Library

Description: 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.

Description: 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.

Description: 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.

Description: 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.

Description: 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).

Description: 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.

Description: 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.

Description: 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.

Description: 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 $\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) \left( ^{ n+r}_{\;\;\,r} \right)$.

Description: 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.