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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.
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.
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.
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.
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.
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 $A_{i}(u,v), B_{i}(u,v)$.
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.
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.
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 $\frac{1}{r+1}\left( ^{n + r - 1}_{\;\;\;\;\,r} \right) \left( ^{ n+r}_{\;\;\,r} \right)$.