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 - 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 - 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 - 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 - 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 -