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Integrability, analyticity, isochrony, equilibria, small oscillations, and Diophantine relations: results from the stationary Burgers hierarchy

M Bruschi1,2, F Calogero1,2 and R Droghei3

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An isochronous system is introduced by modifying the Nth ODE of the stationary Burgers hierarchy, and then, by investigating its behaviour near its equilibria, neat Diophantine relations are identified, involving (well-known) polynomials of arbitrary degree having integer zeros, or equivalently matrices the determinants of which yield such polynomials. The basic idea to arrive at such relations is not new, but the specific application reported in this paper is new, and it is likely to open the way to several analogous new findings.


PACS

02.30.Ik Integrable systems

02.10.De Algebraic structures and number theory

02.10.Yn Matrix theory

02.30.Hq Ordinary differential equations

02.30.Sa Functional analysis

MSC

30D30 Meromorphic functions, general theory

11C20 Matrices, determinants (See also 15A36)

34M55 Painlevé and other special equations; classification, hierarchies; isomonodromic deformations

30C10 Polynomials

34K11 Oscillation theory

34M05 Entire and meromorphic solutions

Subjects

Mathematical physics

Dates

Issue 47 (27 November 2009)

Received 18 July 2009

Published 4 November 2009



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