The Newton polyhedron in the nonlinear Analysis

Alexander D. Bruno Institute of Applied Mathematics, Moscow 125047 Russia (bruno@applmat.msk.su)

Abstract

We give a simple presentation of an algorithm of selecting asymptotical first approximations of equations (algebraic and ordinary differential and partial differential). Here the first approximation of a solution of the initial equation is a solution of the corresponding first approximation of the equation. The algorithm is based on the geometry of power exponents including the Newton polyhedron. We give also a surway of applications of the algorithm in problems of Celestial Mechanics and Hydrodynamics.

Keywords: Newton polyhedron, algebraic equations, differential equations, asymptotics, first approximation, singular perturbation, boundary layer.

Introduction

Let F (X) be a polynomial (may be vectorial) of some variables x_{1}, ..., x_{l} and their derivaties (ordinary or partial). Let X = G(T) be a solution of the equation F (X) = 0, where T = (t_{1}, …, t_{m}) are some parameters. We are interested in solutions for which some of x_{i} → 0 or ∞. And we want to find an asymptotics of such a solution G(T). Let Ĝ(T) be the asymptotical first approximation of the solution G(T) and let F̂(X) be the corresponding first approximation of the polynomial F (X).

General Statement: The first approximation X = Ĝ(T) of the solution X = G(T) is a solution of the corresponding first approximation F̂(X) = 0 of the equation F (X) = 0.

The aim of the talk is to explain the algorithm for selecting first approximations, to show how this approach works for algebraic equations and for differential equations, and to point out some applications.

2. A polyhedron and normal cones of its faces

Let in the space IR^{n} with Cartesian coordinates q_{1}, …, q_{n} we have a finite set D of points Q_{j} = (q_{1j}, …, q_{nj}), j = 1, …, m. Let IR_{∗}^{n} denote the dual space such that for P = (p_{1}, …, p_{n}) ∈ IR_{∗}^{n} and Q = (q_{1}, …, q_{n}) ∈ IR^{n} there is the scalar product ⟨P, Q⟩ = p_{1}q_{1} + … + p_{n}q_{n}. For a fixed vector P ≠ 0 let D_{P} denote such subset of the set D on which the scalar product ⟨P, Q⟩ has the maximal value c, that is

⟨P, Q_{j}⟩ = c for Q_{j} ∈ D_{P} and

⟨P, Q_{j}⟩ < c for Q_{j} ∈ D \ D_{P}. (1)

Now for the given set D and for each P ≠ 0 we will find the corresponding set D_{P}. Let M denote the convex null of the set D. The boundary ∂M of the polyhedron M consists of faces Γ_{k}^{(d)} of different dimension d (0 ≤ d < n). Zerodimensional faces Γ_{k}^{(0)} are verteces of the polyhedron M, onedimensional faces are its edges etc. Accordingly (1) to each vector P ∈ IR_{∗}^{n} there corresponds in IR^{n} a supporting to the set D hyperplane L_{P} = {Q : ⟨P, Q⟩ = c}. It intersects the polyhedron M along a face Γ_{k}^{(d)}. The set U_{k}^{(d)} of all vectors P ∈ IR_{∗}^{n} such, that L_{P} ∩ M = Γ_{k}^{(d)}, is called the normal cone of the face Γ_{k}^{(d)}. The normal cone U_{k}^{(n−1)} of the hyperface Γ_{k}^{(n−1)} is a ray ortogonal to the face and directed out of the polyhedron M. The normal cone U_{k}^{(n−2)} is a sector bounded by rays U_{l}^{(n−1)} and U_{m}^{(n−1)} if Γ_{k}^{(n−2)} = Γ_{l}^{(n−1)} ∩ Γ_{m}^{(n−1)} etc. The union of normal cones of all faces is IR_{∗}^{n} \ {0}. Let us consider the boundary subsets D_{k}^{(d)} = Γ_{k}^{(d)} ∩ D of the set D.

Theorem 1 [1]. If P ∈ U_{k}^{(d)}, then D_{P} = D_{k}^{(d)}.

Example 1. For n = 2 and

D = {(3, 0), (0, 3), (1, 1)}  (2)

the polygon M, faces Γ_{k}^{(d)} and their normal cones are shown in Figures 6 and 7 of the book [1]. Here

\begin{matrix}D_{1}^{(0)} = Q_{1}, & D_{2}^{(0)} = Q_{3}, & D_{3}^{(0)} = Q_{2}; \\ D_{1}^{(1)} = \{Q_{1}, Q_{3}\}, & D_{2}^{(1)} = \{Q_{2}, Q_{3}\}, & D_{3}^{(1)} = \{Q_{1}, Q_{2}\}; \\ U_{1}^{(1)} = −λ(1, 2), & U_{2}^{(1)} = −λ(2, 1), & U_{3}^{(1)} = −λ(1, 1) (λ > 0). \end{matrix}

3. Algebraic equations

Let us consider a polynomial

f(X) = m ; ∑ ; j=1 f_{j}X^{Q_{j}},

where X^{Q} = x_{1}^{q_{1}} … x_{n}^{q_{n}} and D = {Q_{1}, …, Q_{m}}. The convex hull M the set D is called the Newton polyhedron of the polynomial f, and we can construct all accompanying objects as described abowe. Let D_{k}^{(d)} is a boundary subset of the set D. The sum

f̂_{k}^{(d)}(X) = ∑ f_{j}X^{Q_{j}} for Q_{j} ∈ D_{k}^{(d)}  (3)

is called the truncation of the polynomial f(X). Now we consider a curve of the form

x_{i} = b_{i}τ^{p_{i}}(1 + o(1)), b_{i} ≠ 0, i = 1, …, n,  (4)

where τ → ∞. On such curve a monomial

X^{Q} = B^{Q}τ^{⟨P,Q⟩}(1 + o(1)),

where B = (b_{1}, …, b_{n}) and P = (p_{1}, …, p_{n}). If P ∈ U_{k}^{(d)}, then on the curve (4)

f(X) = f̂_{k}^{(d)}(B)τ^{c} + τ^{c}o(1),

where c is from (1). Thus, the truncation (3) is the first approximation of the polynomial f(X) on curves (4) with P ∈ U_{k}^{(d)}. If curve (4) is a solution of the equation f(X) = 0 then f̂_{k}^{(d)}(B) = 0. That is we have

Theorem 2 [1,2,4]. The first approximation x_{i} = b_{i}τ^{p_{i}}, i = 1, …, n of the solution (4) is a solution of the corresponding truncated equation f̂_{k}^{(d)}(X) = 0.

This approach allows to find solutions of the equation f = 0 with any degree of accuracity [1,2,4].

Example 2. Let f = x_{1}^{3} + x_{2}^{3} − 2x_{1}x_{2}, then the set D is (2), and among truncations f̂_{k}^{(d)} there are

\begin{matrix}f̂_{1}^{(0)} = x_{1}^{3}, f̂_{2}^{(0)} = −2x_{1}x_{2}, f̂_{3}^{(0)} = x_{2}^{3}; \\ f̂_{1}^{(1)} = x_{1}^{3} − 2x_{1}x_{2}, f̂_{2}^{(1)} = x_{2}^{3} − 2x_{1}x_{2}. \end{matrix}

If (4) is a solution of the equation f = 0 with P ≠ 0, then P cannot ly in U_{k}^{(0)} because B^{Q} ≠ 0. If on the solution (4) x_{1}, x_{2} → 0 for τ → ∞, then P ∈ U_{1}^{(1)} or U_{2}^{(1)}. If P ∈ U_{1}^{(1)}, i.e. 2p_{1} = p_{2} < 0, then the first approximations x_{i} = b_{i}τ^{p_{i}} (i = 1, 2) satisfies the truncated equation x_{1}^{3} − 2x_{1}x_{2} = 0, i.e x_{2} = x_{1}^{2}/2. Similarly, if P ∈ U_{2}^{(1)}, then the first approximation of the solution (4) is x_{1} = x_{2}^{2}/2. That is two brauches of the curve f = 0 go trough the zero x_{1} = x_{2} = 0 (see [1], Figure 19). Other applications of the Newton polyhedron for algebraic equations see in [1,2,4,20]

4. Ordinary differential equations

Now we consider a differential polynomial

g(z_{1}, …, z_{n}) = ∑ g_{R}Z^{R},  (5)

where Z^{R} = z_{1}^{r_{1}} … z_{n}^{r_{n}} and

z_{1} = x, z_{2} = y, z_{2+l} = d^{l}y/dx^{l}, l = 1, …, n − 2. (6)

To calculate powers of variables x and y we shall consider d^{l}y/dx^{l} as the ratio y/x^{l}. Hence, to each monomial X^{R} we put in correspondance the vector power exponent Q = (q_{1}, q_{2}) = Q(R) with coordinates

q_{1} = r_{1} − n−2 ; ∑ ; l=1 lr_{2+l}, q_{2} = r_{2} + … + r_{n},  (7)

i.e. the point Q of the plane IR^{2}. After the substitution

x = z_{1} = τ^{p_{1}}, y = z_{2} = bτ^{p_{2}}

we have Z^{R} = const · τ^{⟨P,Q⟩}, where P = (p_{1}, p_{2}). Now the set D = D(g) is the set of all points Q(R) with g_{R} ≠ 0. Using boundary subsets D_{k}^{(d)} of the set D, we define truncations

ĝ_{k}^{(d)}(Z) = ∑ g_{R}Z^{R} for R : Q(R) ⊂ D_{k}^{(d)}

of the differential polynomial (4).

Theorem 3 [3]. Let the differential equation g(Z) = 0 have a solution of the form

x_{1} = τ^{p_{1}}, y = τ^{p_{2}}(b + ∞ ; ∑ ; i=1 b_{i}τ^{s_{i}}), b ≠ 0,  (9)

where P = (p_{1}, p_{2}) ≠ 0, τ → ∞ and 0 > s_{i} > s_{i+1}. If P ∈ U_{k}^{(d)} , then the first approximation (8) of the solution (9) is a solution of the corresponding truncated equation f̂_{k}^{(d)}(X) = 0.

Examples and applications of the Newton polyhedron in ordinary differential equations (including singular perturbations) see in [1, 3, 5–10, 13–19, 21–25].

5. Partial differential equations

Let us consider the partial differntial polynomial

h(W) = ∑ h_{R}W^{R},  (10)

where W = (w_{1}, …, w_{n}), w_{1} = x, w_{2} = y and w_{j+2} are partial derivations of the form (∂^{k+l}ψ)/(∂x^{k}∂y^{l}) To calculate the power exponents, we consider such a derivative as the monomial x^{−k}y^{−l}ψ, i.e. to it there corresponds the vector exponent Q = (q_{1}, q_{2}, q_{3}) = (−k, −l, 1) ∈ IR^{3}. To a product of such derivatives there corresponds the sum of such vector exponents. Thus, to each monomial W^{R} there corresponds a point Q = Q(R) ∈ IR^{3}, and to the polynomial (10) there corresponds a set D of such points Q. To each boundary subset D_{k}^{(d)} there corresponds the truncated polynomial

ĥ_{k}^{(d)} = ∑ h_{R}W^{R} for R : Q(R) ∈ D_{k}^{(d)}.

It is the first approximation of the polynomial (10) when vector (log |x|, log |y|, log |ψ|) tends to infinity near the normal cone U_{k}^{(d)} ⊂ IR_{∗}^{3}.

For an expansion ψ = f(x, y) and for a given vector (p_{1}, p_{2}) we can define its first approximation f̂_{(p_{1},p_{2})} as in Section 3. On the curves (4) it has the order of magnitude const·τ^{p_{3}}. To the vector P = (p_{1}, p_{2}, p_{3}) there corresponds the truncation ĥ_{P}(W) of the differential polynomial (10).

Theorem 4 [3]. If ψ = f(x, y) – is a solution of the equation h(W) = 0, then its first approximation ψ = f̂_{(p_{1},p_{2})} is a solution of the corresponding truncated equation ĥ_{P}(W) = 0.

Example 3 [11]. The Navier-Stokes equations for the plane stationary flow of a viscous fluid can be written in the Helmholtz form as one equation for the stream function ψ:

\frac{∂ψ}{∂y} \frac{∂^{3}ψ}{∂x^{3}} + \frac{∂ψ}{∂y} \frac{∂^{3}ψ}{∂x∂y^{2}} − \frac{∂ψ}{∂x} \frac{∂^{3}ψ}{∂x^{2}∂y} − \frac{∂ψ}{∂x} \frac{∂^{3}ψ}{∂y^{3}} =

= ν (\frac{∂^{4}ψ}{∂x^{4}} + 2\frac{∂^{4}ψ}{∂x^{2}∂y^{2}} + \frac{∂^{4}ψ}{∂y^{4}} ) ,  (11)

where the parameter ν is the cinematic viscous. To the monomials of the equation there correspond in IR^{3} the following points

Q_{1} = (−3, −1, 2), Q_{2} = (−1 − 3, 2), Q_{3} = Q_{1}, Q_{4} = Q_{2},

Q_{5} = (−4, 0, 1), Q_{6} = (−2, −2, 1), Q_{7} = (0, −4, 1). All they ly in the plane q_{1} + q_{2} = −4. Their convex hull M is a trapezoid. To each edge of the trapezoid there corresponds a first approximation of the equation (11). It turns out, that to the upper base Γ_{1}^{(1)} with D_{1}^{(1)} = {Q_{1} = Q_{3}, Q_{2} = Q_{4}} there correspond the Euler equations for a nonviscous fluid. To the lower base Γ_{3}^{(1)} with D_{3}^{(1)} = {Q_{5}, Q_{6}, Q_{7}} there correspond the Stokes equations for a creep flow. To the each side edge Γ_{2}^{(1)} with D_{2}^{(1)} = {Q_{2} = Q_{4}, Q_{7}} or Γ_{4}^{(1)} with D_{4}^{(1)} = {Q_{1} = Q_{3}, Q_{5}} there correspond the Prandtl equations for a boundary layer.

Considering the boundary conditions from the geometry of powers point of view, we can obtain asymptoics of the viscous flow around some particular bodies [11].

Other examples and applications of the method for partial differential equations see in [1, 3, 11, 19].

6. Conclusion

This approach is applicable to general systems of equations, which can be algebraic, ordinary differential and partial differential [4, 2, 3]. It uses methods of the geometry of power exponents developed in [1], see also [23]. We have created a program to compute the polyhedron M, boundary subsets D_{k}^{(d)} and their normal cones U_{k}^{(d)} [12]. It was applied to a 6-dimensional problem related to the water-wade problem [13, 18]. For a Hamiltonian system, we have given an algorithm for selecting all such its first approximations, which are Hamiltonian systems themselfs [14].

Here we have considered an analysis of solutions: a selection of first approximations of an equation, each of which can give only a part of a singular solution. To obtain whole singular solution we must match these parts (see Introduction to [22]). Such synthesis will be a subject of future works.

The Newton polygon was introduced by Briot and Bouquet (1856), Newton himself found one its edge only [26]. The Newton polyhedron was introduced in [21] for a system of ODE. See its history in [1,4].

The work was partly supported by the International Science Foundation, Grant M3W300.

References

[1] Bruno, A.D. Local Methods in Nonlinear Differential Equations. Springer -Verlag, Berlin, 1989.

[2] Bruno, A.D. and Soleev, A. First approximations of algebraic equations // Russian Ac. Sci. Doklady. Mathem. 49 (1994).

[3] Bruno, A.D. First approximations of differential equations // Ibid.

[4] Bruno, A.D. and Soleev, A. Local uniformization of branches of a space curve, and Newton polyhedra // St. Petersburg Math. J. 3:1 (1992) 53–82.

[5] Bruno, A.D. and Petrovich, V.Yu. Regularization of oscillations of a satellite on the very stretched orbit //Inst. Appl. Math., Preprint No. 4, Moscow, 1994 (Russian).

[6] Bruno, A.D. and Petrovich, V.Yu. Computation of periodic oscillations of a satellite. Singular case//Ibid. No. 44, 1994 (Russian).

[7] Sadov, S.Yu. Analysis of a function that defines rotational stability of almort symmetric satellite // Ibid. No. 84, 1994 (R).

[8] Sadov, S.Yu. Coefficients of an averaged equation of satellite oscillations // Ibid. No. 27, 1995 (R).

[9] Bruno, A.D. Singular perturbations in Hamiltonian Mechanics // Hamiltonian Mechanics (J. Seimenis, ed.). Plenum Press: N.Y., 1994. 43–49.

[10] Bruno, A.D. Simple (double, multiple) periodic solutions of the restricted three-boody problem in the Sun-Jupiter case // Inst. Appl. Math., Preprints Nos. 66, 67, 68, Moscow, 1993 (Russian).

[11] Bruno, A.D. and Vasil’ev, M.M. Newton polyhedra and the asymptotic analysis of the viscous fluid flow around a flat plate // Ibid. No. 44, 1995 (R).

[12] Soleev, A. and Aranson, A.B. Computation of a polyhedron and of normal cones of its faces. // Ibid . No 36, 1994 (R).

[13] Soleev, A. and Aranson, A.B. First approximations of an invertible ODE system. // Ibid. No. 28, 1995 (R).

[14] Bruno, A.D. and Soleev, A. Hamiltonian truncations of a Hamiltonian system // Ibid. No. 55, 1995 (R) = Russian Ac. Sci. Doklady. Mathem. 51 (1995).

[15] Bruno, A.D. Bifurcation of the periodic solutions in the symmetric case of a multiple pair of imaginary eigenvalues // Selecta Math. formerly Sovietica 12:1 (1993) 1-12.

[16] Sadov, S.Yu. On a dynamical system, arised from a finitedimensional aproximation of the Schrödinger equation // Mathematical Notes 56:3 (1994).

[17] Bruno A.D. and Sadov, S.Yu. On a formal integral of a nondivergent system // Ibid. 57:6 (1995).

[18] Bruno, A.D. and Soleev, A. Local analysis of a singularity of an invertible ODE system // Institute Appl. Mathem. Preprints Nos. 40, 47, 54. Moscow, 1995 (R).

[19] Bruno, A.D. The Newton polyhedron in Nonlinear Aanalysis //Ibid. No. 48, 1995 (R) = Vestnik of the Moscow University (1995).

[20] Khovanskii, A.G. Newton polyhedra (resolution of singularities) // J. Sov. Math. 27 (1984) 2811–2830.

[21] Bruno, A.D. The asymptotic behavior of solutions of nonlinear systems of differential equations // Soviet Math. Dokl. 3(1962) 464–467.

[22] Bruno, A.D. The Restricted 3-Body Problem. Walter de Gruyter, Berlin, 1994.

[23] Bruno, A.D. The Geometry of power exponents // I.H.E.S., Preprint M/90/42, Paris, 1990.

[24] Bruno, A.D. On Periodic flybys of the moon //Celestial Mechanics 24:3 (1981) 255–268.

[25] Bruno, A.D. Local method of nonlinear analysis // Banach Center Publications 20(1988) 103–120.

[26] Newton, I. A treatise of the method of fluxions and infinite series, with its application to the geometry of curve lines // The Mathematical Works of Isaac Newton (Harry Woolf, ed.), v.1, Johnson Reprint Corp., N.Y. and London, 1964, pp. 27-137.