International Journal of Differential Equations

Volume 2015 (2015), Article ID 340715, 13 pages

http://dx.doi.org/10.1155/2015/340715

## Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

Keldysh Institute of Applied Mathematics, Miusskaya Square 4, Moscow 125047, Russia

Received 30 January 2014; Accepted 24 June 2014

Academic Editor: Sining Zheng

Copyright © 2015 Alexander D. Bruno. This is an open access article 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

We consider an ordinary differential equation (ODE) which can be written as a polynomial in variables and derivatives. Several types of asymptotic expansions of its solutions can be found by algorithms of 2D Power Geometry. They are power, power-logarithmic, exotic, and complicated expansions. Here we develop 3D Power Geometry and apply it for calculation power-elliptic expansions of solutions to an ODE. Among them we select regular power-elliptic expansions and give a survey of all such expansions in solutions of the Painlevé equations .

#### 1. Universal Nonlinear Analysis

We develop a new calculus based on Power Geometry [1–4]. Now it allows to compute local and asymptotic expansions of solutions to nonlinear equations of three classes: (A) algebraic, (B) ordinary differential, and (C) partial differential, as well as to systems of such equations.

Principal ideas and algorithms are common for all classes of equations. Computation of asymptotic expansions of solutions consists of the 3 following steps (we describe them for one equation ).(1)Isolation of truncated equations by means of faces of the convex polyhedron which is a generalization of the Newton polyhedron: the first term of the expansion of a solution to the initial equation is a solution to the corresponding truncated equation .(2)Finding solutions to a truncated equation which is quasihomogenous: using power and logarithmic transformations of coordinates we can reduce the equation to such simple form that can be solved. Among the solutions found we must select appropriate ones which give the first terms of asymptotic expansions.(3)Computation of the tail of the asymptotic expansion. Each term in the expansion is a solution to a linear equation which can be written down and solved.

*Applications*

*Class A*. (1) Sets of stability of multiparameter problems [5, 6].

*Class B*. (2) Asymptotic forms and expansions of solutions to the Painlevé equations [4, 7, 8].

(3) Periodic motions of a satellite around its mass center moving along an elliptic orbit [9].

(4) New properties of motion of a top [10].

(5) Families of periodic solutions of the restricted three-body problem and distribution of asteroids [11, 12].

(6) Integrability of ODE systems [13].

*Class C*. (7) Boundary layer on a needle [14].

(8) Evolution of the turbulent flow [15].

For a survey of these applications see [16].

#### 2. Introduction

Let be a formal elliptic asymptotic form of a solution to an ODE; that is, it is a solution of a corresponding truncated equation. The form is suitable if it can be extended into power asymptotic expansion , where are some functions. The expansion is regular, if all are not branching functions of and its derivatives. If all functions have no branching, then they are elliptic functions with the same periods as . Selection of such cases is our aim. For given and fixed point (including infinity), we can compute power-logarithmic expansions of functions near . In these expansions logarithmic branching can appear, only if is a singular point, and algebraic branching (of finite order) can be for subsingular points . To each singular point and suitable asymptotic form , we assign unique regular expansion , so called basic, and we are looking for such basic expansions near singular point , which have no branching.

We propose algorithms for (1) finding all formal elliptic asymptotic forms, (2) finding all suitable elliptic asymptotic forms, and (3) calculation of power-logarithmic expansions of functions near a singular point and selection of basic expansions without branching. All algorithms are based on 3D Power Geometry.

Expansions are formal; their convergence is not considered. Application of these algorithms to the Painlevé equations gives following.(1), , and have continuum of 2-parameter families of elliptic asymptotic forms each, has three, has two of them, and does not have.(2), , and have countable sets of families of suitable asymptotic forms each, and all 5 forms of and are suitable.(3)Basic expansions for all suitable forms have no branching for , for if the independent variable tends to infinity, for if condition C is fulfilled, and for if condition D is fulfilled and , and .

History of calculation of elliptic expansions of solutions to the Painlevé equations is as follows.

A hundred years ago, Boutroux [17] found 2 families of elliptic asymptotic forms of solutions to the Painlevé equations and . During the last 5 years we found 6 additional families of elliptic asymptotic forms of solutions to (three) [18, 19], (one) [20], and (two) [21]. Moreover the Painlevé equations , , and have continuum of families of elliptic asymptotic forms each, and I proposed a criterion for selection suitable asymptotic forms, which can be extended as asymptotic expansions. All 8 known elliptic asymptotic forms are suitable. Solutions to the equation have no elliptic asymptotic forms at all.

Near infinity of the independent variable, the Painlevé equations have 12 families of suitable asymptotic forms and near zero of the independent variable equations , , and have countable sets of such families each. Next I extend these suitable elliptic asymptotic forms into power-elliptic expansions , where coefficients are functions of the corresponding elliptic asymptotic forms and their derivatives. To each family of suitable elliptic asymptotic forms, I put in correspondence unique basic formal power-elliptic expansion near for , near for , and near for . Obstacles (logarithmic branching) in calculations of these basic expansions appeared only for if the independent variable tends to zero, for and for if or .

Thus, near infinity of the independent variable, there are 10 families of regular (i.e., without branching) elliptic expansions of solutions to equations : 4 for , 2 for , 3 for , and 1 for . Existence of these expansions for two Boutroux families of asymptotic forms was proven in [22], and this is all known up-to-date. Near zero of the independent variable there is a countable set of families of such expansions for . The results were obtained by means of algorithms of 3D Power Geometry [18–24], realized in very cumbersome calculations.

Here I introduce the third variant of 3D Power Geometry. The first was in [18, 20, 24], and the second was in [19, 21–23].

In more precise form main results are as in Theorems 12, 14, 15, and 16 and Conditions C and D.

Equation cannot be studied by proposed approach.

#### 3. 3D Power Geometry

Let be independent and be dependent variables, . A* differential monomial * is a product of an ordinary monomial , where , , and a finite number of derivatives of the form , . The sum of differential monomials
is called the* differential sum. *Let be the maximal value of in .

In [2–4] it was shown that as () or as () solutions to the ODE , where is a differential sum, can be found by means of algorithms of Plane (2D) Power Geometry, if where the order on a ray and is the maximal order of derivatives in . Order of the power function with is .

Here we introduce algorithms, which allow to calculate solutions with the property where .

Theorem 1. *.*

For example, for and for . Note that in Plane Power Geometry we had ; that is, . So, new interesting possibilities correspond to .

*Problem 2. *Select leading terms in the sum (1) after substitution with property (4).

Below we describe algorithms for solution of the problem. To each differential monomial , we assign its* (3D) power exponent * by the following rules: = sum of orders of all derivatives; = order of ; = difference of order of and .

Then the 2D vector is the same as in 2D Power Geometry [2–4] and corresponds to the total order of derivatives. The power exponent of the product of differential monomials is the sum of power exponents of factors: .

The set of power exponents of all differential monomials presented in the differential sum is called the* 3D support of the sum *. Obviously, . The convex hull of the support is called the* polyhedron* of the sum . The boundary of the polyhedron consists of the vertices , the edges , and the faces . They are called* (generalized) faces *, where the upper index indicates the dimension of the face, and the lower one is its number. Each face corresponds to the 3D* truncated sum*:

All these definitions are applied to differential equation
Thus, each generalized face corresponds to the truncated equation

Let be the external normal to two-dimensional face . We will consider only normals with .

*Example 3. *Consider the second Painlevé equation :
where is the complex parameter.

If , the 3D support consists of 4 points
They are shown in Figure 1.