TY - GEN A1 - Sokolov, Vladimir V. A1 - Wolf, Thomas T1 - Classification of integrable polynomial vector evolution equations N2 - Several classes of systems of evolution equations with one or two vector unknowns are considered. We investigate also systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are listed. T3 - ZIB-Report - 01-34 Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6601 ER - TY - GEN A1 - Wolf, Thomas T1 - The Symbolic Integration of Exact PDEs N2 - An algorithm is described to decide if a given polynomial differential expression $\Delta$ of multivariate functions is exact, i.e. whether there exists a first integral $P$ such that $D_xP = \Delta$ for any one of a set of variables $x$ and to provide the integral $P$. A generalization is given to allow integration in the case that the exactness is prohibited by terms which contain only functions of not all the independent variables. T3 - ZIB-Report - 00-02 KW - symbolic integration KW - partial differential equations Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5702 ER - TY - GEN A1 - Wolf, Thomas T1 - The integration of systems of linear PDEs using conservation laws of syzygies N2 - The purpose of the paper is to formulate and use syzygies for systems of linear PDEs. The computation of an equivalent of a GCD for linear partial differential operators will save us their factorization which is otherwise only possible algorithmically in special cases. After showing the computation with the new and the traditional method and comparing both in the next three sections, the algorithm is explained in general and an overview is given. T3 - ZIB-Report - 02-08 KW - PDE KW - syzygies KW - conservation laws Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6751 ER - TY - GEN A1 - Tsuchida, Takayuki A1 - Wolf, Thomas T1 - Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I N2 - 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. T3 - ZIB-Report - 05-05 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8391 ER - TY - GEN A1 - Wolf, Thomas T1 - Partial and complete linearization of PDEs based on conservation laws N2 - 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}. T3 - ZIB-Report - 05-06 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8409 ER - TY - GEN A1 - Wolf, Thomas T1 - Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket N2 - 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. T3 - ZIB-Report - 05-07 KW - Hamilton systems KW - Integrability KW - Computer algebra Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8414 ER - TY - GEN A1 - Anco, Stephen A1 - Wolf, Thomas T1 - Some classifications of hyperbolic vector evolution equations N2 - 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. T3 - ZIB-Report - 05-08 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8429 ER - TY - GEN A1 - Sokolov, Vladimir V. A1 - Wolf, Thomas T1 - Integrable quadratic Hamiltonians on so(4) and so(3,1) N2 - We investigate a special class of quadratic Hamiltonians on $so(4)$ and $so(3,1)$ and describe Hamiltonians that have additional polynomial integrals. One of the main results is a new integrable case with an integral of sixth degree. T3 - ZIB-Report - 04-33 KW - integrable quadratic Hamiltonians KW - polynomial integrals KW - classification Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8085 ER - TY - GEN A1 - Wolf, Thomas T1 - Applications of CRACK in the Classification of Integrable Systems N2 - The talk given by the author at the CRM workshop on Superintegrability in Sep.\ 2002 and this related paper report on work in two subjects. One is the collaboration with Vladimir Sokolov and Takayuki Tsuchida in an effort to classify polynomial integrable vector evolution equations. The other is the computer algebra package {\sc Crack} which did the main computations in solving large bi-linear algebraic systems. Although originally designed to solve over-determined systems of partial differential equations a number of extensions made {\sc Crack} a powerful tool for solving systems of bi-linear algebraic equations. Such systems turn up in many different classification problems some of which were investigated by other participants of this workshop. Two additional applications are outlined. In the talk on which this article is based a method to reduce the length of equations was presented which proved to be useful in solving the bi-linear algebraic systems. Due to numerous asked questions about the computer program, a more complete overview is given in the appendix. T3 - ZIB-Report - 02-48 KW - Integrability KW - evolution equations KW - quadratic Hamiltonians KW - Computer algebra KW - Bi-linear algebraic systems KW - conservation laws Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7159 ER - TY - GEN A1 - Tsarev, Sergey A1 - Wolf, Thomas T1 - Hyperdeterminants as integrable discrete systems N2 - 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. T3 - ZIB-Report - 09-17 KW - hyper determinants KW - integrable systems KW - discrete equations KW - large polynomial systems KW - computer algebra KW - Form KW - Reduce KW - Singular Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11292 SN - 1438-0064 ER -