@misc{SokolovWolf, author = {Sokolov, Vladimir V. and Wolf, Thomas}, title = {Classification of integrable polynomial vector evolution equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6601}, number = {01-34}, abstract = {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.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {The Symbolic Integration of Exact PDEs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5702}, number = {00-02}, abstract = {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.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {The integration of systems of linear PDEs using conservation laws of syzygies}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6751}, number = {02-08}, abstract = {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.}, language = {en} } @misc{TsuchidaWolf, author = {Tsuchida, Takayuki and Wolf, Thomas}, title = {Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8391}, number = {05-05}, abstract = {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.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {Partial and complete linearization of PDEs based on conservation laws}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8409}, number = {05-06}, abstract = {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}.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket}, doi = {10.1007/s10714-006-0293-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8414}, number = {05-07}, abstract = {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.}, language = {en} } @misc{AncoWolf, author = {Anco, Stephen and Wolf, Thomas}, title = {Some classifications of hyperbolic vector evolution equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8429}, number = {05-08}, abstract = {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.}, language = {en} } @misc{SokolovWolf, author = {Sokolov, Vladimir V. and Wolf, Thomas}, title = {Integrable quadratic Hamiltonians on so(4) and so(3,1)}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8085}, number = {04-33}, abstract = {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.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {Applications of CRACK in the Classification of Integrable Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7159}, number = {02-48}, abstract = {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.}, language = {en} } @misc{TsarevWolf, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Hyperdeterminants as integrable discrete systems}, issn = {1438-0064}, doi = {10.1088/1751-8113/42/45/454023}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11292}, number = {09-17}, abstract = {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.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {On solving large systems of polynomial equations appearing in Discrete Differential Geometry}, issn = {1438-0064}, doi = {10.1134/S0361768808020047}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10646}, number = {08-10}, abstract = {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.}, language = {en} } @misc{AncoBlumanWolf, author = {Anco, Stephen and Bluman, George and Wolf, Thomas}, title = {Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Admitted Conservation Laws}, organization = {Department of Mathematics, Brock University, St. Catharines, ON Canada L2S 3A1}, issn = {1438-0064}, doi = {10.1007/s10440-008-9205-7}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10652}, number = {08-11}, abstract = {An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists. Previous methods yielded such conditions from admitted point or contact symmetries of the nonlinear PDE. Through examples, these two linearization approaches are contrasted.}, language = {en} } @misc{TsarevWolf, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Classification of 3-dimensional integrable scalar discrete equations}, issn = {1438-0064}, doi = {10.1007/s11005-008-0230-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10667}, number = {08-13}, abstract = {We classify all integrable 3-dimensional scalar discrete affine linear equations \$Q_3=0\$ on an elementary cubic cell of the lattice \${\mathbb Z}^3\$. An equation \$Q_3=0\$ \%of such form is called integrable if it may be consistently imposed on all \$3\$-dimensional elementary faces of the lattice \${\mathbb Z}^4\$. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only ontrivial(non-linearizable) integrable equation from this class is the well-known dBKP-system.}, language = {en} } @misc{WolfNeun, author = {Wolf, Thomas and Neun, Winfried}, title = {About a Computer Algebra based online Mathtest}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10688}, number = {08-14}, abstract = {The article describes the online mathematics test {\tt http://lie.math.brocku.ca/mathtest}, its typical applications and experiences gathered.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {The Parametric Solution of Underdetermined linear ODEs}, issn = {1438-0064}, doi = {10.1134/S0361768811020113}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10693}, number = {08-15}, abstract = {The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are arbitrary analytic functions in the independent variable. A second conceptual aim is to present an algorithm that is in some sense dual to the fundamental Euclids algorithm, and thus an alternative to the special case of a Gr\"{o}bner basis algorithm as it is used for solving linear ODE-systems. In the paper Euclids algorithm and the new dual version' are compared and their complementary strengths are analysed on the task of solving underdetermined ODEs. An implementation of the described algorithm is interactively accessible at http://lie.math.brocku.ca/crack/uode.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {A comparison of four approaches to the calculation of conservation laws}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3896}, number = {SC-99-01}, abstract = {The paper compares computational aspects of four approaches to compute conservation laws of single differential equations or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints may be given. Examples of new conservation laws include non-polynomial expressions, an explicit variable dependence and conservation laws involving arbitrary functions. Examples involve the following equations: Ito, Liouville, Burgers, Kadomtsev-Petviashvili, Karney-Sen-Chu-Verheest, Boussinesq, Tzetzeica, Benney.}, language = {en} } @misc{SokolovWolf, author = {Sokolov, Vladimir V. and Wolf, Thomas}, title = {A symmetry test for quasilinear coupled systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3901}, number = {SC-99-02}, abstract = {It is well known that the following class of systems of evolution equations \begin{eqnarray} \label{nsgen} \cases{ u_{t}=u_{xx}+F(u,v,u_x,v_x),\cr v_{t}=-v_{xx}+G(u,v,u_x,v_x),\cr} \end{eqnarray} is very rich in integrable cases. The complete classification problem is very difficult. Here we consider only the most interesting (from our opinion) subclass of systems (1). Namely, we consider equations linear in all derivatives of the form \begin{eqnarray} \label{kvazgen} \cases{ u_t = u_{xx} + A_{1}(u,v) u_x + A_{2}(u,v) v_x + A_{0}(u,v)\cr v_t = - v_{xx} + B_{1}(u,v) v_x + B_{2}(u,v) u_x + B_{0}(u,v). \cr} \end{eqnarray} without any restrictions on the functions \$A_{i}(u,v), B_{i}(u,v)\$.}, language = {en} } @misc{KiselevWolf, author = {Kiselev, Arthemy V. and Wolf, Thomas}, title = {On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8850}, number = {05-52}, abstract = {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.}, language = {en} } @misc{KiselevWolf, author = {Kiselev, Arthemy V. and Wolf, Thomas}, title = {Supersymmetric representations and integrable super-extensions of the Burgers and Boussinesq equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8869}, number = {05-53}, abstract = {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.}, language = {en} } @misc{Wolf, author = {Wolf, Thomas}, title = {The program CRACK for solving PDEs in General Relativity. Lecture given at the 152. WE-Heraeus-Seminar on RELATIVITY AND SCIENTIFIC CUMPUTING: Computer Algebra, Numerics, Visualization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1881}, number = {SC-95-22}, abstract = {In the introduction an approach to solving differential equations is motivated in which non-linear DEs are not attacked directly but properties like infinitesimal symmetries or the existence of an equivalent variational principle are investigated. In the course of such investigations overdetermined PDE-systems are generated which are to be solved (where the term `overdetermined' just stands for `more conditions than free functions'). In section 2.\ algorithms for simplifying and solving overdetermined PDE systems are given together with examples. References for more details of the corresponding program {\tt CRACK}, written by A.\ Brand and the author, are given. In sections 3.-05.\ applications of the program {\tt CRACK} are discussed. The first application is the investigation of symmetries of space-time metrics by solving Killing equations for Killing vectors and Killing tensors and their integrability conditions. A program {\tt CLASSYM} that formulates these equations, written by G.\ Grebot, is briefly described. In section 4.\ an example of the original application of {\tt CRACK} is discussed which is the determination of symmetries of a PDE system. The problem is to find the symmetries of an unusual unified field theory of gravitational and hadronic interactions. The application of symmetries with a program {\tt APPLYSYM} is the content of section 5.\ where an ODE, resulting from an attempt to generalize Weyl's class of solutions of Einsteins field equations, is solved. The final section is devoted to future work on, first, making a general PDE-solver more flexible and effective, and secondly, on applying it to more advanced applications. This section contains so far unpublished work. An example requiring the extension of {\tt CRACK} to deal with non-polynomial non-linearities results from an investigation of interior solutions of Einstein's field equations for a spherically symmetric perfect fluid in shear-free motion by H.\ Stephani. A possible future application of {\tt CRACK} is the determination of Killing tensors of higher rank. In the last sub-section an algorithm for formulating corresponding integrability conditions has been sketched. The maximal number of Killing tensors of rank \$r\$ in a \$n\$-dimensional Riemannian space has been found to be \$\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) \left( ^{ n+r}_{\;\;\,r} \right)\$.}, language = {en} }