[Paper Review] From integrable equations to Laurent recurrences

Khaled Hamad, Peter H. van der Kamp|arXiv (Cornell University)|Dec 18, 2014

Nonlinear Waves and Solitons39 references3 citations

TL;DR

This paper introduces a recursive factorization method to derive Laurent property recurrences from integrable difference equations. By homogenizing rational recurrences and iteratively factoring polynomial sequences, it constructs non-autonomous Somos-$k$ recurrences with periodic coefficients (period 8 for $k=4$, period 7 for $k=5$) and proves the Laurent property via divisor control. The method is applied to the DTKQ-$N$ equation, yielding higher-order Laurent recurrences with periodic coefficients, including a sixth-order recurrence with period-8 coefficients for $N=3$. The approach provides a constructive proof of the Laurent property through divisor-based recurrence derivation.

ABSTRACT

Based on a recursive factorisation technique we show how integrable difference equations give rise to recurrences which possess the Laurent property. We derive non-autonomous Somos-$k$ sequences, with $k=4,5$, whose coefficients are periodic functions with period 8 for $k=4$, and period 7 for $k=5$, and which possess the Laurent property. We also apply our method to the DTKQ-$N$ equation, with $N=2,3$, and derive Laurent recurrences with $N+2$ terms, of order $N+3$. In the case $N=3$ the recurrence has periodic coefficients with period 8. We demonstrate that recursive factorisation also provides a proof of the Laurent property.

Motivation & Objective

• To establish a systematic method for generating recurrences with the Laurent property from integrable difference equations.
• To prove the Laurent property constructively by controlling divisor multiplicities in polynomial sequences derived from rational recurrences.
• To derive non-autonomous Somos-$k$ recurrences with periodic coefficients for $k=4,5$ using recursive factorization.
• To extend the method to the DTKQ-$N$ equation, producing higher-order Laurent recurrences with periodic coefficients for $N=2,3$.
• To provide a constructive proof of the Laurent property by deriving divisor-based recurrences from homogenized systems.

Proposed method

• Homogenize a rational recurrence $ u_{n+N} = R(u_n, dots, u_{n+N-1}) $ into polynomial sequences $ a_n $ and $ b_n $, where $ u_n = a_n / b_n $.
• Apply ultra-discrete limits to the homogenized system to bound the growth of degrees and multiplicities of common divisors in $ a_n $ and $ b_n $.
• Use recursive factorization: define new sequences $ e_n $ as quotients of $ a_n $ and $ b_n $ after dividing by previously identified common divisors.
• Derive a nonlinear rational recurrence for the divisor sequences $ e_n $, which are polynomial by construction, ensuring the Laurent property.
• Verify the Laurent property by iterating the recurrence and checking that numerator and denominator polynomials are coprime to previous terms.
• For the DTKQ-$N$ equation, conjecture periodicity in divisor multiplicities (e.g., period 8 for $N=3$) and use this to define a Laurent recurrence with periodic coefficients.

Experimental results

Research questions

• RQ1 Can integrable difference equations be systematically transformed into recurrences with the Laurent property using recursive factorization?
• RQ2 What is the structure of non-autonomous Somos-$k$ recurrences derived from integrable systems, and what periodicity do their coefficients exhibit?
• RQ3 How can divisor multiplicities in polynomial sequences derived from rational recurrences be controlled to prove the Laurent property?
• RQ4 Can the recursive factorization method be extended to higher-order integrable equations like the DTKQ-$N$ equation?
• RQ5 What is the role of periodicity in the multiplicities of common divisors in generating Laurent recurrences with periodic coefficients?

Key findings

• The recursive factorization method successfully generates a non-autonomous Somos-4 recurrence with coefficients periodic with period 8.
• A non-autonomous Somos-5 recurrence with coefficients periodic with period 7 is derived, both satisfying the Laurent property.
• For the DTKQ-2 equation, a sixth-order Laurent recurrence with five terms is constructed, with coefficients that are periodic with period 8.
• For the DTKQ-3 equation, a sixth-order Laurent recurrence is derived with coefficients periodic with period 8, supported by a conjecture on periodic divisor multiplicities.
• The conjecture that the difference in multiplicities of $ z_2 $ in $ a_n $ and $ b_n $ is periodic with period 8 ($ heta_n = [0,1,0,-1,-1,2,-1,-1] $) leads to a Laurent recurrence with periodic coefficients.
• The Laurent property of the derived recurrences is verified by iterating the recurrence and confirming that numerators and denominators are coprime to earlier terms, ensuring Laurent polynomial structure.