Alexander Bruno

RUSSIA

The Power Geometry studies the properties of solutions to an equation through the power exponents of its monomials [2]. Below we explain this approach for a partial differential equation with two independent variables x₁, x₂, one unknown function x₃ and without parameters. Let X = (x₁, x₂, x₃). We define a differential monomial a(X) as a product of powers of coordinates X and derivatives  ^k+l x₃/ x^k₁ x^l₂. The point Q = Q(a) ï ÷ ³ corresponds to the monomial a(X) by the following manner: the vector Q = (q₁, q₂, q₃) corresponds to the product const x^q1₁, x^q2₂, x^q3₃, the vector Q = (-k, -l, 1) corresponds to the derivative ^k+l x₃/x^k₁x^l₂, the sum of the vectors Q(a) + Q(b) corresponds to the product of monomials ab. A differential polynomial f(X) is a sum of the differential monomials; to the polynomial f in ÷ ³ there corresponds a set S = S(f) of points Q of its monomials. The set S(f) is called the support of the polynomial f.

Power Geometry studies properties of solution to the equation f(X) = 0 and gives algorithms for finding the solutions basing on the structure of the set S(f) in ÷ ³. For instance, let the polynomial f(X) is quasihomogeneous, i.e. its support S(f) lies ina plane L, ortogonal to vector L = (l ₁, l ₂, l ₃). Then the equation f(X) = 0 admits the Lie operator Ål_i x_i /x_i and has the self-similar solution of the form x₃ = x₁^a y (x₁^-b , x₂), where the support of the solution is in another plane L₁ ortogonal to vector L . It means that points (0,0,1) and (a -b , 1,0) belong to the plane L₁, i.e. if the plane L₁ is defined by the equation l₁q₁ + l₂q₂ + l₃q₃ = c, then l₃ = c, l₁a = l₃, l₁b = l₂. The function y satisfies an ordinary differential equation.

Let logX = (logx₁, logx₂, logx₃)^*, where the asterisk denotes the transposition. After the power transformation logY = AlogX, where A is real nonsingular square matrix, we have f(X) = g(Y)/h(Y), where g(Y) and h(Y) are differential polynomials and the support S(h) consist of one point Q = 0.So, the ratio g(Y)/h(Y) has the support S(g/h) = S(g). Moreover, S(f) = A^* S(g), where A^* is the transposed matrix A. That property allows to simplify the quasigomogeneous differential equation f(X) = 0 by means of a power transformation and a logarithmic transformation of the form log y₁ = z₁.

For the arbitrary differential polynimial f(X), we can find all its quasihomogeneous first approximations _f̂ (X) using its support S(f) . Here the first approximation x₃ = j ̂(x_1, x₂) of a solution x₃ = j (x_1, x₂) to the equation f(X) = 0 is a solution to the corresponding first approximation _f̂ (X) = 0 of the equation, i.e. if f(x_1, x₂,j ) = 0, then _f̂ (x_1, x₂, j ̂)=0. Combining algorithms of the selection of first approximations of the equation, of power transformations and of logarithmic transformations in many problems we can resolve a singularity and obtain the asymptotic expansions of solutions. In [2] these algorithms are given for problems with any number of variables, parameters and equations.

In exapmles the equations describing combustion without a source and with a source are considered. For the last one, a new class of self similar solutions with blow up is found.

More detaled presentation see in Proceedings of the International Conference on Differential and Functional Differential Equations, Moscow, Russia, August 16-21, 1999.