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 - TY - GEN A1 - Wolf, Thomas T1 - On solving large systems of polynomial equations appearing in Discrete Differential Geometry N2 - 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. T3 - ZIB-Report - 08-10 KW - integrable systems KW - discrete equations KW - large polynomial systems KW - computer algebra KW - REDUCE Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10646 SN - 1438-0064 ER - TY - GEN A1 - Anco, Stephen A1 - Bluman, George A1 - Wolf, Thomas T1 - Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Admitted Conservation Laws N2 - 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. T3 - ZIB-Report - 08-11 KW - conservation laws KW - linearization KW - symmetries Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10652 SN - 1438-0064 ER - TY - GEN A1 - Tsarev, Sergey A1 - Wolf, Thomas T1 - Classification of 3-dimensional integrable scalar discrete equations N2 - 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. T3 - ZIB-Report - 08-13 KW - integrable systems KW - discrete equations KW - large polynomial systems KW - computer algebra KW - REDUCE KW - FORM KW - Crack Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10667 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas A1 - Neun, Winfried T1 - About a Computer Algebra based online Mathtest N2 - The article describes the online mathematics test {\tt http://lie.math.brocku.ca/mathtest}, its typical applications and experiences gathered. T3 - ZIB-Report - 08-14 KW - accessment of mathematical skills KW - computer algebra KW - online Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10688 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas T1 - The Parametric Solution of Underdetermined linear ODEs N2 - 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. T3 - ZIB-Report - 08-15 KW - linear ODEs KW - underdetermined systems KW - Euclid's algorithm KW - computer algebra KW - Reduce KW - Crack Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10693 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas T1 - A comparison of four approaches to the calculation of conservation laws N2 - 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. T3 - ZIB-Report - SC-99-01 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3896 ER - TY - GEN A1 - Sokolov, Vladimir V. A1 - Wolf, Thomas T1 - A symmetry test for quasilinear coupled systems N2 - 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 $A_{i}(u,v), B_{i}(u,v)$. T3 - ZIB-Report - SC-99-02 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3901 ER - TY - GEN A1 - Kiselev, Arthemy V. A1 - Wolf, Thomas T1 - On weakly non-local, nilpotent, and super-recursion operators for N=1 super-equations N2 - 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. T3 - ZIB-Report - 05-52 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8850 ER - TY - GEN A1 - Kiselev, Arthemy V. A1 - Wolf, Thomas T1 - Supersymmetric representations and integrable super-extensions of the Burgers and Boussinesq equations N2 - 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. T3 - ZIB-Report - 05-53 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8869 ER - TY - GEN A1 - Wolf, Thomas T1 - 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 N2 - 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 $\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) \left( ^{ n+r}_{\;\;\,r} \right)$. T3 - ZIB-Report - SC-95-22 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1881 ER - TY - GEN A1 - Mars, Marc A1 - Wolf, Thomas T1 - G2 Perfect-Fluid Cosmologies with a proper conformal Killing vector N2 - We study the Einstein field equations for spacetimes admitting a maximal two-dimensional abelian group of isometries acting orthogonally transitively on spacelike surfaces and, in addition, with at least one conformal Killing vector. The three-dimensional conformal group is restricted to the case when the two-dimensional abelian isometry subalgebra is an ideal and it is also assumed to act on non-null hypersurfaces (both, spacelike and timelike cases are studied). We consider both, diagonal and non-diagonal metrics and find all the perfect-fluid solutions under these assumptions (except those already known). We find four families of solutions, each one containing arbitrary parameters for which no differential equations remain to be integrated. We write the line-elements in a simplified form and perform a detailed study for each of these solutions, giving the kinematical quantities of the fluid velocity vector, the energy-density and pressure, values of the parameters for which the energy conditions are fulfilled everywhere, the Petrov type, the singularities in the spacetimes and the Friedmann-Lema\^{\i}tre-Robertson-Walker metrics contained in each family. T3 - ZIB-Report - SC-97-08 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2774 ER - TY - GEN A1 - Wolf, Thomas T1 - Programs for Applying Symmetries of PDEs N2 - In this paper the programs {\tt APPLYSYM}, {\tt QUASILINPDE} and {\tt DETRAFO} are described which aim at the utilization of infinitesimal symmetries of differential equations. The purpose of {\tt QUASILINPDE} is the general solution of quasilinear PDEs. This procedure is used by {\tt APPLYSYM} for the application of point symmetries for either \begin{itemize} \item calculating similarity variables to perform a point transformation which lowers the order of an ODE or effectively reduces the number of explicitly occuring independent variables in a PDE(-system) or for \item generalizing given special solutions of ODEs/PDEs with new constant parameters. \end{itemize} The program {\tt DETRAFO} performs arbitrary point- and contact transformations of ODEs/PDEs and is applied if similarity and symmetry variables have been found. The program {\tt APPLYSYM} is used in connection with the program {\tt LIEPDE} for formulating and solving the conditions for point- and contact symmetries which is described in LIEPDE(1992). The actual problem solving is done in all these programs through a call to the package {\tt CRACK} for solving overdetermined PDE-systems. T3 - ZIB-Report - SC-95-05 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1710 ER - TY - GEN A1 - Wolf, Thomas T1 - Structural equations for Killing tensors of arbitrary rank N2 - An algorithm is given for bringing the equations of monomial first integrals of arbitrary degree of the geodesic motion in a Riemannian space $V_n$ into the form $(F_A)_{;k} = \sum_B \Gamma_{kAB} F_B$. The $F_A$ are the components of a Killing tensor $K_{i_1\ldots i_r}$ of arbitrary rank $r$ and its symmetrized covariant derivatives. Explicit formulas are given for rank 1,2 and 3. %The maximal number of Killing tensors %(reducible + non-reducible) is found to be %$\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) % \left( ^{ n+r}_{\;\;\,r} \right)$. Killing tensor equations in structural form allow the formulation of algebraic integrability conditions and are supposed to be well suited for integration as it is demonstrated in the case of flat space. An alternative proof of the reducibility of these Killing tensors is given which shows the correspondence to structural equations for rank 2 Killing tensors as formulated by Hauser & Malhiot. They used tensors with different symmetry properties. T3 - ZIB-Report - SC-97-62 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3310 ER - TY - GEN A1 - Stephani, Hans A1 - Wolf, Thomas T1 - Spherically symmetric perfect fluids in shearfree motion - the symmetry approach N2 - In General Relativity, the motion of expanding shearfree perfect fluids is governed by the ordinary differential equation $y^{\prime \prime }=$ $% F(x)\,y^2$ , where $F$ is an arbitrary function from which the equation of state can be computed. A complete symmetry analysis of this differential equation is given; its solutions are classified according to this scheme, and in particular the relation to Wyman's Painlev\'e analysis is clarified. T3 - ZIB-Report - SC-96-14 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2253 ER - TY - GEN A1 - Wolf, Thomas T1 - A Study of Genetic Algorithms solving a combinatorial Puzzle N2 - The suitability of Genetic Algorithms (GAs) to solve a combinatorial problem with only one solution is investigated. The dependence of the performance is studied for GA-hard and GA-soft fitness functions, both with a range of different parameter values and different encodings. T3 - ZIB-Report - SC-98-01 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3445 ER - TY - GEN A1 - Wolf, Thomas A1 - Brandt, Andreas A1 - Mohammadzadeh, Majid T1 - Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions N2 - Three different approaches for the determination of conservation laws of differential equations are presented. For three corresponding REDUCE computer algebra programs CONLAW1/2/3 the necessary subroutines are discribed. One of them simplifies general solutions of overdetermined PDE systems so that all remaining free functions and constants correspond to independent conservation laws. It determines redundant functions and constants in differential expressions and is equally useful for the determination of symmetries or the fixing of gauge freedom in differential expressions. T3 - ZIB-Report - SC-98-05 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3482 ER - TY - GEN A1 - Wolf, Thomas T1 - Killing Pairs in Flat Space N2 - In this paper the integrability conditions for Killing pairs in flat spaces are investigated and it is shown that only trivial Killing pairs exist. T3 - ZIB-Report - SC-97-58 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3275 ER - TY - GEN A1 - Wolf, Thomas A1 - Efimovskaya, Olga T1 - On integrability of the Kontsevich non-abelian ODE system N2 - We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich. We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries. T3 - ZIB-Report - 11-24 KW - integrability, Lax pairs, noncommutative ODE, Laurent ODE Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13113 ER - TY - GEN A1 - Euler, Marianna A1 - Euler, Norbert A1 - Wolf, Thomas T1 - The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws N2 - Recently, Holm and Ivanov, proposed and studied a class of multi-component generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, {\it J. Phys A: Math. Theor} {\bf 43}, 492001 (20pp), 2010]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete sent of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2). T3 - ZIB-Report - 12-30 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16087 SN - 1438-0064 ER - TY - GEN A1 - Odesskii, Alexander A1 - Wolf, Thomas T1 - Compatible quadratic Poisson brackets related to a family of elliptic curves N2 - We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in $n$ generators. Explicit formulas for Casimir elements of this elliptic Poisson structure are obtained. T3 - ZIB-Report - 12-31 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16097 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas A1 - Schrüfer, Eberhard A1 - Webster, Kenneth T1 - Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs N2 - The paper reports on a computer algebra program {\sc LSSS} (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M.\ Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14. T3 - ZIB-Report - 12-32 KW - linear algebraic systems, integrability, Lax pairs, noncommutative ODE, Laurent ODE Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16102 SN - 1438-0064 ER - TY - GEN A1 - Bouchareb, Adel A1 - Clément, Gérard A1 - Chen, Chiang-Mei A1 - Wolf, Thomas A1 - Gal'tsov, Dmitri V. A1 - Scherbluk, Nikolai G. T1 - G_2 generating technique for minimal D=5 supergravity and black rings N2 - A solution generating technique is developed for $D=5$ minimal supergravity with two commuting Killing vectors based on the $G_2$ U-duality arising in the reduction of the theory to three dimensions. The target space of the corresponding 3-dimensional sigma-model is the coset $G_{2(2)}/(SL(2,R)\times SL(2,R))$. Its isometries constitute the set of solution generating symmetries. These include two electric and two magnetic Harrison transformations with the corresponding two pairs of gauge transformations, three $SL(2,R) \; S$-duality transformations, and the three gravitational scale, gauge and Ehlers transformations (altogether 14). We construct a representation of the coset in terms of $7\times 7$ matrices realizing the automorphisms of split octonions. Generating a new solution amounts to transforming the coset matrices by one-parametric subgroups of $G_{2(2)}$ and subsequently solving the dualization equations. Using this formalism we derive a new charged black ring solution with two independent parameters of rotation. T3 - ZIB-Report - 10-14 KW - supergravity KW - black rings KW - Killing vectors KW - duality Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11732 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas T1 - Merging solutions of non-linear algebraic systems N2 - In solving large polynomial algebraic systems that are too big for standard Gröbner basis techniques one way to make progress is to introduce case distinctions. This divide and conquer technique can be beneficial if the algorithms and computer programs know how to take advantage of inequalities. A further hurdle is the form of the resulting general solutions which often have unnecessarily many branches. In this paper we discuss a procedure to merge solutions by dropping inequalities which are associated with them and, if necessary, by re-parametrizing solutions. In the appendix the usefulness of the procedure is demonstrated in the classification of quadratic Hamiltonians with a Lie-Poisson bracket $e(3)$. This application required the solution of algebraic systems with over 200 unknowns, 450 equations and between 5000 and 9000 terms. T3 - ZIB-Report - 10-13 KW - computer algebra KW - polynomial systems KW - Lie-Poisson brackets KW - quadratic Hamiltonians Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11769 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas T1 - A Dynamical Systems Approach for Static Evaluation in Go N2 - In the paper arguments are given why the concept of static evaluation (SE) has the potential to be a useful extension to Monte Carlo tree search. A new concept of modeling SE through a dynamical system is introduced and strengths and weaknesses are discussed. The general suitability of this approach is demonstrated. A Remark: Among users of the Internet Go server KGS the abbreviation SE is used for 'Score Estimator'. Although different from 'Static Evaluation' a score estimator is easily obtained from static evaluation by adding up probabilities of chains to be alive at the end of the game or points to be owned by White or Black T3 - ZIB-Report - 10-15 KW - discrete dynamical systems KW - Computer Go KW - static evaluation Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11771 SN - 1438-0064 ER - TY - GEN A1 - Hussin, V. A1 - Kiselev, A. V. A1 - Krutov, A.O. A1 - Wolf, Thomas T1 - N=2 supersymmetric a=4-KdV hierarchy derived via Gardner's deformation of Kaup-Boussinesq equation N2 - We consider the problem of constructing Gardner's deformations for the $N{=}2$ supersymmetric $a{=}4$--\/Korteweg\/--\/de Vries equation; such deformations yield recurrence relations between the super\/-\/Hamiltonians of the hierarchy. We prove the non\/-\/existence %P.~Mathieu's Open problem on constructing for of supersymmetry\/-\/invariant %Gardner's deformations that %solutions, retract to Gardner's formulas for the KdV equation %whenever it is assumed that, under the %respective component reduction. % in the $N{=}2$ super\/-\/field. the solutions . At the same time, we propose a two\/-\/step scheme for the recursive production of the integrals of motion for the $N{=}2$,\ $a{=}4$--\/SKdV. First, we find a new Gardner's deformation of the Kaup\/--\/Boussinesq equation, which is contained in the bosonic limit of the super\/-\/%$N{=}2$,\ $a{=}4$--\/SKdV hierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super\/- /Hamiltonians of the full $N{=}2$, $a{=}4$--\/SKdV hierarchy. T3 - ZIB-Report - 10-16 KW - Korteweg-de Vries equation KW - Kaup-Boussinesq equation KW - supersymmetry KW - Gardner's deformations KW - bi-Hamilltonian hierarchies Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11787 SN - 1438-0064 ER - TY - GEN A1 - Kiselev, A. V. A1 - Wolf, Thomas T1 - Classification of integrable super-systems using the SsTools environment N2 - A classification problem is proposed for supersymmetric %scaling\/-\/in\-va\-ri\-ant evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson\/-\/fermion systems are discovered under the weighting assumption $|f|=|b|=|D_t|=\oh$. The syntax of the \Reduce\ package \SsTools, which was used for intermediate computations, and the applicability of its procedures to the calculus of super\/-\/PDE are described. T3 - ZIB-Report - 10-17 KW - Integrable super-systems KW - symmetries KW - recursions KW - classifiaction Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11792 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas A1 - Efimovskaya, Olya V. T1 - Classification of integrable quadratic Hamiltonians on e(3) N2 - Linear Poisson brackets on e(3) typical of rigid body dynamics are considered. All quadratic Hamiltonians of Kowalevski type having additional first integral of fourth degree are found. Quantum analogs of these Hamiltonians are listed. T3 - ZIB-Report - 10-18 KW - Linear Poisson brackets on e(3) KW - rigid body dynamics KW - Kowalevski type Hamiltonians KW - first integrals Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11807 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas T1 - Positions in the Game of Go as Complex Systems N2 - The paper gathers evidence showing different dimensions of the game of Go: the continuous and discrete nature of the game and different types of relations between state variables happening on ultra local, local, regional, and global scales. Based on these observations a new continuous local model for describing a board position is introduced. This includes the identification of the basic variables describing a board position and the formulation and solution of a dynamical system for their computation. To be usable as a static evaluation function for a game playing program at least group-wide (regional)aspects will have to be incorporated. T3 - ZIB-Report - 10-19 KW - Computer Go KW - static evaluation KW - discrete dynamical systems Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11823 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas A1 - Amadi, Chimaobi T1 - Rational Solutions of Underdetermined Polynomial Equations N2 - In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our work provides a method to compute rational solutions of single polynomial equations that are typically large with 10^2 ... 10^5 terms and that are heavily underdetermined. It was possible to obtain this functionality by adding a number of new modules for a new type of splitting of equations to the existing package CRACK that is normally used to solve polynomial algebraic and differential systems of equations. T3 - ZIB-Report - 17-33 KW - underdetermined polynomial systems, rational solutions, computer algebra, recreational mathematics Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-64488 SN - 1438-0064 ER - TY - JOUR A1 - Anco, Stephen A1 - Mohiuddin, Mohammad A1 - Wolf, Thomas T1 - Traveling waves and conservation laws for complex mKdV-type equations Y1 - 2012 U6 - https://doi.org/10.1016/j.amc.2012.06.061 VL - 219 IS - 2 SP - 679 EP - 698 ER - TY - JOUR A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction JF - SIGMA 7 N2 - A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation. Y1 - 2011 U6 - https://doi.org/10.3842/SIGMA.2011.066 ER - TY - JOUR A1 - Wolf, Thomas A1 - Efimovskaya, Olga T1 - On Integrability of the Kontsevich Non-Abelian ODE System JF - Letters in Mathematical Physics Y1 - 2012 U6 - https://doi.org/10.1007/s11005-011-0527-4 VL - 100 IS - 2 SP - 161 EP - 170 ER - TY - JOUR A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions JF - Journal of Mathematical Analysis and Applications Y1 - 2011 U6 - https://doi.org/10.1016/j.jmaa.2011.01.073 VL - 379 IS - 2 SP - 748 EP - 763 ER - TY - JOUR A1 - Wolf, Thomas T1 - A Dynamical Systems Approach for Static Evaluation in Go JF - Computational Intelligence and AI in Games, IEEE Transactions on N2 - In the paper, arguments are given as to why the concept of static evaluation has the potential to be a useful extension to Monte Carlo tree search. A new concept of modeling static evaluation through a dynamical system is introduced and strengths and weaknesses are discussed. The general suitability of this approach is demonstrated. Y1 - 2011 U6 - https://doi.org/10.1109/TCIAIG.2011.2141669 VL - 3 IS - 2 SP - 129 EP - 141 ER - TY - JOUR A1 - Hussin, V. A1 - Kiselev, A. V. A1 - Krutov, A.O. A1 - Wolf, Thomas T1 - N=2 Supersymmetric a=4 - Korteweg-de Vries hierarchy derived via Gardner's deformation of Kaup-Bousinesq equation JF - Journal of Mathematical Physics Y1 - 2010 U6 - https://doi.org/10.1063/1.3447731 VL - 51 ER - TY - JOUR A1 - Wolf, Thomas T1 - Positions in the Game of Go as Complex Systems JF - Proceedings of 2009 IEEE Toronto International Conference (TIC-STH 2009) - Science and Technology for Humanity Y1 - 2009 SN - 978-1-4244-3878-5 ER - TY - JOUR A1 - Tsarev, Sergey A1 - Wolf, Thomas T1 - Hyperdeterminants as integrable discrete systems JF - Journal of Physics A: Mathematical and Theoretical Y1 - 2009 U6 - https://doi.org//10.1088/1751-8113/42/45/454023 VL - 42 ER - TY - JOUR A1 - Odesskii, Alexander A1 - Wolf, Thomas T1 - Compatible quadratic Poisson brackets related to a family of elliptic curves JF - Journal of Geometry and Physics Y1 - 2013 SN - 0393-0440 VL - 63 IS - 0 SP - 107 EP - 117 ER - TY - JOUR A1 - Anco, Stephen A1 - MacNaughton, Steven A. A1 - Wolf, Thomas T1 - Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions JF - Journal of Mathematical Physics Y1 - 2012 U6 - https://doi.org/10.1063/1.4711814 VL - 53 IS - 5 ER - TY - JOUR A1 - Wolf, Thomas A1 - Schrüfer, Eberhard A1 - Webster, Kenneth T1 - Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs JF - Programming and Computer Software Y1 - 2012 U6 - https://doi.org/10.1134/S0361768812020065 SN - 0361-7688 VL - 38 IS - 2 SP - 73 EP - 83 ER - TY - JOUR A1 - Euler, Marianna A1 - Euler, Norbert A1 - Wolf, Thomas T1 - The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws JF - Journal of Nonlinear Mathematical Physics N2 - Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa–Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor. 43 (2010) 492001 (20pp)]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete set of first-order integrating factors for the systems in Cauchy–Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2). Y1 - 2013 U6 - https://doi.org/10.1142/S1402925112400025 VL - 19 IS - 1 SP - 13 EP - 22 ER - TY - GEN A1 - Anco, Stephen A1 - Mohiuddin, Mohammad A1 - Wolf, Thomas T1 - Travelling waves and conservation laws for complex mKdV-type equations N2 - Travelling waves and conservation laws are studied for a wide class of $U(1)$-invariant complex mKdV equations containing the two known integrable generalizations of the ordinary (real) mKdV equation. The main results on travelling waves include deriving new complex solitary waves and kinks that generalize the well-known mKdV $\sech$ and $\tanh$ solutions. The main results on conservation laws consist of explicitly finding all 1st order conserved densities that yield phase-invariant counterparts of the well-known mKdV conserved densities for momentum, energy, and Galilean energy, and a new conserved density describing the angular twist of complex kink solutions. T3 - ZIB-Report - 12-34 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16124 SN - 1438-0064 ER - TY - GEN A1 - Anco, Stephen A1 - MacNaughton, Steven A. A1 - Wolf, Thomas T1 - Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions N2 - Symmetries and conservation laws are studied for two classes of physically and analytically interesting radial wave equations with power nonlinearities in multi-dimensions. The results consist of two main classifications: all symmetries of point type and all conservation laws of a general energy-momentum type are explicitly determined, including those such as dilations, inversions, similarity energies and conformal energies that exist only for special powers or dimensions. In particular, all variational cases (when a Lagrangian formulation exists) and non-variational cases (when no Lagrangian exists) for these wave equations are considered. As main results, the classification yields generalized energies and radial momenta in certain non-variational cases, which are shown to arise from a new type of Morawetz dilation identity that produces conservation laws for each of the two classes of wave equations in a different way than Noether's theorem. T3 - ZIB-Report - 12-33 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-16111 SN - 1438-0064 ER - TY - GEN A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions N2 - A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in $n>1$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors. T3 - ZIB-Report - 11-26 KW - semilinear heat equation, similarity reduction, exact solutions, group foliation, symmetry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13213 ER - TY - GEN A1 - Anco, Stephen A1 - Ali, Sajid A1 - Wolf, Thomas T1 - Exact solutions of nonlinear partial differential equations by the method of group foliation reduction N2 - A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method is based on group foliation reduction and employs a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation. T3 - ZIB-Report - 11-25 KW - semilinear heat equation, similarity reduction, exact solutions, group foliation, symmetry Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-13171 ER -