On the convergence of an exotic formal series solution of an ODE

TL;DR

This paper establishes a sufficient condition for the convergence of a special type of formal series solution, with complex exponents, to a general ordinary differential equation.

Contribution

It introduces a new convergence criterion for exotic formal series solutions of ODEs, expanding understanding of their analytical properties.

Findings

Proposes a sufficient condition for convergence.

Applies to a broad class of ODEs.

Enhances theoretical understanding of exotic series solutions.

Abstract

A sufficient condition of the convergence of an exotic formal series (a kind of power series with complex exponents) solution to an ODE of a general form is proposed.

Equations

$$F(x,y,\delta y,\dots ,{\delta }^{n}y)=0,$$

$$\Delta =\{|x|<\rho ,|y_j|<{\epsilon }_j,j=0,\dots ,n\}.$$

$$\phi =\sum _{k=0}^{\infty }{\alpha }_k(x^{\mathrm{i}\eta })x^k,\mathrm{i}=\sqrt{-1},\eta \in \mathbb{R}\setminus \{0\},$$

$${\alpha }_k(t)=t^{-r_k}\sum _{\ell =0}^{\infty }{\alpha }_{k\ell }{t}^{\ell },t^{r_k}{\alpha }_k(t)\in \mathbb{C}\{t\},r_k\in \mathbb{Z},$$

$$F(x,\Phi )=0,\Phi :=(\phi ,\delta \phi ,\dots ,{\delta }^n\phi ),$$

$$\frac{\partial F}{\partial y_i}(x,\Phi )=B_i(x^{\mathrm{i}\eta })x^N+D_i(x^{\mathrm{i}\eta })x^{N+1}+\dots ,i=0,1,\dots ,n,$$

$${\mathrm{ord}}_0{B}_i⩾{\mathrm{ord}}_0{B}_n,i=0,1,\dots ,n.$$

$$y=\sum _{k=0}^m{\alpha }_k(t)x^k+x^{m}u$$

$$x^{\mathrm{i}\eta r}\sum _{i=0}^n{A}_i(x^{\mathrm{i}\eta })(\delta +m{)}^{i}u=xM(x^{\mathrm{i}\eta },x,u,\delta u,\dots ,{\delta }^{n}u),$$

$$\psi =\sum _{k=1}^{\infty }{c}_k(x^{\mathrm{i}\eta r})x^k,c_k(t)={\alpha }_{k+m}(t)=t^{-{\nu }_k}\sum _{\ell =0}^{\infty }{c}_{k\ell }{t}^{\ell },$$

$$\sum_{i=0}^{n}A_{i}(t)\Bigl{(}k+m+{\rm i}\,\eta\,t\frac{d}{dt}\Bigr{)}^{i}$$

$$L(\delta )u:=\sum _{i=0}^n{A}_i(0)(\delta +m{)}^{i}u,$$

$$x^{\mathrm{i}\eta }H(x^{\mathrm{i}\eta },u,\delta u\dots ,{\delta }^{n}u):=L(\delta )u-\sum _{i=0}^n{A}_i(x^{\mathrm{i}\eta })(\delta +m{)}^{i}u,$$

$$x^{\mathrm{i}\eta r}L(\delta )u=x^{\mathrm{i}\eta (r+1)}H(x^{\mathrm{i}\eta },u,\delta u\dots ,{\delta }^{n}u)+xM(x^{\mathrm{i}\eta },x,u,\delta u,\dots ,{\delta }^{n}u).$$

$$t^r\sigma v=t^{r+1}\tilde{H}(t,v,v,\dots ,v)+x\tilde{M}(t,x,v,v,\dots ,v),\sigma >0,$$

$$\tilde{\psi }=\sum _{k=1}^{\infty }{C}_k(t)x^k,C_k(t)=t^{-{\nu }_k}\sum _{\ell =0}^{\infty }{C}_{k\ell }{t}^{\ell },C_{k\ell }⩾0,$$

$$H(t,u_0,\dots ,u_n)$$

$$M(t,x,u_0,\dots ,u_n)$$

$$\tilde{H}(t,v,\dots ,v)$$

$$\tilde{M}(t,x,v,\dots ,v)$$

$$\sigma =\underset{\genfrac{}{}{0ex}{}{k\in \mathbb{N}}{\ell \in {\mathbb{Z}}_{+}}}{\inf }|L(k+m+(\ell -kr)\mathrm{i}\eta )|.$$

$$v=\frac{t}{\sigma }\tilde{H}(t,v,\dots ,v)+\frac{x}{\sigma t^r}\tilde{M}(t,x,v,\dots ,v).$$

$$\tilde{\mathcal{D}}=\{0<\tau <|t|<\epsilon ,|x|<\rho ,|v|<{\epsilon }_n\},$$

$$\mathcal{M}=\sup\limits_{\widetilde{\mathcal{D}}}\Bigl{|}\frac{t}{\sigma}\widetilde{H}(t,v,\dots,v)+\frac{x}{\sigma t^{r}}\,\widetilde{M}(t,x,v,\dots,v)\Bigr{|}<+\infty.$$

$$\frac{t_0}{\sigma }\sum _{i=0}^n\sum _{s=0}^{\infty }|h_{s,i}|t_0^s\ne 1,$$

$$\mathcal{V}_{t_{0}}=\Bigl{\{}|x|<\rho\Bigl{(}\frac{\varepsilon_{n}}{\varepsilon_{n}+2\mathcal{M}_{t_{0}}}\Bigr{)}^{2}\Bigr{\}},\qquad\mathcal{M}_{t_{0}}\leqslant\mathcal{M},$$

$$\mathcal{V}=\Bigl{\{}0<\tau<|t|<\tau^{\prime},\;|x|<\rho\Bigl{(}\frac{\varepsilon_{n}}{\varepsilon_{n}+2\mathcal{M}}\Bigr{)}^{2}\Bigr{\}},$$

Full text

On the convergence of an exotic formal series solution of an ODE

R. R. Gontsov111Institute for Information Transmission Problems of RAS, Bolshoy Karetny per. 19, build.1, Moscow 127051 Russia, gontsovrr@gmail.com., I. V. Goryuchkina222Keldysh Institite of Applied Mathematics of RAS, Miusskaya sq. 4, Moscow 125047 Russia, igoryuchkina@gmail.com.

We consider a non-linear ordinary differential equation (ODE)

[TABLE]

where $\delta=x(d/dx)$ and $F(x,y_{0},y_{1},\ldots,y_{n})$ is a holomorphic function in a small polydisc

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Suppose that the equation (1) possesses a formal solution $y=\varphi,$

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where $\alpha_{k}(t)$ are meromorphic functions at the origin:

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with some common punctured disc ${\cal D}=\{0<|t|<\varepsilon\}$ of convergence.

The series (2) will be called exotic, in the terminology of A. D. Bruno [1]. In particular, the Painlevé III, V, VI equations possess formal solutions of such type [2], [3], [8]. Thus, our present goal is to obtain some general condition for the convergence of an exotic formal series solution of the equation (1). In this sense, our work continues a series of articles where similar questions were studied for generalized formal power series [4], [6] and formal Dulac series [7], which were inspired by the original paper of B. Malgrange [9] on the classical formal power series solutions of a non-linear ODE.

Theorem 1. Let $(\ref{eq2})$ be a formal solution of the equation $(\ref{eq1}):$

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such that $\displaystyle\frac{\partial F}{\partial y_{n}}(x,\Phi)\not\equiv 0$. Furthermore, let each exotic formal series $\displaystyle\frac{\partial F}{\partial y_{i}}(x,\Phi)$ be of the form

[TABLE]

with the same $N$ for all $i$, $B_{n}\not\equiv 0$ and

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Then the series $(\ref{eq2})$ converges uniformly in some open sector $S\subset{\mathbb{C}}\setminus{\mathbb{R}}_{+}$ with the vertex at the origin and of sufficiently small radius.

We give a sketch of the proof of Theorem 1 in a series of lemmas. The first one is on the reduction of the initial equation to a special form, with the proof in a spirit of Malgrange’s proof [9] of the corresponding lemma for a classical formal power series solution.

Lemma 1. Under the assumptions of Theorem 1, there exists $m^{\prime}\in\mathbb{N}$ such that for any $m\geqslant m^{\prime}$, the transformation

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reduces the equation (1) to an equation of the form

[TABLE]

where $r\in{\mathbb{Z}}_{+}$, $A_{i}(t)$ are holomorphic near $t=0$, $A_{n}(0)\neq 0,\;$ $M(x,t,u_{0},\dots,u_{n})$ is holomorphic near $0\in{\mathbb{C}}^{n+3}$.

The reduced equation (4) possesses an exotic formal series solution

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moreover, this is its unique formal solution of such a form, since the integer $m$ can be chosen in such a way that for any $k\geqslant 1$, a Fuchsian linear differential operator (near $t=0$)

[TABLE]

has no integer exponents. To prove Theorem 1, one should prove the convergence of the series $\psi$.

Let us denote

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[TABLE]

and write down the equation (4) in the form

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First, by the equation (6) we construct an algebraic equation

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majorant for the equation (6) in the sense that it has a unique formal solution of the form

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and $|c_{k\ell}|\leqslant C_{k\ell}$ for all $k,\ell$. Let us briefly describe this construction.

The functions $H,M$ of the right hand side in (6) are represented by power series convergent in $\mathcal{D}\times\Delta$:

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Then the functions $\widetilde{H},\widetilde{M}$ in the equation (7) are defined as follows:

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The number $\sigma$ is defined by the formula

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This value is strictly positive, as $L(k+m+(\ell-kr){\rm i\,}\eta)\neq 0\;$ for $\;{k\in\mathbb{N}},$ $\;{\ell\in\mathbb{Z}_{+}}$.

Lemma 2. The equation (7) has a uniquely determined formal solution of the form (8), which is majorant for the exotic formal series solution (5) of the equation (6): $|c_{kl}|\leqslant C_{kl}$.

The proof of this lemma is somehow similar to that of Lemma 2 from [5], where a proof of Malgrange’s theorem for a classical power series by the majorant method was proposed.

The second step is to prove the convergence of the series (8). Let us write down the equation (7) in the form

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Its right hand side is holomorphic in the domain

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and we have

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Further we look at the equation (9) as at an equation in two variables $x,$ $v$, with a nonzero parameter $t$. For each fixed value $t=t_{0}$, $\tau<|t_{0}|<\varepsilon$, such that

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we can apply the implicit function theorem to the equation (9). According to this theorem, the equation has a unique holomorphic solution $v=v(x,t_{0})$ in some open disc

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whose power series should coincide with the formal series (8), for $t=t_{0}$. Hence, the latter converges uniformly in some open domain

[TABLE]

which concludes the proof of Theorem 1.