@misc{AncoMohiuddinWolf2012, author = {Anco, Stephen and Mohiuddin, Mohammad and Wolf, Thomas}, title = {Travelling waves and conservation laws for complex mKdV-type equations}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/1110.2403}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16124}, year = {2012}, abstract = {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.}, language = {en} } @misc{WolfAmadi2017, author = {Wolf, Thomas and Amadi, Chimaobi}, title = {Rational Solutions of Underdetermined Polynomial Equations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64488}, year = {2017}, abstract = {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.}, language = {en} } @misc{SokolovWolf2001, 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}, year = {2001}, 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{Wolf2002, 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}, year = {2002}, 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{AncoBlumanWolf2008, 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}, year = {2008}, 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{Wolf2008, 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}, year = {2008}, 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{Wolf2008, 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}, year = {2008}, 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{TsarevWolf2008, author = {Tsarev, Sergey and Wolf, Thomas}, title = {Classification of 3-dimensional integrable scalar discrete equations}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/0706.2464}, doi = {10.1007/s11005-008-0230-2}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10667}, number = {08-13}, year = {2008}, 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} }