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  • 2005-2009  (14)
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  • 1
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    Hyperdeterminants as integrable discrete systems (2009)
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    Publication Date: 2020-12-11
    Description: We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A.~Cayley in 1845. We prove integrability (understood as $4d$-consistency) of a nonlinear difference equation defined by the $2 \times 2 \times 2$ - hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the $2\times 2\times 2\times 2$ - hyperdeterminant.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/postscript
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  • 2
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    On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations (2005)
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    Publication Date: 2014-02-26
    Description: We consider nonlinear, scaling-invariant $N=1$ boson$+$fermion supersymmetric systems whose right-hand sides are homogeneous differential polynomials and satisfy some natural assumptions. We select the super-systems that admit infinitely many higher symmetries generated by recursion operators; we further restrict ourselves to the case when the dilaton dimensions of the bosonic and fermionic super-fields coincide and the weight of the time is half the weight of the spatial variable. We discover five systems that satisfy these assumptions; one system is transformed to the purely bosonic Burgers equation. We construct local, nilpotent, triangular, weakly non-local, and super-recursion operators for their symmetry algebras.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/postscript
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  • 3
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    Supersymmetric representations and integrable super-extensions of the Burgers and Boussinesq equations (2005)
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    Publication Date: 2014-02-26
    Description: New evolutionary supersymmetric systems whose right-hand sides are homogeneous differential polynomials and which possess infinitely many higher symmetries are constructed. Their intrinsic geometry (symmetries, conservation laws, recursion operators, Hamiltonian structures, and exact solutions) is analyzed by using algebraic methods. A supersymmetric $N=1$ representation of the Burgers equation is obtained. An $N=2$ KdV-component system that reduces to the Burgers equation in the diagonal $N=1$ case $\theta^1=\theta^2$ is found; the $N=2$ Burgers equation admits and $N=2$ modified KdV symmetry. A one\/-\/parametric family of $N=0$ super\/-\/systems that exte nd the Burgers equation is described; we relate the systems within this family with the Burgers equation on associative algebras. A supersymmetric boson$+$fermion representation of the dispersionless Boussinesq equation is investigated. We solve this equation explicitly and construct its integrable deformation that generates two infinite sequences of the Hamiltonians. The Boussinesq equation with dispersion is embedded in a one-parametric family of two-component systems with dissipation. We finally construct a three-parametric supersymmetric system that incorporates the Boussinesq equation with dispersion and dissipation but never retracts to it for any values of the parameters.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/postscript
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  • 4
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    Positions in the Game of Go as Complex Systems (2009)
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    Publication Date: 2020-12-11
    Language: English
    Type: article , doc-type:article
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  • 5
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    Hyperdeterminants as integrable discrete systems (2009)
    Details
    Publication Date: 2020-12-11
    Language: English
    Type: article , doc-type:article
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  • 6
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    Partial and complete linearization of PDEs based on conservation laws (2005)
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    Publication Date: 2014-02-26
    Description: A method based on infinite parameter conservation laws is described to factor linear differential operators out of nonlinear partial differential equations (PDEs) or out of differential consequences of nonlinear PDEs. This includes a complete linearization to an equivalent linear PDE (-system) if that is possible. Infinite parameter conservation laws can be computed, for example, with the computer algebra package {\sc ConLaw}.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 7
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    Some classifications of hyperbolic vector evolution equations (2005)
    Details
    Publication Date: 2022-03-11
    Description: Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several $O(N)$-invariant classes of hyperbolic equations $Utx=f(U,Ut,Ux)$ for an $N$-component vector $U(t,x)$ are considered. In each class we find all scaling-homogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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    OPUS
  • 8
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    Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket (2005)
    Details
    Publication Date: 2014-02-26
    Description: Quadratic Hamiltonians with a linear Lie-Poisson bracket have a number of applications in mechanics. For example, the Lie-Poisson bracket $e(3)$ includes the Euler-Poinsot model describing motion of a rigid body around a fixed point under gravity and the Kirchhoff model describes the motion of a rigid body in ideal fluid. Advances in computer algebra algorithms, in implementations and hardware, together allow the computation of Hamiltonians with higher degree first integrals providing new results in the search for integrable models. A computer algebra module enabling related computations in a 3-dimensional vector formalism is described.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 9
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    Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I (2005)
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    Publication Date: 2014-02-26
    Description: We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov--Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of $2^{\mbox{\scriptsize nd }}$order systems with a $3^{\mbox{\scriptsize rd }}$order or a $4^{\mbox{\scriptsize th }}$order symmetry and $3^{\mbox{\scriptsize rd }}$order systems with a $5^{\mbox{\scriptsize th }}$order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 10
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    On solving large systems of polynomial equations appearing in Discrete Differential Geometry (2008)
    Details
    Publication Date: 2020-08-05
    Description: The paper describes a method for solution of very large overdetermined algebraic polynomial systems on an example that appears from a classification of all integrable 3-dimensional scalar discrete quasilinear equations $Q_3=0$ on an elementary cubic cell of the lattice ${\mathbb Z}^3$. The overdetermined polynomial algebraic system that has to be solved is far too large to be formulated. A probing' technique which replaces independent variables by random integers or zero allows to formulate subsets of this system. An automatic alteration of equation formulating steps and equation solving steps leads to an iteration process that solves the computational problem.
    Keywords: ddc:510
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/postscript
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