New Integrable and Linearizable Nonlinear Difference Equations

Sahadevan, R.; Nagavigneshwari, G.

doi:10.1080/14029251.2013.805563

Cited by 2 publications

(2 citation statements)

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“…Thus, it might be interesting to investigate this problem further. Furthermore, there are non-linear affine equations which are integrable in the sense of possessing a Lax pair [27]. It would be worth to study growth of degrees of these equations as well.…”

Section: Discussionmentioning

confidence: 99%

“…Using (33), we can calculate the array of spontaneous gcd degrees corresponding to ( 29) and ( 30)-see the left hand side of, respectively, figures 2 and 3. Using (25) and the gcd values ( 29) and (30), we can also calculate the reduced degree values (16) in each case-see the right hand side of figures 2 and 3.…”

Section: Exponentially Growing Gcd: the Inherited Gcd And The Spontan...mentioning

confidence: 99%

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Algebraic entropy of (integrable) lattice equations and their reductions

Roberts

Tran

2019

Nonlinearity

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We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler-Bobenko-Suris equations and Viallet's QV and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. The resulting degrees satisfy a linear partial difference equation which is universal, i.e. the same for all the integrable lattice equations considered. When we take periodic reductions of these equations, which includes staircase initial conditions, we obtain from this linear partial difference equation an ordinary difference equation for degrees that implies quadratic or linear degree growth. We also study growth of degree of several non-integrable lattice equations. Exponential growth of degrees of these equations, and their mapping reductions, is also proved subject to a conjecture.

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Section: Discussionmentioning

confidence: 99%