International Journal of Differential Equations
Volume 2015 (2015), Article ID 340715, 13 pages
http://dx.doi.org/10.1155/2015/340715

Research Article

Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations

Alexander D. Bruno

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.

Figure 1: 3D support and polyhedron of equation   (8). The grey face is ; the grey edge is . Projection on the plane is shown by dotted lines. Dashed line is the invisible edge.

Their convex hull is a tetrahedron. It has 4 vertices , 6 edges , and 4 faces . Face is distinguished in Figure 1; its external normal and its truncated equation

Edge is also distinguished in Figure 1; its truncated equation

Example 3 is finished.

Let be a solution to (6) with property (4) and , ; then the order of a monomial with is

where and is the scalar product. Leading terms of the sum (1) after substitution are monomials , for which reaches the maximal value on the support . Here and according to Theorem 1. On the support maximum of the scalar product is achieved on a generalized face of the polyhedron .

By we denote the 3D real space, where we put power exponents , and by we denote the space dual (conjugate) to . We will denote points in as . Then we have the scalar product

Each face corresponds to its normal cone [2]:

Thus, normal cone of the face is a ray spanned on the exterior normal of the face , normal cone of the edge is 2D angle spanned on rays and , where ; normal cone of the vertex is a 3D angle spanned on exterior normals of all 2D faces containing the vertex (see [2]).

Thus, selection of the truncated sums can be made by the following method. First we compute the support of the initial sum . Using support , we compute the polyhedron of sum , that is, all its vertices , edges , and faces . Next we compute their normal cones and select only such truncated equations for which the intersection . But truncated equations with can be studied by algorithms of 2D Power Geometry. So 3D Power Geometry studies truncated equations with nonempty intersection .

Example 4 (continuation of Example 3). Polyhedron for equation   (8) has 4 following faces with exterior normal

Only two of them, and , have . Hence, all edges exept and all vertices have vectors with in their normal cones and .

4. Power Transformations

If the face has the normal then the corresponding truncation , where the differential sum contains and its derivatives but does not contain . In that case the full sum can be written as , where and is a differential sum.

Remark 5. If is a solution to the equation with the property

when or , then can be the asymptotic form of the solutions to the full equation (6). Here is a small real number. We call as formal asymptotic form.
Let the power transformation of variables

transform into : .

Theorem 6. Let the face of have the exterior normal with

then the power transformation (17) with , transforms the truncation of into the truncation

of , corresponding to the face of with the exterior normal . Here equals after substitution

instead of .

So, if is a solution to the equation and is bounded from zero and infinity as in (16), then the initial equation can have a solution with the asymptotic form:

Herewith the power transformation (17) induces the following formulas for derivatives:

where .

Theorem 7. Let an equation of order

have a solution of the form

where is the solution to the truncated equation

with the property

Then satisfies the linear equation

where , is a polynomial on depending on , and and for and . is the first variation.

The first variation is the formal Frechet derivative (see [3]).

Solution to the transformed equation is expanded into series (24) with integer only if the transformed equation divided by has form (23) with integer . In that case, solutions to the truncated equation are suitable asymptotic forms for continuation by power expansion (24) and corresponding normal is also suitable.

External normal to 2D face is unique up to positive scalar factor. Hence, power transformation (17) of Theorem 6 is unique and we must only check that the transformed equation has form (23) with integer . The external normal to 1D edge belongs to the normal cone . Hence, in the cone we must select suitable vectors with mentioned property of integer . Things for a vertex are the same, but usually solutions to corresponding equation are so simple, which do not give interesting expansion.

Let , , , , and , . Denote

and .

Theorem 8. The transformed equation (23) has the property of integer if and only if all numbers

are natural.

There are 8 essentially different polyhedrons for Painlevé equations [19]. Each of them has exactly one 2D face in which truncated equation has elliptic solutions. It was shown [23] that all those elliptic asymptotic forms are suitable. Among 8 polyhedrons only 3 have an edge which truncated equation has elliptic solutions. These are , , and . No truncated equations corresponding to vertices of these 8 polyhedrons have elliptic solutions.

Example 9 (continuation of Examples 3 and 4). Polyhedron of equation   (8) has edge with truncated equation . Its first integral is

where is arbitrary constant. If , solutions to (30) are elliptic functions. The same will be true after any power transformation (17). Let us apply Theorem 8 to the edge . The edge . So normal cone is the conic hull of two normals and ; that is, up to positive scalar factor, vectors have the form

Here , , , , and . Conditions of Theorem 8 are

where and are natural numbers. Hence ; that is, , .
We can write . Condition (18) of Theorem 6 means that . If , then ; that is, ; if , then ; that is, . So there is a countable set of suitable normals to edge . According to Theorem 6, here

5. Computation of Expansions

Below we consider the case when the truncated equation has the first integral of the form

Differentiating with respect to and dividing by , we obtain

Here and below the prime denotes the derivative with respect to .

Using (34) and (35), any power series of and its derivatives can be written as the sum , where and are power series only of . Let , where and are functions only of . Then, omitting the index , by (34) and (35), we obtain

Further derivatives of do not need us here, because we consider only (23) of the second order. In our case

Thus, (27) splits in two:

where . Note that in (38) differential operators and are operators on and do not depend on . If polynomial in (34) does not have multiple roots and its degree is greater than one, that is,

where is discriminant of the polynomial , then solution to the truncated equation (25) is periodic (if ), or elliptic (if or ), or hyperelliptic (if ) function.

Near some point we will compute asymptotic expansions of fundations and :

where if and if . If initial equation (23) is a differential sum then according to Theorem  3.1 [3] coefficients and are either constants or polynomial of ; that is, expansions (40) are either power or power-logarithmic [3]. Moreover, according to Theorem  3.4 [3] (see proof in Theorem   . [4]) power expansions (40) converge for small .

If the solutions and to the system (38) have no branching, then they are also periodic or (hyper)elliptic functions. Finally, if for the sequence of (38) with , there exist solutions and without branching, the solutions to (23) have a regular asymptotic expansion (24).

Let operators and be inverse to operators and , respectively. Then the solutions of (27) are of forms

In our case the initial ODE (23) has order two. Hence operators and are of the second order. Moreover, in our case factors of in and of in are the same. Denote it as . Singular points of operators and are roots of . Indeed , where is a simple polynomial. So roots of and will be singular points of operators and , but roots of polynomial different from singular points will be their subsingular points.

Theorem 10. If functions and are regular then the solutions to (41) can have logarithmic branching only at infinity and at singular points of the operators and but they can have algebraic branching and can be in singular and subsingular points only.

For the existence of a regular expansion (24) we need to prove the existence of a sequence of functions and that do not have branching. From the other side, if it is shown that or have branching, then it proves the absence of regular expansion.

In [19, 23], for each polyhedron of the Painlevé equations, we selected suitable 2D faces, for each of them we wrote (23), operators and , and inverse ones and . We found their singular points and the conditions on the parameters of the equation and on the solution under which the functions and do not have logarithmic branching, as well as the conditions under which at least one of these functions has such branching. It is a wonder that for each Painlevé equation the operators and are expressed in the same way in terms of polynomial and different cases distinguish only by this polynomial. At the same time, for all cases of faces of five Painlevé equations , there are only four different pairs of operators and .

Singular point of operators and are for and for and for . To each suitable elliptic asymptotic form and to each singular point we assign one basic formal asymptotic expansion (24). Our aim is to show existence or nonexistence of regular basic expansions by means of calculation of expansions (40) near the singular points.

6. Expansions for

Details of calculation of expansions (24) will be explained for equation

and its truncated equation

First, according to (33) and Theorem 6, we make power transformation ,    (17) using formulas (22) and obtain equation   (42) in the form (23):

where

Here , and is arbitrary complex constant.

Operators and   (41) are

Here [23] and singular points of operators (47) are only infinity. Let us introduce a function

Here the integral is determined by mentioned asymptotic expansion near . Solutions of system (38) or (47) have 4 arbitrary constants :

where and are fixed solutions. Here expansions near are

So we will assume that power expansion for does not contain terms and but expansion for does not contain terms and . If it is necessary we can change constants . Now the functions and are unique and expansion  (24) is called basic if there all . Below we compute this basic expansion only.

Lemma 11. If , then solutions (49) to (47) for are regular in subsingular points (if and are also regular in them).

Let and be power series on decreasing power exponents of and and let be their terms with maximal power exponents and correspondingly, , , . and contain , if

So these numbers are critical for operators and .

We will compute , , and as functions of , for , and also will compute leading terms of and , that is, power exponents and and constants and .

For that we will use following expansions:

Case . According to (45), ; hence, , . According to (46) and (47) we obtain , . Next,

Hence, according to (36),

According to (47), , . Next,

Hence, , according to (36),

According to (47), , .

Next,

Hence, according to (36),

Here power exponent of leading term in is critical for operator but . Hence has no logarithmic branching.

Now we take into account terms and from (45). For power exponents and for and are small enough to neglect them. So

We can write corresponding expansions for , , and . Then

Hence

It means that and have no branching at and and for .

So we neglect for and consider

We have

Hence, according to results after (59),

where and has the logarithmic branching; that is, the regular expansion does not exist.

For , we must add to the computed value of , but it does not change result on existence of logarithmic branching in .

 Case is close to the case and it has branching in . Case was calculated separately. It has no branching. Case corresponds to 2D face . It has no branching.

Thus, for equation (42), basic formal expansions are regular for two suitable asymptotic forms with and when .

Theorem 12. For , the regular basic families of formal power-elliptic expansions exist only for two suitable elliptic asymptotic forms with and , that is, when .

It is possible to prescribe power exponents and of leading terms in and . So we can compute such numbers and , where for and for . Here and are smaller critical values (51) of operators and . And it is enough to calculate and up to .

7. Nonbasic Expansions for

Basic expansions (24) were defined by formulas (47), (49) with . According to Lemma 11, condition guarantees regularity of and in subsingular points. Now we want to study cases with nonzero .

Example 13. Let us show that in gives the logarithmic branching in for . For , we put . According to formulas for case , we obtain

Hence,

Next,

Power exponent is critical for (see (51)). Coefficient for in is . It is equal to zero only for , but ; that is, . But , then has logarithmic branching.

8. Equation

Equation is

Support consists of 3 points , , and . Its polyhedron is a triangle with normal . So the equation is its own truncation. The edge of the triangle corresponds to the truncated equation

which has the first integral

with elliptic solutions.

Suitable normals to the edge are , , and if . Here , , and , ; the transformed equation is

, operators and are again (47) and [23]. Hence there is only one singular point and Lemma 11 is true for . Here and integral critical numbers are and . Formulas (47)–(49) again define basic expansions. If then

Hence, has no logarithmic branching, if .

Similarly to the end of Section 6 (see (59)), we obtain

,   for   and the regular expansion exists. If , then the regular expansion exists; the same is true for . Case corresponds to 2D face and to other , but . Thus, equation has regular basic families of elliptic expansions corresponding to all suitable asymptotic forms. Thus we have the following.

Theorem 14. To each suitable elliptic asymptotic form of there corresponds the basic family of formal power-elliptic expansions, which is regular.

9. Equation

Equation is

which has 3 different polyhedrons depending on values of coefficients [19, 23].

Case . See Figure 2.

Figure 2: 3D support and polyhedron of equation   (74) with all . The grey face is . All dotted lines are in the plane , ; they show projections of on the plane . Dashed lines are invisible edges.

Here only one truncated equation

corresponding to the distinguished 2D face in Figure 2 has elliptic solutions. Here the power transformation (17) is identical.

Equation (74) with is of the form (23) with , where

Solutions to (38) are of the form

Here [23], so there are 2 singular points and . This is true for all cases of . Near the singular point , . So ,   and expansions for and do not contain terms , and , correspondingly. Critical numbers for and are 2, 5 and 0, 3 correspondingly. Moreover, , , and , for . So expansion has no logarithmic branching at .

Near the singular point we have . Here we have 4 constants and basic expansion if all . Here Lemma 11 is correct for .

Condition C. Condition C is .

Theorem 15. If the Condition C is satisfied then basic expansions for are regular.

Case , . After the power transformation , , (74) with takes the form (23) with , where

Formula (77) is valid here. At , and have critical number 2.   and orders of , are less than for .

The same is at . Thus, here formal basic expansion is regular. Lemma 11 and Theorem 15 are true.

Case , . After the power transformation , , (74) with takes the form (23) with , where

At critical values for and are 2, , for . So here basic expansion has no branching.

The same is at . Lemma 11 and Theorem 15 are true.

Each of 3 polyhedrons has exactly one 2D face corresponding to a truncated equation with elliptic solutions [18, 19, 23]. They have different first integrals , but common operators and with singularities in two points and .

10. Equation

Equation is

If complex parameters , its support consists of 6 points; polyhedron is a tetrahedron and has one 2D face and one edge with truncated equations

having elliptic solutions [19, 20, 23]. Normal to is , and suitable normals to are , . After power transformation (17) with , , we obtain (23) with

Here solutions to (38) are

[23], so there are two singular points and . Near   . Critical numbers for and are and correspondingly. If , , ,  , , , and .

Now we compute expansion of the form (59). Then , ,  , ,  , and  . Thus, only if ; that is, , which is impossible. Thus, has logarithmic branching and the regular basic expansion is absent. The same is true for and for , when .

11. Equation

Equation is

where are complex parameters, having two different polyhedrons depending on values of parameter [21, 23]. Each of the polyhedrons has only one 2D face with elliptic solutions.

Case . Here transformation (17) is identical , . So, in (23)  ,

Solutions to (38) are

Here [23], so singular points are . Near the singular point

critical numbers for and are 4 and 3 correspondingly. If , then contains the term and has logarithmic branching. If , then and for all . Thus, the basic expansion is regular. Similarly basic expansions are regular near if and only if and near without restrictions.

Condition D. Condition D is .

Theorem 16. If in equation with and with , Condition D is fulfilled then basic expansions are regular. If one of these conditions is violated then all basic expansions are nonregular.

Case , . After the change , , equation takes the form (23) with , where

Formulas (86) are again valid. Here basic expansions near are regular if and only if , near if and only if , and near are always nonregular.

12. Equation

In generic case has polyhedron with ten 2D faces , but all external normal to them   does not satisfy conditions (18)  , . Moreover, all edges have no suitable normal. The same is true for degenerate cases.

13. Summary

Thus, all basic expansions are regular for without additional restrictions (Theorem 14), for if (Theorem 12), for under Condition C (Theorem 15), and for with and under Condition D (Theorem 16).

As next step it is necessary to study convergence of found regular formal power-elliptic expansions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

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