Self-similar solutions and power geometry

A. D. Bruno

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 55 (2000) no. 1, pp. 1-42

Résumé

The prime application of the ideas and algorithms of power geometry is in the study of parameter-free partial differential equations. To each differential monomial we assign a point in $\mathbb R^n$: the vector exponent of this monomial. To a differential equation corresponds its support, which is the set of vector exponents of the monomials in the equation. The forms of self-similar solutions of an equation can be calculated from the support using the methods of linear algebra. The equations of a combustion process, with or without sources, are used as examples. For a quasihomogeneous ordinary differential equation, this approach enables one to reduce the order and to simplify some boundary-value problems. Next, generalizations are made to systems of differential equations. Moreover, we suggest a classification of levels of complexity for problems in power geometry. This classification contains four levels and is based on the complexity of the geometric objects corresponding to a give problem (in the space of exponents). We give a comparative survey of these objects and of the methods based on them for studying solutions of systems of algebraic equations, ordinary differential equations, and partial differential equations. We list some publications in which the methods of power geometry have been effectively applied.

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