Asymptotic Solution of Nonlinear Algebraic and Differential Equations

Transcription

1 International Mathematical Forum, Vol. 0, 205, no., HIKARI Ltd, Asymptotic Solution of Nonlinear Algebraic and Differential Equations Alexander D. Bruno Keldysh Institute of Applied Mathematics Miusskaya sq. 4, Moscow, 25047, Russia Copyright c 205 Alexander D. Bruno. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Here we present a way of computation of asymptotic expansions of solutions to algebraic and differential equations and present a survey of some of its applications. The way is based on ideas and algorithms of Power Geometry. Power Geometry has applications in Algebraic Geometry, Differential Algebra, Nonstandard Analysis, Microlocal Analysis, Group Analysis, Tropical/Idempotent Mathematics and so on. We also discuss a connection of Power Geometry with Idempotent Mathematics. Keywords: Power Geometry; asymptotic expansion. Introduction We develop a new Calculus based on Power Geometry [, 2, 3, 4]. At present, it allows to compute local and asymptotic expansions of solutions to nonlinear equations of three classes:. algebraic, 2. ordinary differential, 3. partial differential, as well as to systems of such equations. However, it can also be extended to other classes of nonlinear equations: functional, integral, integro-differential etc. Principal ideas and algorithms are common for all classes of equations. Computation of asymptotic expansions of solutions consists of 3 following steps (we describe them for one equation f = 0).

2 536 Alexander D. Bruno. Calculation of truncated equations = 0 by means of generalized faces of the convex polyhedron Γ(f) which is a generalization of the Newton polyhedron. The first term of the expansion of a solution to the initial ˆf (d) j equation f = 0 is a solution to the corresponding truncated equation ˆf (d) j = Finding solutions to a truncated equation = 0, which is quasihomogenous. Using power and logarithmic transformations of coordinates (d) we can reduce the equation ˆf j = 0 to such simple form that can be solved. Among the solutions found we should select appropriate ones which yield the first terms of asymptotic expansions. 3. Computation of the tail of the asymptotic expansion. Each term in the expansion is a solution of a linear equation which can be written down and solved. Indeed Power Geometry (as a basis of Nonlinear Analysis) can be considered as the third level of Differential Calculus (after Classical Analysis and Functional Analysis). Elements of Plane Power Geometry were proposed by Newton for algebraic equation (670) [5]; and by Briot and Bouquet for ordinary differential equation of the first order (856) [6]. Space Power Geometry for a nonlinear autonomous system of Ordinary differential equations (ODEs) was proposed by the author (962) [], and for a linear partial differential equation (PDE), by Mikhailov (963) [7]. Thus, in 202 we could celebrate 50 years of the first publication on the Newton polyhedron. Back in the autumn of 959, I was a third-year student of Department of Mechanics and Mathematics of the Lomonosov Moscow State University, and invented a polyhedron to study asymptotic behavior of solutions to an autonomous system of ODEs near a degenerated stationary point. The polyhedron was described in my work, which was presented at a students works competition in 96. In that year Arnold was a postgraduate student, and he became a referee of my work. He estimated my works as not very good, for the mere reason that geometry of power exponents is useless. In , Arnold wrote reports on some of my articles with the same (rather negative) evaluation of Power Geometry. See details in Section 6 of Chapter 8 of English Edition of my book [3]. However, in 973 Arnold re-introduced my polyhedron as Newton polyhedron and that looked as if he was the inventor of the polyhedron. In fact, he invented only the name [8]. V.P. Maslov was very surprised when I told him (in 990) that the Newton polyhedron was my invention (he thought that it was Arnold s). In this paper we intend to explain basic notions of Power Geometry, present some of its algorithms, results, and applications. It is clear that this calculus cannot be mastered using the present paper alone. Power Geometry was the ˆf (d) j

3 Asymptotic solution of nonlinear algebraic and differential equations 537 subject of a one-year lecture course Nonlinear Analysis, taught by the author at the Lomonosov Moscow State University. 2. Algebraic Equations [2, 3] We consider a polynomial depending on three variables near its singular point where the polynomial vanishes with all the first partial derivatives. We propose a method of computation of asymptotic expansions of all branches of the set of roots of the polynomial near the above mentioned singular point. There are three types of expansions. The method of computation is based on the spatial Power Geometry. Most of our examples are for polynomials in two variables. 2.. The Problem Statement. Let X = (x, y, z) R 3 or C 3 and g(x) be a polynomial. Definition. A point X 0 is called singular for the set G = {X : g(x) = 0} if all the partial derivatives of the first order of the polynomial g vanish in the point X 0 and g(x 0 ) = 0. Problem. Near the singular point X 0 = 0 for each branch of the set G, find a parameter expansion of one of the following three types. Type x = b k v k, y = k= where b k, c k, d k are constants. c k v k, z = k= d k v k, k= Type 2 x = b kl u k v l, y = c kl u k v l, z = d kl u k v l, where b kl, c kl, d kl are constants and integer points (k, l) are in a sector with the angle less than π (see Fig. ). Type 3 x = β k (u)v k, y = γ k (u)v k, z = δ k (u)v k, k=0 k=0 k=0 where β k (u), γ k (u), δ k (u) are rational functions of u and ψ(u), and ψ(u) is a polynomial in u.

4 538 Alexander D. Bruno l 0 k Figure. Set of power exponents k, l in expansion of Type Objects and algorithms of Power Geometry. Consider a finite sum (for example, a polynomial) g(x) = g Q X Q over Q S(g), () where X = (x, y, z) C 3, Q = (q, q 2, q 3 ) R 3 and X Q = x q y q 2 z q 3, g Q = const C\{0}. To each of the terms of sum (), we assign its vector power exponent Q, and to the whole sum (), we assign the set of all vector power exponents of its terms, which is called the support of sum () or of the polynomial g(x), and it is denoted by S(g). The convex hull of the support S(g) is called the Newton polyhedron of the sum g(x), and it is denoted by Γ(g). The boundary Γ of the polyhedron Γ(g) consists of generalized faces Γ (d) j of various dimensions d = 0,, 2. Here j is the number of a face. To each generalized face Γ (d) j, we assign the truncated sum ĝ (d) j (X) = g Q X Q over Q Γ (d) j S(g). Example : support and the Newton polygon. We consider the polynomial g(x, y) = x 3 + y 3 3xy. Support S(g) consists of points Q = (3, 0), Q 2 = (0, 3), Q 3 = (, ). The Newton polygon Γ(g) is the triangle Q Q 2 Q 3 (Fig. 2). Its edges and corresponding truncated polynomials are Γ () : ĝ () = x 3 3xy, Γ () 2 : ĝ () 2 = y 3 3xy, Γ () 3 : ĝ () 3 = x 3 + y 3. Let R 3 be a space dual to the space R 3 and P = (p, p 2, p 3 ) be points of this dual space. The scalar product in other terms, facets Q, P = q p + q 2 p 2 + q 3 p 3 (2)

5 Asymptotic solution of nonlinear algebraic and differential equations 539 is defined for the points Q R 3 and P R 3. In particular, the external normal N k to the generalized face Γ (d) k is a point (vector) in R 3. The scalar product Q, N k reaches the maximum value at the points Q Γ (d) k S(g), i. e. at the points of the generalized face Γ(d) k. The set of all points P R 3, at which the scalar product (2) reaches the maximum over Q S(g) exactly at points Q Γ (d) k, is called normal cone of the generalized face Γ(d) k and is denoted by U (d) k. Example 2: Normal Cones (cont. of Example ). For faces Γ (d) j Newton polygon Γ(g) of Fig. 2, normal cones are shown in Fig. 3. of the q 2 3 Q 2 2 Γ () 2 Γ () 3 0 Q 3 Γ () Q 2 3 q Figure 2. The Newton polygon of polynomial x 3 + y 3 = 3xy. For edge Γ () j, j =, 2, 3, normal cone U () j to this edge. For vertex Γ (0) j is the external ray orthogonal = Q j, j =, 2, 3, normal cone is open sector between external rays orthogonal to edges Γ () k adjacent to vertex Q j. Theorem ([3]). If for t the curve x = bt p ( + o()), y = ct p 2 ( + o()), z = dt p 3 ( + o()), (3) where b, c, d and p i are constants, belongs to the set G = {X : g(x) = 0}, and the vector P = (p, p 2, p 3 ) belongs to U (d) k, then the first approximation x = bt p, y = ct p 2, z = dt p 3 of curve (3) satisfies the truncated equation (X) = 0. ĝ (d) k The truncated sum ĝ (0) j corresponding to the vertex Γ (0) j is a monomial. Such truncations are of no interest and will not be considered. We will consider truncated sums corresponding to edges Γ () j and faces Γ (2) j only.

6 540 Alexander D. Bruno p 2 U (0) 2 U () 3 U () 2 U (0) 3 U (0) p U () Figure 3. Normal cones U (d) j for polygon of Fig. 2 Power transformations are mappings of the form log X = B log X, (4) where log X = (log x, log y, log z) T, log X = (log x, log y, log z ) T, B is a nondegenerate square 3 3 matrix (b ij ) with rational elements b ij (they are often integer). The monomial X Q is transformed to the monomial X Q by power transformation (4), where Q T = B T Q T. Power transformations and multiplications of polynomial by monomial generate the affine geometry in space R 3 of vector power exponents of polynomial monomials. The matrix B with integer elements and det B = ± is called unimodular. Theorem 2 ([3]). For the face Γ (d) j, there exists a power transformation (4) with an unimodular matrix B which transforms the truncated sum ĝ (d) j (X) into the sum in d coordinates, i. e. ĝ (d) j (X) = X Q h(x ), where h(x ) = h(x ) if d =, and h(x ) = h(x, y ) if d = 2. Here Q = (q, q 2, q 3 ) R 3 and other coordinates y, z for d = and z for d = 2 are small. If ĝ (d) j (X) is a polynomial, then the sum h(x ) is a polynomial as well Cone of the problem. The cone of the problem L is a convex cone of such vectors P = (p, p 2, p 3 ) R 3 that curves of form (3) fill those part of the space (x, y, z) which is under consideration, i. e. must be studied.

7 Asymptotic solution of nonlinear algebraic and differential equations 54 So, our initial Problem corresponds to the cone of the problem L = {P = (p, p 2, p 3 ) : P < 0} in R 3, since x, y, z 0 (and x, y, z as in (3)). If x then p > 0 in the cone of the problem L. Example 3: Cont. of Examples and 2. For variables x, y near origin x = y = 0 cone of the problem is the quadrant III: L 3 = {p, p 2 < 0}, near infinity x = y = cone of the problem is the quadrant I: L = {p, p 2 > 0}, near point x = 0, y = cone of the problem is the quadrant II: L 2 = {p < 0, p 2 > 0} (Fig. 4). In Fig. 3 some cones of the problem L i intersects several normal cones U (2) j. E.g. L 3 intersects U (), U () 2 and U (0), U (0) 2, U (0) 3. L intersects U () 3, U (0), U (0) 2. p 2 II I p III IV Figure 4. Cones of the problem Steps for Problem solving. Step. We compute the support S(g), the Newton polyhedron Γ(g), its twodimensional faces Γ (2) j and their external normals N j. Using normals N j we compute the normal cones U () k to edges Γ () k. Step 2. We find all the edges Γ () k and faces Γ (2) j, whose normal cones intersect the cone of the problem L. It is enough to select all the faces Γ (2) j, whose external normals N j intersect the cone of the problem L, and then add all the edges Γ () k of these faces.

8 542 Alexander D. Bruno Step 3. Step 4. For each of the selected edges Γ () k, we perform a power transformation X X of Theorem 2 and we get the truncated equation in a form h(x ) = 0. We find the roots of this equation. Let x 0 be one of its roots. We perform the power transformation X X in the whole polynomial g(x) and we get the polynomial g (X ) = g(x). We make the shift of variables x 2 = x x 0, y 2 = y, z 2 = z in the polynomial g (X) and get the polynomial g 2 (X 2 ) = g (X ). If x 0 is a simple root of the equation h(x ) = 0 then, according to the Implicit Function Theorem, it corresponds to an expansion of the form x 2 = a kl y k z l, where a kl are constants. It gives an expansion of type 2 in coordinates X. If x 0 is a multiple root of the equation h(x ) = 0 then we compute the Newton polyhedron of the polynomial g 2 (X 2 ), compute the new cone of the problem L 2 as the convex cone generated by vector (, 0, 0) and two external normals of faces adjacent the edge, and we continue as above and as follows. For each of the selected faces Γ (2) j, we perform a power transformation X X of Theorem 2 and we get a truncated equation in the form h(x, y ) = 0. We factorize h(x, y ) into prime factors. Let h(x, y ) be one of such factors and its algebraic curve has genus ρ. If ρ = 0 then there exists birational uniformization x = ξ(y 2 ), y = η(y 2 ) of this curve. We change variables x = ξ(y 2 ) + x 2, y = η(y 2 ) and then h is divided by x 2. We change variables in the whole polynomial g(x) and get the polynomial g 2 (X 2 ) def = g (X ) = g(x). If h(x, y ) is simple factor of h(x, y ) then roots of the polynomial g 2 (X 2 ) are expanded into series of the form x 2 = α k (y 2 )z2, k (5) k= where α k (y 2 ) are rational functions of y 2. It gives an expansion of type 3 in original coordinates X.

9 Asymptotic solution of nonlinear algebraic and differential equations 543 If h(x, y ) is a multiple factor of h(x, y ) then we compute the Newton polyhedron of the polynomial g 2 (X 2 ), compute the cone of the problem L 2 = {P : p 2, p 3 < 0} and continue computations. If ρ = (elliptic curve), there exists the birational change of variables x, y x 2, y 2 transforming h(x, y ) into the normal form x 2 2 ψ(y 2 ), where ψ is a polynomial of order 3 or 4. If ρ >, we distinguish hyper-elliptic and non hyper-elliptic curves. The hyper-elliptic curve is birationally equivalent x, y x 2, y 2 to its normal form x 2 2 ψ(y 2 ), where ψ is a polynomial of order 2ρ + or 2ρ + 2. If ρ and we have the (hyper)elliptic curve and factor h of h is simple we get expansions of solutions of equation g 2 (X 2 ) = 0 into series (5), where α k are rational functions of y 2 and ψ(y 2 ). We get the expansion of type 3 in original coordinates X. If ρ and we have the (hyper)elliptic curve and h(x, y ) is a multiple factor of h(x, y ) then we continue for g 2 (X 2 ) as above. In this procedure we distinguish two cases: Case. Truncated polynomial contains linear part of one of the variables or x 0 is a simple root of h(x ) or h(x, y ) is simple factor of h(x, y ). Then a generalization of Implicit Function Theorem is applicable and it is possible to compute parametric expansion of set of roots of full polynomial. Case 2. Truncated polynomial does not contain linear part of any variable and x 0 is a multiple root of h(x ) or h(x, y ) is a multiple factor of h(x, y ). Then the Newton polyhedron for full polynomial must be built and we must consider new truncated polynomials taking into account the new cone of the problem L. Example 4 (cont. of Examples 3). For edge Γ (), we get truncated equation x 2 3y = 0, i. e. y = x 2 /3. It is the case, and this asymptotic form is continued into power expansion of branch y = x 2 /3 + k=2 b kx 2k near the origin x = y = 0. For edge Γ () 2, we get truncated equation y 2 3x = 0, i. e. y = ± 3x. It is the case, and these asymptotic forms are continued into power expansions of branches y = ± 3x+ k=2 b kx k/2 near the origin x = y = 0.

10 544 Alexander D. Bruno For edge Γ () 3, we get truncated equation x 3 + y 3 = 0. It has the simple factor x + y = 0, i. e. y = x. It is again case of simple root, and the power expansion at infinity y = x + k= b kx k gives the asymptote x + y = for the curve g(x, y) = 0 called folium of Descartes (Fig. 5). y 2 2 x Figure 5. Folium of Descartes Asymptotic description of a subset of singular points of G can be obtained by the same procedure, but we have to select only singular points in each truncated equation. As result we obtain expansions of type Results. Theorem 3 ([4]). The algorithm of Subsection 2.4, in the case when all curves formed by roots of corresponding two-dimensional truncated equations with positive genus are elliptic or hyper-elliptic, yields a local description of all components of the set G, adjacent to the starting point X 0, in form of expansions of types Implementation and Application. Implementation of the described algorithm see in [9]. Its application to computation of a set of stability of a certain ODE system depending on several parameters see in [0]. Example 5 [9]. g(x) = 52z z 5 y 768z 5 x z 4 y z 4 yx + 52z 4 x z 3 y z 3 y 2 x 2752z 3 yx 2 92z 3 x z 2 y 4

11 Asymptotic solution of nonlinear algebraic and differential equations z 2 y 3 x z 2 y 2 x z 2 yx z 2 x zy 5 700zy 4 x 4376zy 3 x 2 904zy 2 x 3 92zyx 4 4zx y 5 x + 200y 4 x y 3 x y 2 x 4 + 4yx 5 256z z 4 y + 344z 4 x 4976z 3 y z 3 yx 344z 3 x z 2 y z 2 y 2 x + 544z 2 yx z 2 x zy zy 3 x 6784zy 2 x 2 52zyx 3 64zx y y 4 x + 356y 3 x y 2 x yx 4 + 2x z z 3 y 5088z 3 x 35496z 2 y z 2 yx +2200z 2 x zy zy 2 x 576zyx 2 344zx y y 3 x y 2 x yx 3 + 3x z z 2 y + 692z 2 x 9332zy zyx 52zx y y 2 x + 728yx x 3. The structure of solutions of the algebraic equation g(x) = 0 near its singular points (including infinity). The Newton polyhedron of this equation is shown on Fig. 6. Figure 6. The Newton polyhedron for polynomial g(x). Near origin X = 0 we obtain x = β k (u)v k, y = γ k (u)v k, z = δ k (u)v k, k=0 k=0 k=0

12 546 Alexander D. Bruno where β k (u), γ k (u), δ k (u) are rational functions of u. More precisely, we have: ( ) x = Ω (u) 2v 3 + 8Ω 2 (u) 27u2 + 0u 5 v 4 + o(v 4 ), u + here Ω (u) = y = Ω (u) ( 2v 2 + 2Ω 2 (u)(5u )v 3) + o(v 3 ), ( ) z = Ω (u) 3v 2 + 4Ω 2 (u) 7u2 + 3u 4 v 3 + o(v 3 ) 3u + 54(3u + )3 (7u + )(u + ) 2, Ω 2(u) = 3u + (7u + )(u + ). 3. Ordinary Differential Equations. Algebraic Approach 3.. Plane Power Geometry []. First, consider one differential equation and power expansions of its solutions (later we consider more complicated expansions). Let x be independent and y be dependent variables, x, y C. A differential monomial a(x, y) is a product of an ordinary monomial cx q y q 2, where c = const C, (q, q 2 ) R 2, and a finite number of derivatives of the form d l y/dx l, l N. A sum of differential monomials is called the differential sum. Problem 2. Let a differential equation be given f(x, y) = a i (x, y) (6) f(x, y) = 0, (7) where f(x, y) is a differential sum. As x 0, or as x, for solutions y = ϕ(x) to equation (7), find all expansions of the form y = c r x r + c s x s, c r = const C, c r 0, (8) where c s are polynomials in log x, and power exponents r, s R, ωr > ωs, and ω =, if x 0, or ω =, if x. The procedure to compute expansions (8) consists of two steps: computation of the first approximations y = c r x r, c r 0 and computation of further expansion terms in (8). To each differential monomial a(x, y), we assign its (vector) power exponent Q(a) = (q, q 2 ) R 2 by the following rules: Q(cx q y q 2 ) = (q, q 2 ); Q(d l y/dx l ) = ( l, );

13 Asymptotic solution of nonlinear algebraic and differential equations 547 when differential monomials are multiplied, their power exponents must be added as vectors Q(a a 2 ) = Q(a ) + Q(a 2 ). The set S(f) of power exponents Q(a i ) of all differential monomials a i (x, y) presented in differential sum (6) is called the support of the sum f(x, y). Obviously, S(f) R 2. The convex hull Γ(f) of the support S(f) is called the polygon of the sum f(x, y). The boundary Γ(f) of the polygon Γ(f) consists of the vertices Γ (0) j and the edges Γ () j. They are called (generalized) faces Γ (d) j, where the upper index indicates the dimension of the face, and the lower one is its number. Each face Γ (d) j corresponds to the truncated sum ˆf (d) j (x, y) = a i (x, y) over i : Q(a i ) Γ (d) j S(f) (9) and to truncated equation ˆf (d) j (x, y) = 0 of equation (7). Example 6. Consider the third Painlevé equation f(x, y) def = xyy + xy 2 yy + ay 3 + by + cxy 4 + dx = 0, (0) assuming the complex parameters a, b, c, d 0. Here the first three differential monomials have the same power exponent Q = (, 2), then Q 2 = (0, 3), Q 3 = (0, ), Q 4 = (, 4), Q 5 = (, 0). They are shown in Fig. 7 in coordinates q, q 2. q 2 Γ () 2 Q Q 2 Q 4 Γ () 3 Γ () Q 3 Q 5 0 q Figure 7. The Newton polygon of equation (0) Their convex hull Γ(f) is the triangle with three vertices Γ (0) = Q, Γ (0) 2 = Q 4, Γ (0) 3 = Q 5, and with three edges Γ (), Γ () 2, Γ () 3. The vertex Γ (0) = Q corresponds to the truncation ˆf (0) (x, y) = xyy + xy 2 yy,

14 548 Alexander D. Bruno and the edge Γ () corresponds to the truncation ˆf () (x, y) = ˆf (0) (x, y) + by + dx. () Let the plane R 2 be dual to the plane R 2 such that for P = (p, p 2 ) R 2 and Q = (q, q 2 ) R 2, the scalar product P, Q def = p q + p 2 q 2 is defined. Each face Γ (d) j corresponds to its own normal cone U (d) j R 2 formed by the external normal vectors P to the face Γ (d) j. For the edge Γ () j, the normal cone U () j is the ray orthogonal to the edge Γ () j the polygon Γ(f). For the vertex Γ (0) j, the normal cone U (0) j and directed outward is the open sector (angle) in the plane R 2 with the vertex at the origin P = 0 and limited by the rays which are the normal cones of the edges adjacent to the vertex Γ (0) j. Example 7 (cont. of Example 6). For equation (0), the normal cones of the faces Γ (d) j are shown in Fig. 8. U (d) j U () 2 p 2 U (0) 2 U (0) U () 3 p U () U (0) 3 Figure 8. The normal cones U (d) j for polygon of Fig. 7 Thus, each face Γ (d) j and to the truncated equation corresponds to the normal cone U (d) j ˆf (d) j (x, y) = 0. in the plane R 2 Theorem 4 ([]). If expansion (8) satisfies equation (7), and ω(, r) U (d) j, then the truncation y = c r x r of solution (8) is the solution to truncated equation (9).

15 Asymptotic solution of nonlinear algebraic and differential equations 549 As truncated equation is quasi homogeneous it is not difficult to find its power solutions. Hence, to find all truncated solutions y = c r x r to equation (7), we need to compute: the support S(f), the polygon Γ(f), all its faces Γ (d) j, and their normal cones U (d) (d) j. Then for each truncated equation ˆf j (x, y) = 0, we need to find all its solutions y = c r x r which have one of the vectors ±(, r) lying in the normal cone U (d) j. For each power solution y = c r x r of the truncated equation = 0, we can compute its characteristic polynomial ν(k). Roots k j of ν(k) with ωk j < ωr are critical numbers of the solution c r x r. (0) Example 8 (cont. of Examples 6, 7). For the truncated equation ˆf = 0 with ω =, we have solutions y = c r x r with arbitrary constants r and c r. But according to Theorem 4, y = c r x r can be a formal asymptotic form of a solution to full equation (7) if ω(, r) U (0), i. e. < r <. Corresponding characteristic polynomial ν(k) = (k r) 2. Hence, here we have no critical numbers. () For the truncated equation ˆf = 0, we have power solution y = dx/b. Its characteristic polynomial is ν(k) = (k ) 2 + b 2 /d.. Using support S(f) of the equation (7) and critical numbers k,..., k κ with ωr > ωk i, we can find the set K(k,..., k κ ) R, support of expansion (8). Its elements s satisfy the inequality ωr > ωs. Theorem 5 ([]). If y = c r x r, c = const, ω(, r) U (d) j, is a power solution to truncated equation (9), then equation (7) has an expansion of solutions of the form z = c r x r + c s x s over s K(k,..., k κ ), (2) where k,..., k κ are critical numbers of the truncated solution y = c r x r ; c s are polynomials in log x, which are uniquely determined for s k i. If all critical numbers k,..., k κ are simple roots, and each k i does not lie in the set K(k,..., k i, k i+,..., k κ ), then all coefficients c s are constant; for s k i, they are uniquely determined; and for s = k i, they are arbitrary. Example 9 (cont. of Examples 6 8). For the truncated solution y = cx r (0) to ˆf = 0, arbitrary c 0, r (, ) K = {s = r + l( r) + m( + r), l, m N {0}, l + m > 0}. (3) Since there are no critical numbers, then all c s are constant and uniquely determined in expansion (2). () For the truncated solution y = dx/b to ˆf = 0, we have K = {s = + 2l, integral l > 0}. If I(b/ d) 0, then there are no real critical numbers, ˆf (d) j

16 550 Alexander D. Bruno and all power exponents s are odd integers greater than in expansion (2), and coefficients c s are constant and uniquely determined. If I(b/ d) = 0, then there is a unique real critical number k = + b/ d, and K(k ) = {s = + 2l + m(k ), l, m N {0}, l + m > 0}. (4) Hence, if the number k is not odd, then all c s are constant and uniquely determined in expansion (2) for s k, and c k is arbitrary. Finally, if k is odd, then K(k ) = K, and in expansion (2) c s is a uniquely determined constant if s < k ; c k is a linear function of log x with an arbitrary constant term; c s is a uniquely determined polynomial in log x if s > k. (d) The truncated equation ˆf j (x, y) = 0 can have non-power solutions y = ϕ(x) which are the asymptotic forms for solutions to the initial equation f(x, y) = 0. These non-power solutions y = ϕ(x) may be found using power and logarithmic transformations. Power transformation is linear in logarithms and defined by log x = b log u + b 2 log v, log y = b 2 log u + b 22 log v, ( ) b b B = 2, b b 2 b ij R, det B It induces linear dual transformations in spaces R 2 and R 2. Logarithmic transformation has the forms ξ = log u or η = log v. Example 0 (cont. of Examples 6 9). For the truncated equation () corresponding to the edge Γ () with the normal vector (, ), we make power transformation log x = log u, log y = log u + log v, ( ) 0 B =, i. e. x = u, y = uv. Since y = xv + v, y = xv + 2v, then, canceling x and collecting similar terms, the equation (), becomes x 2 vv + x 2 v 2 xvv + bv + d = 0. (5) Its support consists of three points Q = (0, 2), Q2 = (0, ), Q3 = 0 on the axis q = 0 (see Fig. 9). Now we make the logarithmic transformation ξ = log x. Since v = v/x, v = ( v v)/x 2, where = d/dξ, then, collecting similar terms, the equation (5) takes the form v v + v 2 + bv + d = 0. Its support and polygon

17 Asymptotic solution of nonlinear algebraic and differential equations 55 q q Figure 9. The support and Newton polygon for the third Painlevé equation (5) Γ () q Figure 0. Support and Newton polygon of equation v v + v 2 + bv + d = 0. are shown in Fig. 0. Applying the technique described before to this equation, we obtain the expansion of its solutions v = (b/2)ξ 2 + cξ + k=0 c kξ k, where c is an arbitrary constant, and the constants c k are uniquely determined. In original [ variables, we obtain the family of] non-power asymptotic forms y x (b/2)(log x) 2 + c log x + c k (log x) k of solutions to the initial equation (0), when x 0. k=0 q 3.2. Complex power exponents []. Indeed, the described method allows to calculate solutions with complex power exponents as well.

18 552 Alexander D. Bruno Thus, by the algebraic approach, expansions of solutions y = c r x r + s c s x s, (6) with complex power exponents r and s, where ωrr ωrs, and coefficients c r and c s are power series in log x, log log x and so on, are found in a similar way. In classical analysis, we encounter expansions in fractional powers and with constant coefficients, but here we obtain more complicated expansions of solutions Algorithms of Power Geometry.. Computation of truncated equations and accompanying objects. 2. Solution of truncated equations. 3. Power transformations. 4. Logarithmic transformations. 5. Introducing independent variable x i instead of x. 6. Computation of the first variation of a sum. 7. Computation of expansions of solutions to the initial equation, beginning by solutions to a truncated equation. All these algorithms, except for 4 and 5, can be applied to solve algebraic equations. Similar technique is used for equations having small or big parameters. The power exponents of these parameters are accounted in the same way as power exponents of variables tending to zero or infinity The sixth Painlevé equation [2]. It has the form ( y = (y ) 2 2 y + y + ) ( y y x x + x + ) + y x [ y(y )(y x) + a + b x x 2 (x ) 2 y + c x ] ) + dx(x, 2 (y ) 2 (y x) 2 (7) where a, b, c, d are complex parameters, x and y are complex variables, y = dy/dx. Equation (7) has three singular points x = 0, x =, and x =. After multiplying by the common denominator, we obtain the equation as a differential sum. Its support and its polygon, in the case a 0, b 0, are shown in Fig..

19 Asymptotic solution of nonlinear algebraic and differential equations 553 We found all formal asymptotic expansions (6) of solutions to equation (7) near its three singular points. They comprise 08 families. In particular, for a = /2 and c = 0, there is an expansion of the form y = cos[log(c x)] + c s x s, (8) Rs where C is an arbitrary constant, the coefficients c s are uniquely determined constants. Here /cos[log(x)] = 2/(x i + x i ) = 2x i k=0 ( x 2i ) k = 2x i ( ) x 2i k. For C = and real x > 0, solution (8) has infinitely many poles accumulating at the point x = 0. We also found all expansions of solutions to equation (7) near its nonsingular points. They comprise 7 families [3]. k=0 q 2 0 q Figure. Support and Newton polygon for the sixth Painlevé equation (7) That approach was applied to ODE systems [4, 5, 6, 7] Applications.. Asymptotic forms and expansions of solutions to the Painlevé equations [2, 8, 9]. 2. Periodic motions of a satellite around its mass center moving along an elliptic orbit [20]. 3. New properties of motion of a top (rigid body with a fixed point) [2]. 4. Families of periodic solutions of the restricted three-body problem and distribution of asteroids [22, 23].

20 554 Alexander D. Bruno 5. Integrability of ODE systems [24]. 6. Surface waves on water [3, Ch. 5]. 4. Ordinary Differential Equations. Differential Approach 4.. Orders of solutions and their derivatives. All solutions of the form (6) have the following property: p ω ( y (k) ) = p ω (y) k, k =, 2,..., n, where n is the maximal order of derivative in the initial ODE, and p ω (ϕ) = ω lim sup x ω log ϕ(x) ω log x along fixed arg x [0, 2π) is the order of the function ϕ(x) Differential Approach [25, 26, 27, 28, 29, 30]. To each differential monomial a(x, y) we put in correspondence the 3D point Q = (q, q 2, q 3 ), where q and q 2 as before, but q 3 is the total order of derivatives in the monomial. We obtain the 3D support S(f) of the initial ODE f = 0 and polyhedron Γ(f) as convex hull of S(f). Using truncated equations, corresponding to its faces and edges, we can find their solutions in the form of elliptic or hyperelliptic functions ϕ 0 (x) and continue them into power-(hyper)elliptic expansion y = ϕ 0 (x) + ϕ l (x)x ωl, (9) l= where all ϕ l (x) are elliptic or hyperelliptic functions. This differential approach allows to find expansions (9) for solutions with property p ω ( y (k) ) = p ω (y) γ ω k, k =, 2,..., n, where γ ω. Example (cont. of Examples 6 0) [27]. The 3D support S of the third Painleve equation (0) consists of 6 points: Q = (, 2, 2), Q 2 = (, 2, ), Q 3 = (0, 3, 0), Q 4 = (0,, 0), Q 5 = (, 4, 0), Q 6 = (, 0, 0). Their convex hull Γ is a pentahedron (Fig. 2). It has 2D upper face Γ (2) with external normal N = (, 0, ) containing 3 points of the support S: Q, Q 5, Q 6. Corresponding truncated equation is ˇf (2) def = xyy + xy 2 + cxy 4 + dx = 0.

21 Asymptotic solution of nonlinear algebraic and differential equations 555 Figure 2. 3D support S(f) and polyhedron Γ(f) of equation (0) with all a, b, c, d 0. The grey face is Γ (2). All dotted lines are in the plane q, q 2, they show projections of Γ(f) on the plane (q, q 2 ). It has the first integral y 2 = cy 4 + C 0 y 2 d def = P (y), (20) where C 0 C is an arbitrary constant. Discriminant of the polynomial P (y) is (P ) = cd(cd + C0/4) 2 2. If (P ) 0, then solutions y = ϕ 0 (x) of the equation (20) are elliptic functions. So we can look for expansions (9), where all ϕ l (x) are regular functions of ϕ 0 (x). Here ω = and γ = Partial Differential Equations. Algebraic Approach 5.. Theory [3]. Let X = (x,..., x m ) C m be independent and Y = (y,..., y n ) C n be dependent variables. Suppose Z = (X, Y ) C m+n. A differential monomial a(z) is the product of an ordinary monomial cz R = cz r z r m+n m+n, where c = const C, R = (r,..., r m+n ) R m+n, and a finite number of derivatives of the form l y j x l x lm m def = l y j X L, l j 0, m l j = l, L = (l,..., l m ). j=

22 556 Alexander D. Bruno A differential monomial a(z) corresponds to its vector power exponent Q(a) R m+n formed by the following rules Q(cZ R ) = R, Q( l y j / X L ) = ( L, E j ), where E j is unit vector. A product of monomials a b corresponds to the sum of their vector power exponents: Q(ab) = Q(a) + Q(b). A differential sum is a sum of differential monomials f(z) = a k (Z). A set S(f) of vector power exponents Q(a k ) is called the support of the sum f(z). The closure of the convex hull Γ(f) of the support S(f) is called the polyhedron of the sum f(z). Consider a system of equations f i (X, Y ) = 0, i =,..., n, (2) where f i are differential sums. Each equation f i = 0 corresponds to: its support S(f i ), its polyhedron Γ(f i ) with the set of faces Γ (d i) ij in the main space R m+n, the set of normal cones U (d i) ij to faces Γ (d i) ij of truncated equations The set of truncated equations ˆf (d i) ij (X, Y ) = 0. in the dual space R m+n, and the set ˆf (d i) ij i (X, Y ) = 0, i =,..., n (22) is the truncated system if the intersection of corresponding normal cones U (d ) j U (dn) nj n (23) is not empty. A solution y i = ϕ i (X), i =,..., n to system (2) is associated to its normal cone u R m+n. If the normal cone u intersects with cone (23), then the asymptotic form y i = ˆϕ i (X), i =,..., n of this solution satisfies truncated system (22), which is quasi-homogeneous Applications. Boundary layer on a needle [3]. The theory of the boundary layer on a plate for a stream of viscous incompressible fluid was developed by Prandtl (904) [32] and Blasius (908) [33]. However a similar theory for the boundary layer on a needle was not known until recently, since no-slip conditions on the needle correspond to a more strong singularity as for the plate. This theory was developed with the help of Power Geometry (2004). Let x be an axis in three-dimensional real space, r be the distance from the axis, and semi-infinite needle be placed on the half-axis x 0, r = 0. We studied stationary axisymmetric flows of viscous fluid which had constant velocity at x = parallel to the axis x, and which satisfied no-slip conditions

23 Asymptotic solution of nonlinear algebraic and differential equations 557 r u 0 x Figure 3. The needle and flow along it. on the needle (Fig. 3). We considered two cases: () incompressible fluid and (2) compressible heat-conducting gas. First case: incompressible fluid. For it, the Navier-Stokes equations in independent variables x, r are equivalent to the system of two equations for the stream function ψ and the pressure p def g = ( ) ψ ψ + ( ) ψ ψ + p r x r r r r r x r r ρ x ( ( ν r ( )) ( )) ψ + 2 ψ = 0, r r r r r x 2 r r = ( ) ψ ψ ( ) ψ ψ + r x r r x r r x r x ρ ( ( ) ( )) 2 ψ + ν + 2 ψ = 0, r r x r x 2 r x g 2 def where ρ, ν = const, with the boundary conditions p r + (24) ψ = ψ 0 r 2 for x =, ψ 0 = const; (25) ψ/ x = ψ/ r = 2 ψ/ x r = 2 ψ/ r 2 = 0 for x 0, r = 0. (26) Here m = n = 2 and m + n = 4. Hence the supports of equations (24) must be considered in R 4. It turned out that polyhedra Γ(g ) and Γ(g 2 ) of equations (24) are three-dimensional tetrahedra, which can be moved by translation in one linear three-dimensional subspace, that simplified the isolation of the truncated systems. An analysis of truncated systems and of the results of their matching revealed that system (24) had no solution with p 0 satisfying both boundary conditions (25), (26). Second case: compressible heat-conducting gas. For this case, the Navier-Stokes equations in independent variables x, r are equivalent to the

24 558 Alexander D. Bruno system of three equations for the stream function ψ, the density ρ, and the enthalpy h (an analog of the temperature) def f = ( ) ψ ψ + ( ) ψ ψ A r x r ρr x r r x ρr x r (ρh)+ + 2 ( ( )) h N ψ 3 CN 2 ( ( )) h N ψ r r r ρ x 3 CN r r x ρ r ( 2CN h N r ( )) ψ + C N ( h N ( )) ψ r r r ρr x x r ρr r C N ( h N ( )) ψ + 2CN h N ψ x x ρr x ρr 3 x = 0, def f 2 = ( ) ψ ψ ( ) ψ ψ A r x r ρr r r r x ρr r x (ρh)+ + 2 ( ( )) h N ψ 3 CN 2 ( ( )) h N ψ x r r ρ x 3 CN + x r x ρ r ( + CN h N r ( )) ( ψ CN h N r ( )) ψ + r r r ρr r r r x ρr x + 2C N ( h N ( )) ψ = 0, x x ρr r def f 3 = ψ h r x r ψ h r r x A ψ (ρh) + A ψ (ρh) ρr x r ρr r x + ( ( )) 2 ( ) 2 ( ( )) 2 + 2C N h N ψ + 2C N h N ψ + 2C N h N ψ + r ρr x r 2 ρ x x ρr r ( ( )) 2 + C N h N ψ C N h N ( ) ( ) ψ ψ + x ρr x x ρr x r ρr r ( ( )) 2 + C N h N ψ 2 ( ( )) 2 r ρr r 3 CN h N ψ + r r ρ x ( ) ( ) + 4CN h N ψ ψ 2 ( ( )) 2 3r r ρ x x ρr r 3 CN h N ψ + x ρr r ( + CN rh N h ) ( + CN h N h ) = 0, σr r r σ x x (27) where parameters A, C, σ > 0 and N [0, ], with the boundary conditions ψ = ψ 0 r 2, ρ = ρ 0, h = h 0 for x =, ψ 0, ρ 0, h 0 = const (28) and (26). Here m = 2, n = 3, and m + n = 5. In the space R 5, all polyhedrons Γ(f ), Γ(f 2 ), Γ(f 3 ) of equations (27) are three-dimensional, and they can be translated into one linear 3D subspace. In coordinates Q = ( q, q 2, q 3) of this

25 Asymptotic solution of nonlinear algebraic and differential equations 559 three-dimensional space, the supports and polyhedra are shown in Figures 4(a) 4(c). The supports of sums f, f 2 and f 3 are following: S(f ) = { Q 0 = 0, Q = (, 0, 0), Q 2 = (0,, 0), Q 3 = (0, 0, )}, S(f 2 ) = { Q 0 = 0, Q = (,, ), Q 2 = (0,, 0), Q 3 = (0, 0, )}, S(f 3 ) = { Q 0 = 0, Q = (0,, 0), Q 2 = (, 2, ), Q 3 = (0, 0, ), Q 4 = (, 0, ), Q 5 = (,, 0)}. The truncated system corresponding to the boundary layer on the needle corresponds to the vertex Q, to faces [ Q 0, Q, Q 2] and [ Q 0, Q, Q 2] of polyhedrons Γ(f ), Γ(f 2 ) and Γ(f 3 ) respectively. In Figs. 4(a) 4(c) they are distinguished. (a) The Newton polyhedron of f in (27) (b) The Newton polyhedron of f 2 in (27) (c) The Newton polyhedron of f 3 in (27) Figure 4. Newton polyhedrons of f, f 2, f 3 in (27) Thus, the truncated system is ˆf (0) def 2 = A (ρh)/ r = 0, ( ) ψ ψ ( ) ψ ψ A x r ρr r r r x ρr r x (ρh)+ ( + CN h N r ( )) ψ = 0, r r r ρr r ψ h x r ψ h r r x A ψ (ρh) + A ψ (ρh) ρr x r ρr r x + ( ( )) 2 ( + C N h N ψ + CN rh N h ) = 0, r ρr r σr r r ˆf (2) def 22 = r ˆf (2) def 32 = r (29)

26 560 Alexander D. Bruno with the boundary conditions ψ = ψ 0 r 2, ρ = ρ 0, h = h 0 ; ψ 0, ρ 0, h 0 = const, for r. An analysis of solutions to the latter problem (29) by methods of Power Geometry revealed that for N (0, ) it has solutions of the form ψ c r 2 log ξ /N, ρ c 2 log ξ /N, h c 3 log ξ /N, (30) where ξ = r 2 /x 0 and c, c 2, c 3 are arbitrary real constants. Thus, for N (0, ), in the boundary layer r 2 /x < const, as x + and ξ = r 2 /x 0, we obtained the asymptotic form of the flow (30), i.e. near the needle, the density ρ tends to zero, and the temperature h increases to infinity as the distance x to the initial point of the needle tends to Other applications of Power Geometry. Evolution of the turbulent flow [34, 35] and Thermodynamics [36] and power-elliptic expansions of solutions to Painlevé equations [37]. 6. Connection with Idempotent Mathematics For polynomial f(x), let us define the function and its subdifferential ˆf(S) = lim h log f(exp(s/h)) h +0 ˆf = {Q R n : Q, S ˆf(S) for all S R n }. Theorem 6 ([38]). If f(x) is a polynomial, then the subdifferential ˆf of f(x) at the origin coincides with the Newton polyhedron Γ(f). V. P. Maslov and his colleagues developed Idempotent Analysis [38, 39]. However, as a method of finding leading terms in nonlinear problems, it is too complicated. Theorem 6 shows that in algebraic problems Idempotent Analysis gives the Newton polyhedron. This observation can be generalized to other classes of problems, but if we just begin with appropriate generalization of the Newton polyhedron (or Power Geometry), then we do not really need Idempotent Analysis (see Sections 4 and 5). Indeed, Idempotent Analysis [39] is useful in problems with bad solutions (for instance, discontinuous or nonsmooth). References [] A.D. Bruno, The asymptotic behavior of solutions of nonlinear systems of differential equations, DAN SSSR, 43, no.4 (962), (in Russian). English translation: Soviet Math. Dokl., 3 (962),

27 Asymptotic solution of nonlinear algebraic and differential equations 56 [2] A.D. Bruno, Local Methods in Nonlinear Differential Equations, Nauka, Moscow, 979, 252 p. (in Russian). English translation: Springer Verlag, Berlin New York, 989, 350 p. [3] A.D. Bruno, Power Geometry in Algebraic and Differential Equations, Fizmatlit, Moscow, 998, 288 p. (in Russian). English translation: Elsevier, Amsterdam, 2000, 385 p. [4] A.D. Bruno, A.B. Batkhin, Asymptotic solution of an algebraic equation, DAN, 440 (20), no. 3, (in Russian). English translation: Doklady Mathematics, 84 (20), no. 2, [5] I. Newton, The Method of Fluxions and Infinite Series with its Applications to the Geometry of Curve Lines, Johnson Reprint Corp., New York, London, 964. [6] C. Briot, T. Bouquet, Recherches sur les propriétés des équations differentielles, J. Ecole Polytechn. Paris, 2 (856), no. 36. [7] V.P. Mikhailov, On the first boundary value problem for some semibounded operators, Dokl. Akad. Nauk SSSR, 5 (963), (in Russian). English translation: Sov. Math. Dokl., 4 (963), [8] A.D. Bruno, On geometric methods in works by V.I. Arnold and V.V. Kozlov. Preprint ArXiv: v. [9] A.D. Bruno, A.B. Batkhin, Resolution of an algebraic singularity by power geometry algorithms, Programmirovanie, 38 (202), no. 2, 2 30 (in Russian). English translation: Programming and Computer Software, 38 (202), no. 2, [0] A.D. Bruno, A.B. Batkhin, V.P. Varin, Sets of stability of multiparameter Hamiltonian systems, PMM, 76 (202), no., (in Russian). English translation: J. Appl. Math. Mech., 76 (202), no., [] A.D. Bruno, Asymptotic behaviour and expansions of solutions of an ordinary differential equation, UMN, 59 (2004), no. 3, 3 80 (in Russian). English translation: Russian Mathem. Surveys, 59 (2004), no. 3, [2] A.D. Bruno, I.V. Goryuchkina, Asymptotic expansions of solutions of the sixth Painlevé equation, Trudy Mosk. Mat. Obs., 7 (200), 6 8 (in

28 562 Alexander D. Bruno Russian). English translation: Transactions of Moscow Math. Soc., 7 (200), [3] A.D. Bruno, I.V. Goryuchkina, All expansions of solutions to the sixth Painlevé equation near its nonsingular point, Doklady Akademii Nauk, 426 (2009), no. 5, (in Russian). English translation: Doklady Mathematics, 79 (2009), no. 3, [4] A.D. Bruno, Power asymptotics of solutions to an ODE system, DAN, 40 (2006), no. 5, (in Russian). English translation: Doklady Mathematics, 74 (2006), no. 2, [5] A.D. Bruno, Power-logarithmic expansions of solutions to a system of ordinary differential equations, DAN, 49 (2008), no. 3, (in Russian). English translation: Doklady Mathematics, 77 (2008), no. 2, [6] A.D. Bruno, Nonpower asymptotic forms of solutions to a system of ordinary differential equations, DAN, 420 (2008), no., 7 0 (in Russian). English translation: Doklady Mathematics, 77 (2008), no. 3, [7] A.D. Bruno, Complicated expansions of solutions to a system of ordinary differential equations, DAN, 42 (2008), no., 7 0 (in Russian). English translation: Doklady Mathematics, 78 (2008), no., [8] A.D. Bruno, A.V. Parusnikova, Local expansions of solutions to the fifth Painlevé equation, DAN, 438 (20),no. 4, (in Russian). English translation: Doklady Mathematics, 83 (20), no. 3, [9] A.D. Bruno, A.V. Parusnikova, Expansions of solutions to the fifth Painlevé equation near its nonsingular point, DAN, 442 (202), no. 5, (in Russian). English translation: Doklady Mathematics, 85 (202), no., [20] A.D. Bruno, Families of periodic solutions to the Beletsky equation, Kosmicheskie Issledovanija, 40 (2002), no. 3, (in Russian). English translation: Cosmic Research, 40 (2002), no. 3,

29 Asymptotic solution of nonlinear algebraic and differential equations 563 [2] A.D. Bruno, Analysis of the Euler-Poisson equations by methods of Power Geometry and Normal Form, PMM, 7 (2007), no. 2, (in Russian). English translation: J. Appl. Math. Mech., 7 (2007), no. 2, [22] A.D. Bruno, V.P. Varin, Periodic solutions of the restricted three-body problem for a small mass ratio, PMM, 7 (2007), no. 6, (in Russian). English translation: J. Appl. Math. Mech., 7 (2007), no. 6, [23] A.D. Bruno, V.P. Varin, On asteroid distribution, Astronomicheskii Vestnik, 45 (20), no. 4, (in Russian). English translation: Solar System Research, 45 (20), no. 4, [24] A.D. Bruno, V.F. Edneral, Algorithmic analysis of local integrability, DAN, 424 (2009), no. 3, (in Russian). English translation: Doklady Mathematics, 79 (2009), no., [25] A.D. Bruno, Space Power Geometry for an ODE and Painleve equations, Proceedings of the International Conference Painleve Equations and Related Topics, St. Petersburg, June, 20, [26] A.D. Bruno, Power-exponential expansions of solutions to an ordinary differential equation, DAN, 444 (202), no. 2, (in Russian). English translation: Doklady Mathematics, 85 (202), no. 3, [27] A.D. Bruno, Space Power Geometry for an ODE and P P 4, P 6, Painleve Equations and Related Topics, (eds. A.D. Bruno and A.B. Batkhin), De Gruyter, Berlin/Boston, 202, [28] A.D. Bruno, A.V. Parusnikova, Elliptic and Periodic Asymptotic Forms of Solutions to P 5, Painleve Equations and Related Topics, (eds. A.D. Bruno and A.B. Batkhin), De Gruyter, Berlin/Boston, 202, [29] A.D. Bruno, Regular Asymptotic Expansions of Solutions to one ODE and P P 5, Painleve Equations and Related Topics, (eds. A.D. Bruno and A.B. Batkhin), De Gruyter, Berlin/Boston, 202,

30 564 Alexander D. Bruno [30] A.D. Bruno, Power-elliptic expansions of solutions to an ordinary differential equation, Zhurnal Vychisl. Matem. i Matem. Fiziki, 52 (202), no. 2, (in Russian). English translation: Comp. Mathem. Math. Phys., 52 (202), no. 2, [3] A.D. Bruno, T.V. Shadrina, An axisymmetric boundary layer on a needle, Trudy Mosk. Mat. Obsch. 68 (2007), (in Russian). English translation: Transactions of Moscow Math. Soc., 68 (2007), [32] L. Prandtl, Über Flüssigkeitsbewegung bei sehr kleiner Reibung, in: Verhandl. III, Intern. Math. Kongr., Heidelberg, 904. [33] H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Zeit. für Math. und Phys., 56 (908), 37. [34] A.D. Bruno, Power geometry in nonlinear partial differential equations, Ukrainean Mathem. Bulletin, 5 (2008), no., [35] A.D. Bruno, Power geometry in differential equations, Contemporary Problems of Mathematics and Mechanics, Mathematics, Dynamical Systems, MGU, Moscow, 4 (2009), no. 2, (in Russian). [36] A.D. Bruno, Self-similar solutions and power geometry, Uspekhi Mat. Nauk, 55 (2000), no., 3 44 (in Russian). English translation: Russian Math. Surveys, 55 (2000), no., [37] A.D. Bruno, Power geometry and elliptic expansions of solutions to the Painlevé equations, International Journal of Differential Equations, 205 (205), Article ID 34075, 3. [38] G.L. Litvinov, G.B. Shpiz, The dequantization transform and generalized Newton polytopes, Idempotent Mathematics and Mathematical Physics (eds. G. L. Litvinov and V. P. Maslov), Contemporary Mathematics, AMS, Providence, RI, 377 (2005), [39] V.P. Maslov, V.N. Kolokoltsov, Idempotent Analysis and Its Applications to the Optimal Control, Nauka, Moscow, 994 (in Russian). Received: October 6, 205; Published: October 6, 205