79 research outputs found

    Growth of degrees of integrable mappings

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    We study mappings obtained as s-periodic reductions of the lattice Korteweg-De Vries equation. For small s=(s1,s2) we establish upper bounds on the growth of the degree of the numerator of their iterates. These upper bounds appear to be exact. Moreover, we conjecture that for any s1,s2 that are co-prime the growth is ~n^2/(2s1s2), except when s1+s2=4 where the growth is linear ~n. Also, we conjecture the degree of the n-th iterate in projective space to be ~n^2(s1+s2)/(2s1s2).Comment: 14 pages, submitted to Journal of difference equations and application

    Initial value problems for quad equations

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    We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem (not all quad-graphs admit well-posed initial value problems) it is effective in the class of rhombic embeddable quad-graphs.Comment: 22 pages, 34 figures, submitted to Discrete & Computational Geometr

    On the Fourier transform of the greatest common divisor

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    The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan's sum, and on the other hand it can be written as a generalised convolution product of the identity with the totient function. We show that this arithmetic function of two integers (a,m) counts the number of elements in the set of ordered pairs (i,j) such that i*j is equivalent to a modulo m. Furthermore we generalise a dozen known identities for the totient function, to identities which involve the discrete Fourier transform of the greatest common divisor, including its partial sums, and its Lambert series.Comment: 3 figures, submitted to Proceedings of the American Mathematical Societ

    Symmetry condition in terms of Lie brackets

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    A passive orthonomic system of PDEs defines a submanifold in the corresponding jet manifold, coordinated by so called parametric derivatives. We restrict the total differential operators and the prolongation of an evolutionary vector field v to this submanifold. We show that the vanishing of their commutators is equivalent to v being a generalized symmetry of the system.Comment: 10 pages, no figures, unpublishe

    Somos-4 and Somos-5 are arithmetic divisibility sequences

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    We provide an elementary proof to a conjecture by Robinson that multiples of (powers of) primes in the Somos-4 sequence are equally spaced. We also show, almost as a corollary, for the generalised Somos-4 sequence defined by Ο„n+2Ο„nβˆ’2=Ξ±Ο„n+1Ο„nβˆ’1+Ξ²Ο„n2\tau_{n+2}\tau_{n-2}=\alpha\tau_{n+1}\tau_{n-1}+\beta\tau_n^2 and initial values Ο„1=Ο„2=Ο„3=Ο„4=1\tau_1=\tau_2=\tau_3=\tau_4=1, that the polynomial Ο„n(Ξ±,Ξ²)\tau_n(\alpha,\beta) is a divisor of Ο„n+k(2nβˆ’5)(Ξ±,Ξ²)\tau_{n+k(2n-5)}(\alpha,\beta) for all n,k∈Zn,k\in\mathbb Z and establish a similar result for the generalized Somos-5 sequence.Comment: 9 page

    From integrable equations to Laurent recurrences

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    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-kk sequences, with k=4,5k=4,5, whose coefficients are periodic functions with period 8 for k=4k=4, and period 7 for k=5k=5, and which possess the Laurent property. We also apply our method to the DTKQ-NN equation, with N=2,3N=2,3, and derive Laurent recurrences with N+2N+2 terms, of order N+3N+3. In the case N=3N=3 the recurrence has periodic coefficients with period 8. We demonstrate that recursive factorisation also provides a proof of the Laurent property

    New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves

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    For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil

    Discrete Painlev\'e equations and their Lax pairs as reductions of integrable lattice equations

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    We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the Lax integrability for entire hierarchies of reductions. A main result is, as an example of this framework, how we may obtain the q-Painlev\'e equation whose group of B\"acklund transformations is an affine Weyl group of type E_6^{(1)} as a similarity reduction of the discrete Schwarzian Korteweg-de Vries equation.Comment: 23 pages, 5 figure

    Duality for discrete integrable systems II

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    We generalise the concept of duality to lattice equations. We derive a novel 3 dimensional lattice equation, which is dual to the lattice AKP equation. Reductions of this equation include Rutishauser's quotient-difference (QD) algorithm, the higher analogue of the discrete time Toda (HADT) equation and its corresponding quotient-quotient-difference (QQD) system, the discrete hungry Lotka-Volterra system, discrete hungry QD, as well as the hungry forms of HADT and QQD. We provide three conservation laws, we conjecture the equation admits N-soliton solutions and that reductions have the Laurent property and vanishing algebraic entropy.Comment: 11 pages, 2 figure

    Symbolic Computation of Lax Pairs of Partial Difference Equations Using Consistency Around the Cube

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    A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P\Delta Es) is reviewed. The method assumes that the P\Delta Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P\Delta Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable P\Delta Es classified by Adler, Bobenko, and Suris and systems of P\Delta Es including the integrable 2-component potential Korteweg-de Vries lattice system, as well as nonlinear Schroedinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for P\Delta Es recently derived by Hietarinta (J. Phys. A: Math. Theor., 44, 2011, Art. No. 165204). The method is algorithmic and is being implemented in Mathematica.Comment: Paper dedicated to Peter Olver as part of a special issue of FoCM in honor of his 60th birthda
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