On Divergence of Puiseux Series Asymptotic Expansions of Solutions to the Third Painlev\'{e} Equation

TL;DR

This paper investigates the divergence of Puiseux series solutions to the third Painlevé equation, identifying parameter conditions where these series are Gevrey order one and approximating solutions in sectors at infinity.

Contribution

It establishes the divergence of specific Puiseux series solutions and constructs analytic functions approximated by these series in sectors at infinity.

Findings

Puiseux series are of exact Gevrey order one under certain parameters.

The series diverge, but approximate solutions are constructed in sectors.

Provides insight into the asymptotic behavior of solutions at infinity.

Abstract

In this paper we present a family of values of the parameters of the third Painlev\'{e} equation such that Puiseux series formally satisfying this equation -- considered as series of $z^{2/3}$ -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them in sectors with the vertices at infinity.

Equations

$$w^{\prime \prime }=\frac{(w^{\prime }{)}^2}{w}-\frac{w^{\prime }}{z}+\frac{\alpha w^2+\beta }{z}+\gamma w^3+\frac{\delta }{w},$$

$$w(z)=z^{1/3}u(x),x=\frac{3}{2}{z}^{2/3},$$

$$u_{xx}^{\prime \prime }=\frac{(u_x^{\prime }{)}^2}{u}-\frac{u_x^{\prime }}{x}+\alpha u^2+\frac{3}{2}\frac{\beta }{x}+\frac{\delta }{u}.$$

$$\sum _{n=0}^{\infty }{a}_n{x}^{-n},$$

$$a_{0}=(-\delta/\alpha)^{1/3};\quad a_{1}=-\dfrac{\beta}{2\alpha a_{0}};\quad a_{2}=0;\quad a_{3}=-\dfrac{\beta}{6\alpha^{2}a_{0}^{2}}\biggl{(}\dfrac{\beta^{2}}{4\delta}+1\biggr{)};$$

$$a_n=\frac{1}{3\alpha a_0^2}(-\alpha \sum _{k=1}^{n-1}(a_k\sum _{j=0}^{n-k}{a}_j{a}_{n-k-j})-\alpha a_0\sum _{k=1}^{n-1}{a}_k{a}_{n-k}-\frac{3}{2}\beta a_{n-1}+$$

$$+\sum _{k=1}^{n-2}(2k^2+k(2-n))a_k{a}_{n-2-k}),n⩾4.$$

$$|a_n|⩽C(n!{)}^{1/k}{M}^n\text{leavevmode}\forall n\in \mathbb{N}.$$

$$a_0=-1,a_1=4,a_2=0,a_3=\frac{64}{3},a_4=\frac{256}{3},a_5=2048.$$

$$a_n=-\frac{1}{3}\sum _{k=1}^{n-1}(a_k\sum _{j=0}^{n-k}{a}_j{a}_{n-k-j})+\frac{1}{3}\sum _{k=1}^{n-1}{a}_k{a}_{n-k}-4a_{n-1}-\frac{32}{3}\sum _{k=1}^{n-2}(2k^2+k(2-n))a_k{a}_{n-2-k};$$

$$a_n=4a_{n-1}-16a_{n-2}-\frac{1}{3}\sum _{k=3}^{n-6}\left(a_k\sum _{j=3}^{n-k-3}{a}_j{a}_{n-k-j}\right)+\sum _{k=3}^{n-3}{a}_k{a}_{n-k}-4\sum _{k=3}^{n-4}{a}_k{a}_{n-1-k}-$$

$$-\frac{32}{3}\sum _{k=3}^{n-5}(2k^2+k(2-n))a_k{a}_{n-2-k}-\frac{128}{3}(n-4{)}^2{a}_{n-3}+\frac{32}{3}(n-2{)}^2{a}_{n-2}.$$

$$(A)\text{leavevmode}\text{leavevmode}{a}_n⩽n!{\left(\frac{32}{3}\right)}^{\frac{n}{2}-1},\text{leavevmode}n\ge 4;$$

$$(B)\text{leavevmode}\text{leavevmode}{a}_n⩾\frac{32}{3}(n-3)(n-2)a_{n-2},\text{leavevmode}n\ge 5;$$

$$(C)\text{leavevmode}\text{leavevmode}{a}_n⩾\sum _{k=3}^{n-2}{a}_k{a}_{n-k+1},\text{leavevmode}n\ge 5;$$

$$(D)\text{leavevmode}\text{leavevmode}{a}_n>0,\text{leavevmode}n\ge 3.$$

$$(A)\text{leavevmode}\text{leavevmode}$$

$$(B)\text{leavevmode}\text{leavevmode}$$

$$(C)\text{leavevmode}\text{leavevmode}$$

$$(D)\text{leavevmode}\text{leavevmode}$$

$$\sum _{k=3}^{n-5}(2k^2+k(2-n))a_k{a}_{n-2-k}⩾0,n⩾20.$$

$$2k^2+k(2-n)+2(n-2-k{)}^2+(n-2-k)(2-n)=(2k-(n-2){)}^2⩾0,$$

$$k!(n-k)!⩽4!(n-4)!\text{mbox}withk=4,\dots ,n-4.$$

$$\frac{n!}{k!(n-k)!}⩾\frac{n!}{4!(n-4)!},k=4,\dots ,n-4;$$

$$a_n⩽4a_{n-1}+\sum _{k=3}^{n-3}{a}_k{a}_{n-k}+\frac{32}{3}(n-2{)}^2{a}_{n-2}=4a_{n-1}+\frac{128}{3}{a}_{n-3}+\sum _{k=4}^{n-4}{a}_k{a}_{n-k}+\frac{32}{3}(n-2{)}^2{a}_{n-2}⩽$$

$$⩽4{\left(\frac{32}{3}\right)}^{\frac{n-3}{2}}(n-1)!+\frac{128}{3}{\left(\frac{32}{3}\right)}^{\frac{n-5}{2}}(n-3)!+{\left(\frac{32}{3}\right)}^{\frac{n-4}{2}}\sum _{k=4}^{n-4}k!(n-k)!+\frac{32}{3}{\left(\frac{32}{3}\right)}^{\frac{n-4}{2}}(n-2{)}^2(n-2)!⩽$$

$$⩽{\left(\frac{32}{3}\right)}^{\frac{n-4}{2}}\left(4\sqrt{\frac{32}{3}}(n-1)!+4\sqrt{\frac{32}{3}}(n-3)!+4!(n-4)!(n-7)+\frac{32}{3}(n-2{)}^2(n-2)!\right)=\tilde{A}.$$

$$\tilde{A}⩽n!{\left(\frac{32}{3}\right)}^{\frac{n}{2}-1}.$$

$$3(n-7)⩽\frac{4}{3}(n-3)\left(n(n-1)(n-2)-\sqrt{\frac{3}{2}}(n-2)(n-1)-\sqrt{\frac{3}{2}}-(n-2{)}^3\right)=$$

$$=\frac{4}{3}(n-3)\left(\left(3-\sqrt{\frac{3}{2}}\right)n^2-\left(10-3\sqrt{\frac{3}{2}}\right)n+8-3\sqrt{\frac{3}{2}}\right).$$

$$f(x)=\left(3-\sqrt{\frac{3}{2}}\right)x^3-\left(19-6\sqrt{\frac{3}{2}}\right)x^2+\left(\frac{143}{4}-12\sqrt{\frac{3}{2}}\right)x+9\sqrt{\frac{3}{2}}-\frac{33}{4},$$

$$\text{mbox}2)\text{leavevmode}\sum _{k=3}^{n-4}{a}_k{a}_{n-k-1}⩽a_{n-2}\text{mbox}\text{leavevmode}byinductionhypothesis(C)with\text{leavevmode}m=n-2;$$

$$\text{mbox}3)\text{leavevmode}\sum _{k=3}^{n-6}\left(a_k\sum _{j=3}^{n-k-3}{a}_j{a}_{n-k-j}\right)\stackrel{\text{mbox}withm=5,\dots ,n-4\text{mbox}\text{leavevmode}ininequality(C)}{⩽}$$

Full text

Keywords: Painlevé equations, asymptotic expansions, summability.

MSC classes: 34m25, 34m55.

1 Introduction

Consider the third Painlevé equation

[TABLE]

with $\gamma=0$, $\alpha,\beta,\delta\in\mathbb{C}$, $\alpha\delta\neq 0$. By making the following transform

[TABLE]

we obtain the equation:

[TABLE]

All formal power series satisfying equation (1) near infinity have the form

[TABLE]

where coefficients $a_{n}$ are determined by the formulae:

[TABLE]

[TABLE]

[TABLE]

Definition 1**.**

The series $\sum\limits_{n=0}^{\infty}a_{n}z^{-n}$ is called a series of Gevrey order $k$ [1] if there exist constants $k,M,C>0$ such that

[TABLE]

Definition 2**.**

The series $\sum\limits_{n=0}^{\infty}a_{n}z^{-n}$ is called a series of exact Gevrey order $k$ [4] if it is of Gevrey order $k$ and there exists no $k^{\prime}>k$ such that it is of Gevrey order $k^{\prime}$.

In [2] it is shown that series (2) is a series of Gevrey order one and as is proved in the book [3], the series (2) is a rational function iff $\beta\neq 0,\delta=-\beta^{2}/(4k)^{2},k\in\mathbb{Z}\setminus\{0\}$ or $\beta=0$: for such values of parameters the estimates (4) are not precise and can be improved. The aim of the present paper is to show that there also exist the values of the parameters of the third Painlevé equation for which the series (2) is of exact Gevrey order one, hence the series (2) diverges.

In the second section we find such values of parameters and prove the following

Theorem 1**.**

Series (2) with coefficients (1) with parameters $\alpha,\beta\in\mathbb{C}$, $\alpha\beta\neq 0$, $\delta=-\beta^{2}/2$ is of exact Gevrey order one.

In the third section we construct analytic functions being approximated by series (2) for parameters of the third Painlevé equation $\alpha=-1/32,\ \beta=-1/4,\ \delta=-1/32$.

2 Proof of Theorem 1

Here we prove the divergence of power series (2) with coefficients (1) considering the fixed values of the parameters of equation (1) $\alpha=-1/32,\ \beta=-1/4,\ \delta=-1/32$. Assume that the branch of the cube root is fixed so that $a_{0}=-1$ with the parameters given. Note that under the above conditions all the coefficients $a_{n}\in\mathbb{R}$.

Write out the first coefficients of the series needed for the further calculations:

[TABLE]

With $n\geqslant 6$ the coefficients have the form

[TABLE]

we rearrange them assuming that $n\geqslant 9$ and considering the values of the first coefficients calculated earlier. We obtain

[TABLE]

[TABLE]

Assertion 1**.**

For the coefficients $a_{n}$ $(\ref{coef_simplify})$ of series $(\ref{series})$, the following holds:

[TABLE]

Proof of Assertion 1 (induction proof). Inequality (A) holds with $n=4,\dots,8$, inequalities (B) and (C) hold with $n=5,\dots,8$, inequality (D) holds with $n=3,\dots,8$.

Let $n\geqslant 9$ and

[TABLE]

We prove that $a_{n}\leqslant n!\left(\dfrac{32}{3}\right)^{\frac{n}{2}-1};\leavevmode\nobreak\ \leavevmode\nobreak\ a_{n}\geqslant\dfrac{32}{3}(n-3)(n-2)a_{n-2};\leavevmode\nobreak\ \leavevmode\nobreak\ a_{n}\geqslant\sum\limits_{k=3}^{n-2}a_{k}a_{n-k+1};\leavevmode\nobreak\ \leavevmode\nobreak\ a_{n}>0.$

Lemma 1**.**

[TABLE]

Proof of Lemma 1. Split the monomials on the left-hand side (LHS) of inequality (8) into pairs and sum up the monomial corresponding to index $k$ and the monomial corresponding to index $n-2-k$, where $k<(n-2)/2$. If $n$ is odd, then the LHS sum of inequality (8) splits into the above pairs; if $n$ is even, then, besides the above pairs, there remains the expression $a^{2}_{(n-2)/2}(2((n-2)/2)^{2}-(n-2)/2\cdot(2-n))=0$; that is, in order to prove inequality (8), it is sufficient to check non-negativity of the coefficient of $a_{k}a_{n-2-k}$, since, by induction hypothesis and base, $a_{k}>0,a_{n-2-k}>0$. This coefficient is equal to

[TABLE]

which completes the proof of Lemma 1. ∎

Lemma 2**.**

[TABLE]

Proof of Lemma 2. $C_{n}^{k}\geq C_{n}^{4},k=4,\dots,n-4$ that is

[TABLE]

therefore $\dfrac{1}{k!(n-k)!}\geqslant\dfrac{1}{4!(n-4)!},\>k=4,\dots,n-4$, whence the statement of Lemma 2 follows. ∎

Proof of inequality (A) of Assertion 1. Replacing terms on the right-hand side (RHS) of equality (2) with equal or greater expressions, and using statements of Lemmas 1 and 2 and induction hypothesis (A), we obtain

[TABLE]

[TABLE]

[TABLE]

Thus, to prove inequality (A) it is sufficient to check that

[TABLE]

Dividing the LHS and RHS of inequality (9) by the positive $8(n-4)!\left(\dfrac{32}{3}\right)^{\frac{n}{2}-2}$, and rearranging the terms, we see that to complete the proof of inequality (A) of Assertion 1 it remains to show that

[TABLE]

[TABLE]

The last inequality holds, since for the function

[TABLE]

it is true that $f(x)>0$ with $x>0$, therefore, $f(n)>0$ with $n\geq 9$, which finishes the proof of inequality (A).

Proof of inequality (B) of Assertion 1. First we prove the following inequalities:

1. $a_{n-1}\geqslant\dfrac{32}{3}(n-4)(n-3)a_{n-3}\geqslant\dfrac{32}{3}(n-4)^{2}a_{n-3}$ with $n\geqslant 4$ by induction hypothesis (B) with $m=n-1$ and by inequality (D) with $m=n-3$;

[TABLE]

[TABLE]

[TABLE]

[TABLE]

To prove (10), we decrease its LHS by subtracting from it two positive (by induction hypothesis (D) (7)) monomials with $k=n-4$, $k=n-3$; then we use inequality (B) (7) for every $a_{n-k}$ in its LHS by taking $m=5,\dots,n-3$, that is

[TABLE]

Thus, to prove (10) it is sufficient to check that

[TABLE]

[TABLE]

Split the monomials in the LHS of (11) into pairs and sum up the monomial corresponding to index $k$ and the monomial corresponding to index $n-2-k$, where $k<(n-2)/2$. Suppose $n$ odd, then the LHS sum of inequality (11) splits into the above pairs. Suppose $n$ even, then, besides the above pairs, there remains the expression $\dfrac{a^{2}_{(n-2)/2}}{4}\left(n-2\right)(n-4)\geqslant 0$ with $n\geqslant 4$, that is in order to prove inequality (10), it is sufficient to check non-negativity of the coefficient of $a_{k}a_{n-2-k}$, $3\leq k<(n-2)/2$   since, by induction hypothesis, $a_{k}>0,a_{n-2-k}>0$. This coefficient is equal to

[TABLE]

[TABLE]

for the indices $k$ considered in (11).

Due to inequalities 1) – 4) and formula (2), the coefficient

[TABLE]

which proves inequality (B) of Assertion 1.

Lemma 3**.**

[TABLE]

Proof of Lemma 3. Assume $n$ is even. Then we use $\left(\frac{n}{2}-2\right)$ times with $m=6,8,\dots,n$ the already proven inequality (B) of Assertion 1:

[TABLE]

If $n$ is odd, we use $\frac{n-3}{2}$ times inequality (B) of Assertion 1 with $m=5,7,\dots,n$:

[TABLE]

From 2\biggl{(}\dfrac{32}{3}\biggr{)}^{\frac{n-1}{2}}(n-2)!>4\biggl{(}\dfrac{32}{3}\biggr{)}^{\frac{n}{2}-1}(n-2)!\mbox{\; with \>}n\geqslant 5, we obtain that inequality (13) holds with all $n\geqslant 5$. Lemma 3 is proved. ∎

To prove inequality (C) of Assertion 1, we use inequality (12). As we have

$a_{n}\geqslant\left(\dfrac{32}{3}(n-2)^{2}-\dfrac{61}{3}\right)a_{n-2}$, it is sufficient to check that

[TABLE]

[TABLE]

For upper estimate of the RHS of inequality (14) we use induction hypothesis (A) for each $a_{\ell}$, and also use an analogue of Lemma 2:

[TABLE]

[TABLE]

For lower estimate of the LHS of inequality (14) we use Lemma 3 and obtain

[TABLE]

Due to above inequalities (2), (16) to check inequality (14) it is sufficient to prove that the following inequality holds with $n\geqslant 9$:

[TABLE]

Divide both LHS and RHS of inequality (17) by $(n-4)!\left(\dfrac{32}{3}\right)^{\frac{n-3}{2}}$, carry all the terms to the LHS; we prove non-negativity of the expression

[TABLE]

Inequality (D) of Assertion 1 follows from inequality (C) and induction hypothesis (D), since

[TABLE]

This completes the proof of Assertion 1. ∎

Corollary 1**.**

For the coefficients $a_{n}$ (1) of series (2) with the fixed values of the parameters $\alpha=-1/32,\ \beta=-1/4,\ \delta=-1/32$ the following holds:

[TABLE]

Lemma 4**.**

If another branch of the cube root is fixed in the calculation of coefficients (1) of series (2), that is $\tilde{a}_{0}=\omega a_{0}$, where $\omega^{3}=1$, then

[TABLE]

The proof is by induction on $n$. ∎

Lemma 4 together with estimates (18) immediately imply

Corollary 2**.**

*Series (2) with coefficients (1) with the fixed values of the parameters

$\alpha=-1/32,\ \beta=-1/4,\ \delta=-1/32$ diverges for any branch of the cube root.*

Corollary 3**.**

Series (2) with coefficients (1) with parameters $\alpha,\beta\in\mathbb{C}$, $\alpha\beta\neq 0$, $\delta=-\beta^{2}/2$ diverges.

The proof can be easily obtained from the following assertion [3]:

Let $w=\varphi(z)$ be a solution of the third Painlevé equation with given values $\alpha,\beta,\gamma,\delta$ of the parameters, then the function $\sigma_{1}\varphi(\sigma_{2}z)$ where $\sigma_{1}$, $\sigma_{2}\in\mathbb{C},\>\sigma_{1}\sigma_{2}\neq 0$ is a solution of the third Painlevé equation with the parameters $\alpha\sigma_{1}^{-1}\sigma_{2},$ $\beta\sigma_{1}\sigma_{2},$ $\gamma\sigma_{1}^{-2}\sigma_{2}^{2},$ $\delta\sigma_{1}^{2}\sigma_{2}^{2}.$ ∎

3 On Borel and Laplace transforms

As is proved in [2] solutions to the third Painlevé equation considered as functions of $z^{2/3}$ are asymptotically approximated of Gevrey order one by the series (2) in the sectors with the vertices at infinity with opening not larger then $\pi$. As we see from Corollary 3 series (2) with parameters of the equation $\alpha,\beta\in\mathbb{C}$, $\alpha\beta\neq 0$, $\delta=-\beta^{2}/2$ diverges and does not present an analytic solution to the third Painlevé equation. We construct an analytic functions being approximated of Gevrey order one by series (2) in the same sectors as mentioned above obtaining the Borel sum of the series (2) and then applying formal Laplace transform to it.

Firstly, consider series

[TABLE]

with $b_{n}=\dfrac{a_{n}}{n!}$, this series is a formal Borel transform of series (2). Series (19) converges for $|x|>R$. We calculate $R$ using the estimates from Corollary 1 and applying Cauchy-Hadamard theorem:

[TABLE]

Then we apply Borel-Ritt-Gevrey theorem [1]: given an arbitrary sector $S$ of opening at most $\pi$ with $d\in\mathbb{R}$ being the bisecting direction of $S$ and $\rho\in\mathbb{R},|\rho|>R$ the formal finite Laplace transform to series (19)

[TABLE]

where integrating is along $\textrm{Arg}\,u=d$, gives us an analytic function $g(x)$ asymptotically approximated by series (2) in the sector $S\cap\{x:|x|>R\}$.

Hence, we conclude that the difference between $g(x)$ and a solution to the third Painlevé equation in the given sector $S$ is a series of Gevrey order one.