Asymptotic Solving Essentially Nonlinear Problems

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.


Introduction
We develop a new Calculus based on Power Geometry [1][2][3][4]. At present, it allows to compute local and asymptotic expansions of solutions to nonlinear equations of three classes: 1. 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). 3. Computation of the tail of the asymptotic expansion.

Calculation of truncated equationsf
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 (1670) [5]; and by Briot and Bouquet for ordinary differential equation of the first order (1856) [6]. Space Power Geometry for a nonlinear autonomous system of Ordinary differential equations (ODEs) was proposed by the author (1962) [1], and for a linear partial differential equation (PDE), by Mikhailov (1963) [7]. Thus, in 2012 we could celebrate 50 years of the first publication on the Newton polyhedron.
Back in the autumn of 1959, 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 1961. 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 1962-1970, 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 1973 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 1990) 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 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.

The Problem Statement
Let X = (x, y, z) ∈ R 3 or C 3 and g(X) be a polynomial.
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 1 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.
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. 1).
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.

Objects and algorithms of Power Geometry
Consider a finite sum (for example, a polynomial) where X = (x, y, z) ∈ C 3 , Q = (q 1 , q 2 , q 3 ) ∈ R 3 and X Q = x q1 y q2 z q3 , g Q = const ∈ C\{0}. To each of the terms of sum (1), we assign its vector power exponent Q, and to the whole sum (1), we assign the set of all vector power exponents of its terms, which is called the support of sum (1) 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 1 Γ Here j is the number of a face. To each generalized face Γ (d) j , we assign the truncated sumĝ

Example 1: support and the Newton polygon
We consider the polynomial g(x, y) = x 3 + y 3 − 3xy. Support S(g) consists of points Q 1 = (3, 0), Q 2 = (0, 3), (Fig. 2). Its edges and corresponding truncated polynomials are * be a space dual to the space R 3 and P = (p 1 , p 2 , p 3 ) be points of this dual space. The scalar product is defined for the points Q ∈ R 3 and P ∈ R 3 * . In particular, the external normal N k to the generalized face Γ 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       j is a monomial. Such truncations are of no interest and will not be considered. We will consider truncated sums corresponding to edges Γ where log X = (log x, log y, log z) T , log X 1 = (log x 1 , log y 1 , log z 1 ) T , B is a non-degenerate 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 Q1 Here Q = (q 1 , q 2 , q 3 ) ∈ R 3 and other coordinates y 1 , z 1 for d = 1 and z 1 for d = 2 are small. Ifĝ (d) j (X) is a polynomial, then the sum h(X 1 ) is a polynomial as well.

Cone of the problem
The cone of the problem L is a convex cone of such vectors P = (p 1 , 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.
So, our initial Problem 1 corresponds to the cone of the problem in R 3 * , since x, y, z → 0 (and x, y, z as in (3)). If x → ∞ then p 1 > 0 in the cone of the problem L. For variables x, y near origin x = y = 0 cone of the problem is the quadrant III: L 3 = {p 1 , p 2 < 0}, near infinity x = y = ∞ cone of the problem is the quadrant I: L 1 = {p 1 , p 2 > 0}, near point x = 0, y = ∞ cone of the problem is the quadrant II: L 2 = {p 1 < 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.

Steps for Problem solving
Step 1. We compute the support S(g), the Newton polyhedron Γ(g), its two-dimensional faces Γ (2) j and their external normals N j . Using normals N j we compute the normal cones U (1) Step 2. We find all the edges Γ (1) 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 (1) k of these faces.
• For each of the selected edges Γ (1) k , we perform a power transformation X → X 1 of Theorem 2 and we get the truncated equation in a form h(x 1 ) = 0.
• We find the roots of this equation. Let x 0 1 be one of its roots.
• We perform the power transformation X → X 1 in the whole polynomial g(X) and we get the polynomial g 1 (X 1 ) = g(X). • We make the shift of variables x 2 = x 1 − x 0 1 , y 2 = y 1 , z 2 = z 1 in the polynomial g 1 (X) and get the polynomial g 2 (X 2 ) = g 1 (X 1 ).
• If x 0 1 is a simple root of the equation h(x 1 ) = 0 then, according to the Implicit Function Theorem, it corresponds to an expansion of the form x 2 = ∑ a kl y k 1 z l 1 , where a kl are constants. It gives an expansion of type 2 in coordinates X.
• If x 0 1 is a multiple root of the equation h(x 1 ) = 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 (−1, 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 1 of Theorem 2 and we get a truncated equation in the form h(x 1 , y 1 ) = 0.
• We factorize h(x 1 , y 1 ) into prime factors. Let h(x 1 , y 1 ) be one of such factors and its algebraic curve has genus ρ.
and thenh is divided by x 2 . We change variables in the whole polynomial g(X) and get the polyno- • Ifh(x 1 , y 1 ) is simple factor of h(x 1 , y 1 ) then roots of the polynomial g 2 (X 2 ) are expanded into series of the form where α k (y 2 ) are rational functions of y 2 . It gives an expansion of type 3 in original coordinates X.
we compute the Newton polyhedron of the polynomial g 2 (X 2 ), compute the cone of the problem , where ψ is a polynomial of order 3 or 4.
• If ρ 1 and we have the (hyper)elliptic curve and factorh 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 1 and we have the (hyper)elliptic curve and h(x 1 , y 1 ) is a multiple factor of h(x 1 , y 1 ) then we continue for g 2 (X 2 ) as above.
In this procedure we distinguish two cases: Case 1. Truncated polynomial contains linear part of one of the variables or x 0 1 is a simple root of h(x 1 ) orh(x 1 , y 1 ) is simple factor of h(x 1 , y 1 ). Then a generalization of Implicit Function Theorem is applicable and it is possible to compute parametric expansion of set of roots of full polynomial.  1 , we get truncated equation x 2 − 3y = 0, i. e. y = x 2 /3. It is the case 1, and this asymptotic form is continued into power expansion of branch y = It is the case 1, and these asymptotic forms are continued into power expansions of branches 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 1 of simple root, and the power expansion at 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 1.

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 [10]. Example 5 [9] g(X) = 512z 6 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. Near origin X = 0 we obtain where β k (u), γ k (u), δ k (u) are rational functions of u. More precisely, we have: here 3 Ordinary Differential Equations. Algebraic Approach 3.1 Plane Power Geometry [11] 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 q1 y q2 , where c = const ∈ C, (q 1 , 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 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 where c s are polynomials in log x, and power exponents r, s ∈ R, ωr > ωs, and 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).
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 j ∩ S(f ) (9) and to truncated equationf j (x, y) = 0 of equation (7).

Example 6
Consider the third Painlevé equation

32
Asymptotic Solving Essentially Nonlinear Problems Figure 7. The Newton polygon of equation (10) assuming the complex parameters a, b, c, d ̸ = 0. Here the first three differential monomials have the same power exponent Q 1 = (−1, 2), then Q 2 = (0, 3), Q 3 = (0, 1), 0). They are shown in Fig. 7 in coordinates q 1 , q 2 . Their convex hull Γ(f ) is the triangle with three vertices Γ 3 . The vertex Γ Let the plane R 2 * be dual to the plane R 2 such that for P = (p 1 , p 2 ) ∈ R 2 * and Q = (q 1 , q 2 ) ∈ R 2 , the scalar product corresponds to its own normal cone U (d) j ⊂ R 2 * formed by the external normal vectors P to the face Γ   (7), and ω(1, r) ∈ U (d) j , then the truncation y = c r x r of solution (8) is the solution to truncated equation (9).
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

Example 8 (cont. of Examples 6, 7)
For the truncated equationf For the truncated equationf . Using support S(f ) of the equation (7) and critical numbers k 1 , . . . , k κ with ωr > ωk i , we can find the set K(k 1 , . . . , k κ ) ⊂ R, support of expansion (8). Its elements s satisfy the inequality ωr > ωs.  1), l, m ∈ N∪{0}, l+m > 0}. (14) Hence, if the number k 1 is not odd, then all c s are constant and uniquely determined in expansion (12) for s ̸ = k 1 , and c k1 is arbitrary. Finally, if k 1 is odd, then K(k 1 ) = K, and in expansion (12) c s is a uniquely determined constant if s < k 1 ; c k1 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 1 .
The truncated equationf 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 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 10 (cont. of Examples 6-9)
For the truncated equation (11) corresponding to the edge Γ i. e. x = u, y = uv. Since y ′ = xv ′ +v, y ′′ = xv ′′ +2v ′ , then, canceling x and collecting similar terms, the equation (11), becomes Its support consists of three points Q 1 = (0, 2), Q 2 = (0, 1), Q 3 = 0 on the axisq 1 = 0 (see Fig. 9).  Figure 9. The support and Newton polygon for the third Painlevé equation (15) Now we make the logarithmic transformation ξ = log x. then, collecting similar terms, the equation (15) takes the form −vv +v 2 + bv + d = 0. Its support and polygon are shown in Fig. 10. Applying the technique described before to this equation, we obtain the expansion of its wherec is an arbitrary constant, and the constants c k are uniquely determined. In original variables, we obtain the family of non-power asymptotic of solutions to the initial equation (10), when x → 0.

Complex power exponents [11]
Indeed, the described method allows to calculate solutions with complex power exponents as well.
Thus, by the algebraic approach, expansions of solutions 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.

Computation of truncated equations and accompanying
objects.

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 [12]
It has the form where a, b, c, d are complex parameters, x and y are complex variables, y ′ = dy/dx. Equation (17) has three singular points x = 0, x = 1, 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. 11. We found all formal asymptotic expansions (16) of solutions to equation (17) near its three singular points. They comprise 108 families. In particular, for a = 1/2 and c = 0, there is an expansion of the form where C 1 is an arbitrary constant, the coefficients c s are uniquely determined constants. Here For C 1 = 1 and real x > 0, solution (18) has infinitely many poles accumulating at the point x = 0. We also found all expansions of solutions to equation (17) near its nonsingular points. They comprise 17 families [13]. q 2 q 1 0 1 1 Figure 11. Support and Newton polygon for the sixth Painlevé equation (17) That approach was applied to ODE systems [14][15][16][17].
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) [21].
4. Families of periodic solutions of the restricted threebody problem and distribution of asteroids [22,23].
4 Ordinary Differential Equations. Differential Approach

Orders of solutions and their derivatives
All solutions of the form (16) have the following property: where n is the maximal order of derivative in the initial ODE, and

Differential Approach [25-30]
To each differential monomial a(x, y) we put in correspondence the 3D point Q = (q 1 , q 2 , q 3 ), where q 1 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 ofS(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 where all φ l (x) are elliptic or hyperelliptic functions. This differential approach allows to find expansions (19) for solutions with property p ω ( y (k) ) = p ω (y) − γ ω k, k = 1, 2, . . . , n, where γ ω ̸ = 1.
5 Partial Differential Equations. Algebraic Approach
A differential monomial a(Z) corresponds to its vector power exponent Q(a) ∈ R m+n formed by the following rules where E j is unit vector. A product of monomials a · b corresponds to the sum of their vector power exponents: 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 where f i are differential sums.
is the truncated system if the intersection of corresponding normal cones U is not empty. A solution y i = φ i (X), i = 1, . . . , n to system (21) 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 =φ i (X), i = 1, . . . , n of this solution satisfies truncated system (22), which is quasi-homogeneous.

Applications. Boundary layer on a needle [31]
The theory of the boundary layer on a plate for a stream of viscous incompressible fluid was developed by Prandtl (1904) [32] and Blasius (1908) [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 noslip conditions on the needle (Fig. 13). We considered two cases: (1) incompressible fluid and (2) compressible heatconducting gas.
Hence the supports of equations (24) must be considered in R 4 . It turned out that polyhedra Γ(g 1 ) 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).
An analysis of solutions to the latter problem (31) by methods of Power Geometry revealed that for N ∈ (0, 1) it has Figure 16. The Newton polyhedron of f 3 in (29) solutions of the form ψ ∼ c 1 r 2 |log ξ| −1/N , ρ ∼ c 2 |log ξ| −1/N , h ∼ c 3 |log ξ| 1/N , (32) where ξ = r 2 /x → 0 and c 1 , c 2 , c 3 are arbitrary real constants. Thus, for N ∈ (0, 1), in the boundary layer r 2 /x < const, as x → +∞ and ξ = r 2 /x → 0, we obtained the asymptotic form of the flow (32), 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]. 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 non-smooth).