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16
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 43 (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...
Review of Symbolic Software for Lie Symmetry Analysis
 CRC HANDBOOK OF LIE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS, VOLUME 3: NEW TRENDS IN THEORETICAL DEVELOPMENT AND COMPUTATIONAL METHODS, CHAPTER 13
, 1995
"... Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are brifley outlined. Examples illustrate the use of the symbolic software. ..."
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Cited by 42 (13 self)
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Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are brifley outlined. Examples illustrate the use of the symbolic software.
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 26 (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
On the Arbitrariness of the General Solution of an Involutive Partial Differential Equation
 J. Math. Phys
, 1994
"... The relationship between the strength of a differential equation as introduced by Einstein, its Cartan characters and its Hilbert polynomial is studied. Using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems. The p ..."
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Cited by 19 (11 self)
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The relationship between the strength of a differential equation as introduced by Einstein, its Cartan characters and its Hilbert polynomial is studied. Using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems. The problem of computing the number of arbitrary functions in the general solution is treated. Finally, the effect of gauge symmetries is considered. PACS: 02.30.J Partial Differential Equations 1 Introduction Usually, it is not possible to construct the general solution of a nonlinear partial differential equation. Nevertheless one can often deduce many properties of solutions and/or the solution space without explicitly solving the equation. In this paper, we study several methods for measuring the "richness" or arbitrariness of the solution space. For systems with gauge symmetries, we can refine the question by considering only "physically distinguishable" solutions, i.e. we identify soluti...
Involution and constrained dynamics I: The Dirac approach
 J. Phys. A
, 1995
"... We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finitedimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification ..."
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Cited by 14 (11 self)
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We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finitedimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the Dirac approach. We also derive intrinsic expressions for the number of degrees of freedom. 1
Involution and Symmetry Reductions
 Math. Comp. Model
, 1995
"... After reviewing some notions of the formal theory of differential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reducti ..."
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Cited by 6 (5 self)
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After reviewing some notions of the formal theory of differential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reductions and more general reductions using differential constraints.
Indices and solvability for general systems of differential equations
 Computer Algebra in Scienti Computing  CASC '99
, 1999
"... We consider general systems of ordinary and partial differential equations from a geometric point of view. This leads to simple interpretations of various index concepts introduced for differential algebraic equations. Especially, we obtain natural generalisations of these concepts to partial diff ..."
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Cited by 6 (2 self)
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We consider general systems of ordinary and partial differential equations from a geometric point of view. This leads to simple interpretations of various index concepts introduced for differential algebraic equations. Especially, we obtain natural generalisations of these concepts to partial differential equations.
Arbitrariness Of The General Solution And Symmetries
 Acta Appl. Math
, 1995
"... . The computation of the number of arbitrary functions in the general solution is briefly reviewed. The results are used to study normal systems and their symmetry reduction. We discuss the treatment of gauge systems, especially the analysis of gauge fixing conditions. As examples the YangMills equ ..."
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Cited by 5 (5 self)
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. The computation of the number of arbitrary functions in the general solution is briefly reviewed. The results are used to study normal systems and their symmetry reduction. We discuss the treatment of gauge systems, especially the analysis of gauge fixing conditions. As examples the YangMills equations with the Lorentz gauge and Einstein's vacuum field equations with harmonic coordinates are considered. 1. Introduction If one can not determine the general solution of an equation, as it is usually the case with differential equations, one wants to know at least the dimension of the solution space. This is, however, not trivial for systems, especially if they are overdetermined. It turns out, that involution [10] provides the key. We showed in a recent paper [13], how to compute the number of arbitrary functions and their arity for closed representations of the general solution of an involutive system. The purpose of this paper is to provide more applications of these results. Normal...
Computer Algebra and Differential Equations  An Overview
"... We present an informal overview of a number of approaches to differential equations which are popular in computer algebra. This includes symmetry and completion theory, local analysis, differential ideal and Galois theory, dynamical systems and numerical analysis. A large bibliography is provided. 1 ..."
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Cited by 4 (0 self)
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We present an informal overview of a number of approaches to differential equations which are popular in computer algebra. This includes symmetry and completion theory, local analysis, differential ideal and Galois theory, dynamical systems and numerical analysis. A large bibliography is provided. 1 Introduction Differential equations represent one of the largest fields within mathematics. Besides being an interesting subject of their own right one can hardly overestimate their importance for applications. They appear in natural and engineering sciences and increasingly often in economics and social sciences. Whenever a continuous process is modeled mathematically, chances are high that differential equations are used. Thus it is not surprising that differential equations also play an important role in computer algebra and most general purpose computer algebra systems provide some kind of solve command. Many casual users believe that designing and improving such procedures is a centra...
Generalized Tableaux and Formally WellPosed Initial Value Problems
, 1995
"... this article we present an intrinsic description of initial value problems where the Cauchy data is completely unconstrained. This leads us to a formulation where it is prescribed on a flag of submanifolds of different dimensions. Such formulations are known from JanetRiquier Theory (Jan20). But th ..."
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Cited by 3 (3 self)
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this article we present an intrinsic description of initial value problems where the Cauchy data is completely unconstrained. This leads us to a formulation where it is prescribed on a flag of submanifolds of different dimensions. Such formulations are known from JanetRiquier Theory (Jan20). But this approach depends decisively on the coordinate system. In a geometric setting we know of only two articles (Kor90a; Kor90b) using such an approach. But the author always assumed that the problem was formally wellposed without studying the conditions for it. In the CauchyKowalevsky Theorem one requires that the data is not given on a characteristic submanifold. We encounter similar conditions. To formulate them in a concise and intrinsic manner we must introduce a generalization of the notion of a tableau of a differential equation. The classical concept of a noncharacteristic oneform must be extended to nonsystatic bases of T