Reduction of systems of nonlinear partial differential equations to simplified involutive forms

We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

[1]  Elizabeth L. Mansfield,et al.  A Simple Criterion for Involutivity , 1996 .

[2]  C. Rogers,et al.  Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation , 1996 .

[3]  Toshinori Oaku,et al.  Algorithms for finding the structure of solutions of a system of linear partial differential equations , 1994, ISSAC '94.

[4]  Volker Weispfenning,et al.  Differential term-orders , 1993, ISSAC '93.

[5]  Fritz Schwarz,et al.  Reduction and completion algorithms for partial differential equations , 1992, ISSAC '92.

[6]  Gregory J. Reid,et al.  Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs , 1992, ISSAC '92.

[7]  Gregory J. Reid,et al.  Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution , 1991, European Journal of Applied Mathematics.

[8]  Gregory J. Reid,et al.  Finding abstract Lie symmetry algebras of differential equations without integrating determining equations , 1991, European Journal of Applied Mathematics.

[9]  David Hartley,et al.  A Constructive Implementation of the Cartan-Kähler Theory of Exterior Differential Systems , 1991, J. Symb. Comput..

[10]  François Ollivier,et al.  Standard Bases of Differential Ideals , 1990, AAECC.

[11]  M. Kruskal,et al.  New similarity reductions of the Boussinesq equation , 1989 .

[12]  V. L. Topunov.  Reducing systems of linear differential equations to a passive form , 1989 .

[13]  A. Bocharov,et al.  Efficiently implementing two methods of the geometrical theory of differential equations: An experience in algorithm and software design , 1989 .

[14]  Patrizia M. Gianni,et al.  Gröbner Bases and Primary Decomposition of Polynomial Ideals , 1988, J. Symb. Comput..

[15]  Giuseppa Carra'Ferro,et al.  Groebner Bases and Differential Algebra , 1987 .

[16]  Thomas Wolf,et al.  A package for the analytic investigation and exact solution of differential equations , 1987, EUROCAL.

[17]  Fritz Schwarz,et al.  The Riquier-Janet theory and its application to nonlinear evolution equations , 1984 .

[18]  M. Tajiri,et al.  On similarity solutions of the Boussinesq equation , 1982 .

[19]  J. Meinhardt,et al.  Symmetries and differential equations , 1981 .

[20]  Donald Clayton Spencer,et al.  Overdetermined systems of linear partial differential equations , 1969 .

[21]  H. Lewy.  Erratum: An Example of a Smooth Linear Partial Differential Equation Without Solution , 1957 .

[22]  Hans Hornich,et al.  Zur Theorie der partiellen Differentialgleichungen , 1951 .

[23]  E. Cartan,et al.  Les systèmes différentiels extérieurs et leurs applications géométriques , 1945 .

[24]  E. Kahler.  Einführung in die Theorie der Systeme von Differentialgleichungen , 1937 .

[25]  Willy Hereman,et al.  Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations , 1994 .

[26]  S. Duzhin,et al.  Differential-Algebraic and Differential-Geometric Approach to the Study of Involutive Symbols , 1993 .

[27]  G. Reid,et al.  A Point Symmetry Group of a Differential Equation which cannot be Found Using Infinitesimal Methods , 1993 .

[28]  Jacques Calmet,et al.  Algorithmic Methods for Lie Pseudogroups , 1993 .

[29]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[30]  O. K. Yoon,et al.  Introduction to differentiable manifolds , 1993 .

[31]  Keith O. Geddes,et al.  Algorithms for computer algebra , 1992 .

[32]  Gunter Czichowski,et al.  Grobner Bases, Standard Forms of Differential Equations and Symmetry Computation , 1992 .

[33]  J. Carminati,et al.  An explicit determination of the space-times on which the conformally invariant scalar wave equation satisfies Huygens' principle. Part III : Petrov type III space-times , 1988 .

[34]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[35]  J. Pommaret.  Systems of partial differential equations and Lie pseudogroups , 1978 .

[36]  Hubert Goldschmidt,et al.  Integrability criteria for systems of nonlinear partial differential equations , 1967 .

[37]  M. Kuranishi.  ON E. CARTAN'S PROLONGATION THEOREM OF EXTERIOR DIFFERENTIAL SYSTEMS.* , 1957 .

[38]  E. Cartan.  Les problèmes d'équivalence , 1937 .

[39]  J. Thomas,et al.  Riquier's Existence Theorems , 1928 .

[40]  E. Vessiot.  Sur une théorie nouvelle des problèmes généraux d'intégration , 1924 .

[41]  Maurice Janet,et al.  Sur les systèmes d'équations aux dérivées partielles , 1920 .

[42]  Ar. Tresse,et al.  Sur les invariants différentiels des groupes continus de transformations , 1894 .