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10
Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Forms
, 1996
"... We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a consta ..."
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Cited by 42 (14 self)
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We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a constant rank condition. The algorithm is useful for classifying initial value problems for determined pde systems and can yield dramatic simplifications of complex overdetermined nonlinear pde systems. Such overdetermined systems arise in analysis of physical pdes for reductions to odes using the Nonclassical Method, the search for exact solutions of Einstein's field equations and the determination of discrete symmetries of differential equations. Application of the algorithm to the associated nonlinear overdetermined system of 856 pdes arising when the Nonclassical Method is applied to a cubic nonlinear Schrodinger system yields new results. Our algorithm combines features of geometric involutiv...
Algorithms for Symmetric Differential Systems
, 2001
"... Over determined systems of partial dierential equations may be studied using differentialelimination algorithms as a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are eectively intractable by these methods due to the ex ..."
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Cited by 25 (0 self)
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Over determined systems of partial dierential equations may be studied using differentialelimination algorithms as a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are eectively intractable by these methods due to the expression swell incurred in the intermediate stages of the calculations. This can happen when, for example, the input system depends on many variables and is invariant under a large rotation group, so that there is no natural choice of term ordering in the elimination and reduction processes. This article
Existence and Uniqueness Theorems for Formal Power Series Solutions of Analytic Differential Systems
, 1999
"... We present Existence and Uniqueness Theorems for formal power series solutions of analytic systems of pde in a certain form. This form can be obtained by a finite number of differentiations and eliminations of the original system, and allows its formal power series solutions to be computed in an alg ..."
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Cited by 12 (6 self)
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We present Existence and Uniqueness Theorems for formal power series solutions of analytic systems of pde in a certain form. This form can be obtained by a finite number of differentiations and eliminations of the original system, and allows its formal power series solutions to be computed in an algorithmic fashion. The resulting reduced involutive form (rif 0 form) produced by our rif 0 algorithm is a generalization of the classical form of Riquier and Janet, and that of Cauchy Kovalevskaya. We weaken the assumption of linearity in the highest derivatives in those approaches to allow for systems which are nonlinear in their highest derivatives. A new formal development of Riquier's theory is given, with proofs, modeled after those in Grobner Basis Theory. For the nonlinear theory, the concept of relative Riquier Bases is introduced. This allows for the easy extension of ideas from the linear to the nonlinear theory. The essential idea is that an arbitrary nonlinear system can ...
Fast Differential Elimination in C: The CDiffElim Environment
, 2000
"... We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in different ..."
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Cited by 9 (7 self)
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We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in differential elimination algorithms, such as exhaustion of computer memory due to intermediate expression swell, and failure to complete due to the massive number of calculations involved. These strategies include lowlevel memory management strategies and data representations that are tailored for efficient differential elimination algorithms. These strategies, which are coded in a lowlevel C implementation, seem much more difficult to implement in highlevel general purpose computer algebra systems. A differential elimination algorithm written in this environment is applied to the determination of symmetry properties of classes of n+1dimensional coupled nonlinear partial differential equations of form iut+r2u+ i a(t)jxj2 + b(t) \Delta x + c(t) + djuj 4n j u = 0; where u is an mcomponent vectorvalued function. The resulting systems of differential equations for the symmetries have been made available on the web, to be used as benchmark systems for other researchers. The new differential elimination algorithm in C, runs on the test suite an average of 400 times faster than our RifSimp algorithm in Maple.
Universal characteristic decomposition of radical differential ideals
, 2006
"... We call a differential ideal universally characterizable, if it is characterizable w.r.t. any ranking on partial derivatives. We propose a factorizationfree algorithm that represents a radical differential ideal as a finite intersection of universally characterizable ideals. The algorithm also cons ..."
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Cited by 2 (1 self)
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We call a differential ideal universally characterizable, if it is characterizable w.r.t. any ranking on partial derivatives. We propose a factorizationfree algorithm that represents a radical differential ideal as a finite intersection of universally characterizable ideals. The algorithm also constructs a universal characteristic set for each universally characterizable component, i.e., a finite set of differential polynomials that contains a characterizing set of the ideal w.r.t. any ranking. As a part of the proposed algorithm, the following problem of satisfiability by a ranking is efficiently solved: given a finite set of differential polynomials with a derivative selected in each polynomial, determine whether there exists a ranking w.r.t. which the selected derivatives are leading derivatives and, if so, construct such a ranking. Key words: differential algebra, radical differential ideals, factorizationfree algorithms, characteristic decomposition, universal characteristic sets, differential rankings 1.
Symmetry condition in terms of Lie brackets
, 2009
"... A passive orthonomic system of PDEs defines a submanifold in the corresponding jet manifold, coordinated by so called parametric derivatives. We restrict the total differential operators and the prolongation of an evolutionary vector field v to this submanifold. We show that the vanishing of their c ..."
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Cited by 1 (1 self)
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A passive orthonomic system of PDEs defines a submanifold in the corresponding jet manifold, coordinated by so called parametric derivatives. We restrict the total differential operators and the prolongation of an evolutionary vector field v to this submanifold. We show that the vanishing of their commutators is equivalent to v being a generalized symmetry of the system.
JanetRiquier Theory and the RiemannLanczos Problems in 2 and 3 Dimensions
, 2002
"... The RiemannLanczos problem for 4dimensional manifolds was discussed by Bampi and Caviglia. Using exterior differential systems they showed that it was not an involutory differential system until a suitable prolongation was made. Here, we introduce the alternative JanetRiquier theory and use it to ..."
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Cited by 1 (1 self)
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The RiemannLanczos problem for 4dimensional manifolds was discussed by Bampi and Caviglia. Using exterior differential systems they showed that it was not an involutory differential system until a suitable prolongation was made. Here, we introduce the alternative JanetRiquier theory and use it to consider the RiemannLanczos problem in 2 and 3 dimensions. We find that in 2 dimensions, the RiemannLanczos problem is a differential system in involution. It depends on one arbitrary function of 2 independent variables when no differential gauge condition is imposed but on 2 arbitrary functions of one independent variable when the differential gauge condition is imposed. For each of the two possible signatures we give the general solution in both instances to show that the occurrence of characteristic coordinates need not affect the result. In 3 dimensions, the RiemannLanczos problem is not in involution as a socalled “internal” identity occurs. This does not prevent the existence of singular solutions. A prolongation of this problem, where an integrability condition is added, leads to an involutory prolonged system and thereby generates nonsingular solutions of the prolonged RiemannLanczos problem. We give a singular solution for the unprolonged RiemannLanczos problem for the 3dimensional reduced Gödel spacetime.
Notes on Triangular . . . II: Differential Systems
 SYMBOLIC AND NUMERICAL SCIENTIFIC COMPUTING
, 2003
"... This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We em ..."
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This is the second in a series of two tutorial articles devoted to triangulationdecomposition algorithms. The value of these notes resides in the uniform presentation of triangulationdecomposition of polynomial and differential radical ideals with detailed proofs of all the presented results.We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.
REDUCTION OF FEYNMAN GRAPH AMPLITUDES TO A MINIMAL SET OF BASIC INTEGRALS ∗
, 2008
"... An algorithm for the reduction of massive Feynman integrals with any number of loops and external momenta to a minimal set of basic integrals is proposed. The method is based on the new algorithm for evaluating tensor integrals, representation of generalized recurrence relations [1] for a given kind ..."
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An algorithm for the reduction of massive Feynman integrals with any number of loops and external momenta to a minimal set of basic integrals is proposed. The method is based on the new algorithm for evaluating tensor integrals, representation of generalized recurrence relations [1] for a given kind of integrals as a linear system of PDEs and the reduction of this system to a standard form using algorithms proposed in [2], [3]. Basic integrals reveal as parametric derivatives of the system in the standard form and the number of basic integrals in the minimal set is determined by the dimension of the solution space of the system of PDEs. 1.