrable systems
- discrete Painlevé equations

### ASJC Scopus subject areas

- Applied Mathematics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Nonlinearity*,

*28*(7), 2379-2396. https://doi.org/10.1088/0951-7715/28/7/2379

**Height growth of solutions and a discrete Painlevé equation.** / Al-Ghassani, A.; Halburd, R. G.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 28, no. 7, pp. 2379-2396. https://doi.org/10.1088/0951-7715/28/7/2379

}

TY - JOUR

T1 - Height growth of solutions and a discrete Painlevé equation

AU - Al-Ghassani, A.

AU - Halburd, R. G.

PY - 2015/7/1

Y1 - 2015/7/1

N2 - Consider the discrete equation y n+1 + y n-1 = an + bnyn + cny2n/1-y2n where the right side is of degree two in yn and where the coefficients an, bn and cn are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, and the height of yn dominates the height of the coefficient functions an, bn and cn. We show that if the logarithmic height of yn grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dPII or its autonomous version or yn is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.

AB - Consider the discrete equation y n+1 + y n-1 = an + bnyn + cny2n/1-y2n where the right side is of degree two in yn and where the coefficients an, bn and cn are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, and the height of yn dominates the height of the coefficient functions an, bn and cn. We show that if the logarithmic height of yn grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dPII or its autonomous version or yn is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.

KW - Diophantine integrability

KW - discrete integrable systems

KW - discrete Painlevé equations

UR - http://www.scopus.com/inward/record.url?scp=84933060193&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84933060193&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/28/7/2379

DO - 10.1088/0951-7715/28/7/2379

M3 - Article

VL - 28

SP - 2379

EP - 2396

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 7

ER -