Analytical Aspects of the Painlevé Test

Jishan Hu ^{∗}  Min Yan ^{†}

July 10, 2000

Contents

1 Introduction 2

2 A Mathematical Foundation for the Painlevé Test 5 2.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Balance and Kowalevskain Matrix . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Resonance and Principal Balance . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Resonance in Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . 12

3 The Mirror Transform for ODE Systems 15 3.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The Essence of the Mirror Transform . . . . . . . . . . . . . . . . . . . . . 21 3.3 Localness and Local Invertibility of the Mirror Transform . . . . . . . . . . 24

4 The Mirror Transform for Other Equations 25 4.1 For Single, High-Order ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Equations with Rational Right Sides . . . . . . . . . . . . . . . . . . . . . 27 4.3 For Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.4 On Non-autonomous Systems . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 For PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

References 40

∗Address: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email: majhu@ust.hk. Fax: (852)2358-1643.

†Address: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email: mamyan@ust.hk. Fax: (852)2358-1643.

1 Introduction

Since Kowalevskaya’s monumental work [11] more than a century ago, the Painlevé test has been the most widely used and the most successful technique for detecting integrable differential equations [1] [2] [14]. The test has been applied to many differential equations and, for those passing the Painlevé test, the indicators of integrability such as the existence of enough conservation laws, the Lax pair, the Darboux transform, the Bäcklund transform can always be found. However, despite the overwhelming evidence that the integrability of a differential equation should be closely related to the behavior of its solutions near movable singularities, the rigorous study of such a relation has been lacking. We feel the immediate difficulty for such a rigorous study is the fact that the Painlevé test exploits only the formal algebraic consequence of such a relation and fails to provide much analytical information. Thus the reason behind the test appears to be heuristic at best.

In this paper, we demonstrate that the Painlevé test actually provides some analytical information about the existence of a transform that regularizes the solutions at the movable singularities and still converts the equation to a regular one. We believe this is the key analytical ingredient that can help us to provide a rigorous explanation for why passing the Painlevé test often leads to many phenomena associated with integrability. Such an investigation is a necessary step in understanding the true nature of integrability.

To explain our main results, let us consider the first Painlevé equation: y′′ = 6y^{2} + x (the notation is the same as the one used by Painlevé in [13]). The equation passes the Painlevé test, with the following formal Laurent series solution:

y = \frac{1}{(x − x_{0})^{2}} − \frac{x_{0}}{10}(x − x_{0})^{2} − \frac{1}{6}(x − x_{0})^{3} + r(x − x_{0})^{4} + \frac{x_{0}^{2}}{18}(x − x_{0})^{6} + ⋯, (1)

where x_{0} and r can be any number. In his proof that all solutions of the equation are meromorphic, Painlevé introduced the change of variable

\{ \begin{matrix}y & = & \frac{1}{z^{2}} , \\ y′ & = & −\frac{2}{z^{3}} − \frac{xz}{2} − \frac{z^{2}}{2} + uz^{3}, \end{matrix}  (2)

which converts the first Painlevé equation into the regular system

\{ \begin{matrix}z′ & = & 1 + \frac{xz^{4}}{4} + \frac{z^{5}}{4} − \frac{uz^{6}}{2} , \\ u′ & = & \frac{x^{2}z}{8} + \frac{3xz^{2}}{8} + (\frac{1}{4} − xu) z^{3} − \frac{5uz^{4}}{4} − \frac{3u^{2}z^{5}}{2} . \end{matrix}  (3)

Moreover, the formal Laurent series solution (1) of the original equation is converted to the formal power series solution of the new system (3) with the initial data

z(x_{0}) = 0, u(x_{0}) = 7r.

An immediate consequence is the following: by applying the Cauchy theorem to the initial value problem of the new system, we find that the power series for z and u are convergent and analytically depend on x_{0} and r. Consequently, the formal Laurent series (1) is also convergent and analytically dependent on x_{0} and r. In other words, what the Painlevé test produces is a realistic solution instead of merely a formal one.

It turns out that what Painlevé created one hundred years ago is actually a general property for differential equations passing the Painlevé test. In [7], we presented an algorithm for constructing the transform like (2) for single, high-order (ordinary as well as partial) differential equations. The algorithm was extended to systems of differential equations in [6]. We further demonstrated in [8] that our algorithm is compatible with the Hamiltonian systems. The purpose of this paper is to justify rigorously and theoretically what we accomplished and observed in these papers.

We will call the transform constructed by our algorithm the mirror transform; we will call the new system obtained by applying the mirror transform the mirror system.

In order to develop our theory, we need to define rigorously some basic concepts in the Painlevé test. This is done in Section 2, with the terminologies borrowed from [4]. Our definitions are by no means the most general ones. They mainly serve as a reasonable working platform. As a matter of fact, we will also discuss some possible extensions in Sections 3.2 and 4.2.

Section 3 contains the primary ideas and results of this paper. For more concrete examples for the section, please see [6]. Theorem 7 in Section 3.1 states that if the right side of a system of ODEs is polynomial of dependent variables, then for any principal balance (in the sense of Definition 3), we can find a transform in a certain special shape, such that the system of ODEs is converted to another regular system of ODEs and such that the Laurent series in the principal balance is converted to a power series. In Section 3.2, we explain that the mirror transform is essentially a concept for (principal) collections of Laurent series. The polynomial condition in Theorem 7 is merely a convenient assumption to make it easy to prove the regularity of the mirror system. Thus, Theorem 7 can be generalized to many systems with non-polynomial functions on the right side. In Section 3.3, we discuss the localness of the mirror transform and the circumstances when the mirror transform is locally one-to-one.

In Section 4, we discuss the mirror transforms for other types of differential equations. Theorem 8 in Section 4.1 is a version of Theorem 7 for single, high-order ODEs. Instead of blindly translating the general results for the systems, we modify the conditions to be more suitable for single, high-order ODEs. We further explain in Section 4.2 how to extend the result to the case of rational functions on the right-side of the equation. Examples for Theorem 8 can be found in [7].

Theorem 9 in Section 4.3 indicates that it is possible to make the mirror transform canonical for Hamiltonian systems (at least for those that are almost weighted homogeneous). In addition to the example in Section 2.1, more examples can be found in [8]. Proposition 10 in Section 4.4 presents a rigorous explanation of non-autonomous Hamiltonian functions under the mirror transform, an observation we made in Section 4 of [8].

Finally, in Section 4.5, we explain how the mirror transform can be extended to PDEs. Instead of stating a general theorem, we choose to present the key ideas through the example of a single, third-order PDE. More concrete examples can be found in [6] and [7]. We would like to point out that our results on PDEs are less comprehensive than on ODEs because of the following feature of the mirror system: although we can still find the mirror transform and the mirror system is still regular, the mirror system is often not in the form to which one can directly apply the Cauchy-Kowalevski theorem (however, see Section 6 of [7] for an alternative approach). Moreover, the examples in Section 5 of [8] indicate that the compatibility between the mirror transform and infinite dimensional Hamiltonian systems is also problematic (we believe a more fundamental study of the change of the symplectic structure under a singular transform is necessary in order to understand this).

An immediate consequence of the existence of the mirror transform, at least for ODEs, is that the Laurent series in any principal balance is convergent and analytically dependent on the location of the movable singularity and the free parameters in the coefficients of the Laurent series. However, the mirror transform is more useful than merely the convergence of the Laurent series. In fact, Painlevé made substantial use of the mirror transform in his subsequent proof that all solutions of the equation y′′ = 6y^{2} + x are meromorphic on the whole complex plane. The method was further extended by Hinkkanen and Laine to the second Painlevé equation [5]. In [9], we combine the mirror transform and the idea from [10] to give an elementary and direct proof that all solutions of the first, second, and fourth Painlevé equations are meromorphic on the whole complex plane. We hope further research in this direction can lead to a unified and elementary proof for certain classes of integrable equations. Such a proof should give us some insight into the relation between the behavior at singularity and the integrability.

The mirror transform has another useful application in the completion of the phase space, especially for the study of algebraically completely integrable systems [3] [4]. In such completions, a complete phase space is constructed by adding all possible movable singularities to the usual phase space through certain patching maps. The Laurent series in the balances was interpreted as the patching map in [3]. More refined and often canonical maps were found (in an ad hoc way) and used as the patching maps for several examples in [4]. Our mirror transform provides a more systematic and theoretical way to construct the patching maps in [4]. So far, we have considered only the theory of canonical patching maps for principal balances. Further research is needed in order to cover the so-called lower balances, which may occur in algebraic complete integrable systems.

The mirror transform is significant for the following philosophical reason. The integrability of differential equations is actually the problem of globally characterizing the solutions. The Cauchy theorem locally characterizes the solutions near regular points. The existence of the mirror transform simply means that, with an appropriate change of coordinates in the phase space, the solutions can also be characterized near movable pole singularities. Therefore, for a differential equation, if the only movable singularities are poles and if the equation passes the Painlevé test (which implies the existence of the mirror transform), then we can patch the local integrations at the regular points with the pole singularities to have the global characterization of the solutions. In other words, the original differential equation, which governs the behaviors of the solution at regular points, and the mirror system, which governs the behaviors of the solution at pole singularities, together form a pair that ultimately makes the equation integrable. As a linkage between this pair, the mirror transform carries rich information about the movable singularities and should be a useful tool in the study of integrable equations.

2 A Mathematical Foundation for the Painlevé Test

2.1 An Example

In order to provide a better explanation of the basic concepts in the Painlevé analysis, we provide a concrete example in this section. More examples can be found in [6], [7] and [8].

The Gelfand-Dikii hierarchy with 2 degrees of freedom can be written as the following Hamiltonian system

\{ \begin{matrix}q′_{1} & = & −2p_{2}, \\ q′_{2} & = & −2q_{1}p_{2} − 2p_{1}, \\ p′_{1} & = & p_{2}^{2} − 6q_{1}q_{2} + 4q_{1}^{3}, \\ p′_{2} & = & −3q_{1}^{2} + 2q_{2} \end{matrix}  (4)

given by the Hamiltonian function

H = −q_{1}p_{2}^{2} − 2p_{2}p_{1} + 3q_{2}q_{1}^{2} − q_{1}^{4} − q_{2}^{2}.

The system passes the Painlevé test, with one of the principal balances as follows:

\{ \begin{matrix}q_{1} & = & t^{−2} + \frac{1}{3}r_{2} − \frac{1}{3}r_{2}^{2}t^{2} − \frac{2}{3}r_{3}t^{3} − \frac{10}{27}r_{2}^{3}t^{4} − \frac{1}{3}r_{2}r_{3}t^{5} − \frac{1}{3}r_{4}t^{6} + ⋯, \\ q_{2} & = & r_{2}t^{−2} − \frac{2}{3}r_{2}^{2} − r_{3}t − \frac{1}{3}r_{2}^{3}t^{2} + (−\frac{11}{54}r_{2}^{4} + \frac{3}{2}r_{4}) t^{4} + ⋯, \\ p_{1} & = & −t^{−5} + \frac{2}{3}r_{2}t^{−3} + \frac{1}{6}r_{3} − \frac{4}{27}r_{2}^{3}t − \frac{5}{6}r_{2}r_{3}t^{2} + (\frac{22}{81}r_{2}^{4} − \frac{11}{3}r_{4}) t^{3} + ⋯, \\ p_{2} & = & t^{−3} + \frac{1}{3}r_{2}^{2}t + r_{3}t^{2} + \frac{20}{27}r_{2}^{3}t^{3} + \frac{5}{6}r_{2}r_{3}t^{4} + r_{4}t^{5} + ⋯, \end{matrix}  (5)

where we abbreviate (t − t_{0}) as t because the system is autonomous. Regarding this balance, we introduce the indicial normalization q_{1} = θ^{−2} and find the following Laurent series in θ:

\{ \begin{matrix}θ′ & = & 1 + r_{2}θ^{2} + \frac{7}{2}θ^{4} + r_{3}θ^{5} − \frac{23}{2}r_{2}^{3}θ^{6} − 2r_{3}r_{3}θ^{7} + r_{4}θ^{8} + ⋯, \\ q_{2} & = & −2r_{2}θ^{−2} − 4r_{2}^{2} − \frac{1}{2}r_{3}θ − \frac{1}{2}r_{2}r_{3}θ^{3} + (−\frac{43}{8}r_{2}^{4} + r_{4}) θ^{4} + ⋯, \\ p_{1} & = & −θ^{−5} − 3r_{2}θ^{−3} − \frac{11}{2}r_{2}^{2}θ^{−1} − \frac{3}{4}r_{3} + \frac{9}{2}r_{2}^{3}θ + r_{2}r_{3}θ^{2} + (\frac{135}{4}r_{2}^{4} − 3r_{4}) θ^{3} + ⋯, \\ p_{2} & = & θ^{−3} + r_{2}θ^{−1} + \frac{7}{2}r_{2}^{2}θ + r_{3}θ^{2} − \frac{23}{2}r_{2}^{3}θ^{3} − 2r_{2}r_{3}θ^{4} + r_{4}θ^{5} + · · · . \end{matrix}

This leads to the introduction of new variables ξ_{2}, ξ_{3}, and ξ_{4} by successively truncating q_{2}, p_{2}, and p_{1} at the resonances

\{ \begin{matrix}q_{2} & = & ξ_{2}θ^{−2}, \\ p_{1} & = & −θ^{−5} + \frac{3}{2}ξ_{2}θ^{−3} + \frac{1}{8}ξ_{2}^{2}θ^{−1} − \frac{5}{16}ξ_{2}^{3}θ − ξ_{2}ξ_{3}θ^{2} + ξ_{4}θ^{3}, \\ p_{2} & = & θ^{−3} − \frac{1}{2}ξ_{2}θ^{−1} + \frac{3}{8}ξ_{2}^{2}θ + ξ_{3}θ^{2}. \end{matrix}

Then, the new variables θ, ξ_{2}, ξ_{3}, and ξ_{4} satisfy a regular equation.

It is easy to see that

dq_{1} ∧ dp_{1} + dq_{2} ∧ dp_{2} = −2dθ ∧ dξ_{4} + dξ_{2} ∧ dξ_{3}.

Therefore, if we denote Q_{1} = θ, Q_{2} = ξ_{2}, P_{1} = −2ξ_{4}, P_{2} = ξ_{3}, then the following transform

\{ \begin{matrix}q_{1} & = & Q_{1}^{−2}, \\ q_{2} & = & Q_{1}^{−2}Q_{2}, \\ p_{1} & = & −Q_{1}^{−5} + \frac{3}{2}Q_{1}^{−3}Q_{2} + \frac{1}{8}Q_{1}^{−1}Q_{2}^{2} − \frac{5}{16}Q_{1}Q_{2}^{3} − Q_{1}^{2}Q_{2}P_{2} − \frac{1}{2}Q_{1}^{3}P_{1}, \\ p_{2} & = & Q_{1}^{−3} − \frac{1}{2}Q_{1}^{−1}Q_{2} + \frac{3}{8}Q_{1}Q_{2}^{2} + Q_{1}^{2}P_{2} \end{matrix}  (6)

preserves the symplectic form. Moreover, the transform converts the Hamiltonian system (4) into a Hamiltonian system given again by a polynomial

H̄(Q_{1}, Q_{2}, P_{1}, P_{2}) = H(q_{1}, p_{1}, q_{2}, p_{2})
= P_{1} − \frac{35}{64}Q_{2}^{4} − 2Q_{1}Q_{2}^{2}P_{2} + Q_{1}^{2} (−P_{2}^{2} − \frac{1}{2}Q_{2}P_{1} + \frac{15}{64}Q_{2}^{5})

+\frac{11}{8}Q_{1}^{3}Q_{2}^{3}P_{2} + Q_{1}^{4} (2Q_{2}P_{2}^{2} + \frac{3}{8}Q_{2}^{2}P_{1}) + Q_{1}^{5}P_{1}P_{2}.

2.2 Balance and Kowalevskain Matrix

Consider an ODE system

u′ = F (t, u), u = (u_{1}, ⋯, u_{n}), F = (F_{1}, ⋯, F_{n}),  (7)

where each F_{i} is a polynomial in u and analytic in t.

Definition 1 A balance for system (7) is the formal Laurent series solution

u_{i} = c_{i}(t − t_{0})^{−g_{i}} + u_{i,1}(t − t_{0})^{1−g_{i}} + ⋯ + u_{i,j}(t − t_{0})^{j−g_{i}} + ⋯, i = 1, ⋯, n (8)

such that 1. the singularity t_{0} can be any point;

2. the coefficient functions c_{i} and u_{i,j} are analytic;

3. the leading exponents g_{i} are integers; and

4. the vector

(g_{1}c_{1}, ⋯, g_{n}c_{n}) ≠ 0.  (9)

By formal solution, we mean that if we substitute the series into the equation and expand both sides by powers of (t − t_{0}), then, for each power p, the coefficients of (t − t_{0})^{p} on both sides are the same.

We emphasize that some leading coefficients, c_{i}, are allowed to vanish. In other words, some leading exponents do not have to be the natural ones. Such relaxation in our requirement is necessary for including many interesting examples.

For example, (5) is a principal balance of system (4), with the leading coefficients 1, r_{2}, −1, 1, and the leading exponents 2, 2, 5, 3.

We may rewrite the balance (8) in vector form:

( u_{1}(t − t_{0})^{g_{1}} ; ⋮ ; u_{n}(t − t_{0})^{g_{n}} ) = ( c_{1} ; ⋮ ; c_{n} ) + ( u_{1,1} ; ⋮ ; u_{n,1} ) (t − t_{0}) + ⋯ + ( u_{1,j} ; ⋮ ; u_{n,j} ) (t − t_{0})^{j} + · · · .

Then, c_{∗} = (c_{1}, ⋯, c_{n}) is the leading coefficient vector, and u_{∗,j} = (u_{1,j}, ⋯, u_{n,j}) is the jth coefficient vector. In the usual formulation of the Painlevé test, the coefficient vectors are supposed to be related by specific recursive relations involving a certain so-called Kowalevskian matrix. To make this into a rigorous case, we introduce the following definition (the g_{∗}-weighted degree of polynomials in u_{1}, ⋯, u_{n} are obtained by thinking of the variable u_{i} to be of degree g_{i}).

Definition 2 We call the exponents g_{1}, ⋯, g_{n} Fuchsian for the ODE system (7) if the g_{∗}-weighted degree of F_{i} is ≤ g_{i} + 1.

Note that if we choose the natural leading exponents 2, 2, 5, 3 for the balance (5), then the Fuchsian condition is not satisfied. To fit the example into the Fuchsian framework, we consider

q_{2} = 0t^{−4} + 0t^{−3} + r_{2}t^{−2} − \frac{2}{3}r_{2}^{2} − r_{3}t − \frac{1}{3}r_{2}^{3}t^{2} + (−\frac{11}{54}r_{2}^{4} + \frac{3}{2}r_{4}) t^{4} + ⋯  (10)

as the Laurent series of q_{2}. Then (5) is a principal balance, with leading coefficients 1, 0, −1, 1, leading exponents 2, 4, 5, 3, and satisfying the Fuchsian condition.

Under the Fuchsian assumption, we denote

Fi^{D} = ∑{terms in F_{i} with g_{∗}-weighted degree = g_{i} + 1};
Fi^{E} = ∑{terms in F_{i} with g_{∗}-weighted degree < g_{i} + 1}.

Then, F_{i} = Fi^{D} +Fi^{E}. We substitute the balance into u′_{i} = Fi^{D}(t, u)+Fi^{E}(t, u) and compare the coefficients of (t − t_{0})^{−g_{i}−1} on both sides to get

−g_{i}c_{i} = Fi^{D}(t_{0}, c_{1}, ⋯, c_{n}),  i = 1, ⋯, n.  (11)

This is the (formal) dominant balance for the ODE system (7) at the (movable) singularity t_{0}. For j > 0, we further compare the coefficients of (t − t_{0})^{j−g_{i}−1} to get

(j − g_{i})u_{i,j} = \frac{∂Fi^{D}}{∂u_{1}} (t_{0}, c_{1}, ⋯, c_{n})u_{1,j} + ⋯ + \frac{∂Fi^{D}}{∂u_{n}} (t_{0}, c_{1}, ⋯, c_{n})u_{n,j}
+[an expression involving c_{∗}, u_{∗,1}, ⋯, u_{∗,j−1}].

In other words, if we define the Kowalevskian matrix as

K = ( \begin{matrix}\frac{∂F1^{D}}{∂u_{1}} + g_{1} & \frac{∂F1^{D}}{∂u_{2}} & ⋯ & \frac{∂F1^{D}}{∂u_{n}} \\ \frac{∂F2^{D}}{∂u_{1}} & \frac{∂F2^{D}}{∂u_{2}} + g_{2} & ⋯ & \frac{∂F2^{D}}{∂u_{n}} \\ ⋮ & ⋮ & & ⋮ \\ \frac{∂Fn^{D}}{∂u_{1}} & \frac{∂Fn^{D}}{∂u_{2}} & ⋯ & \frac{∂Fn^{D}}{∂u_{n}} + g_{n} \end{matrix} )

= \frac{∂F^{D}}{∂u} (t_{0}, c_{1}, ⋯, c_{n}) + ( \begin{matrix}g_{1} \\ & . & . & . \\ & & & & g_{n} \end{matrix} )

= \frac{∂F^{D}}{∂u} + Γ,

then the coefficient vector u_{∗,j} is given by the recursive relation

(K − jI)u_{∗,j} = [an expression involving c_{∗}, u_{∗,1}, ⋯, u_{∗,j−1}].  (12)

A formal balance is then equivalent to solutions of (11) and (12) for j = 1, 2, · · ·.

We emphasize that the Fuchsian condition is needed in the definition of the Kowalevskian matrix. In [4], Ercolani and Siggia made a similar observation regarding the balance (5). They suggested that one should not blindly proceed to use the linearized equation as the constraint on the resonance vectors and specifically pointed out that (10) should be introduced in order to define the Kowalevskian matrix for (4). In our viewpoint, this is exactly the pitfall of Lochak’s work [12], in which he discussed the linearized equation but failed to discuss rigorously the connection between the linearization and the Painlevé test. In the subsequent sections, we will provide a solid foundation for the connection.

In practice, choosing Fuchsian leading exponents of a balance is often guesswork. Next, we will describe a simple case in which the natural leading exponents are already Fuchsian. For the given leading exponents g_{1}, ⋯, g_{n} in a balance, denote

D_{i}(t, u_{1}, ⋯, u_{n}) = ∑{terms in F_{i} with highest g_{∗}-weighted degree},
E_{i}(t, u_{1}, ⋯, u_{n}) = F_{i} − D_{i}(t, u_{1}, ⋯, u_{n}).

Proposition 1 Suppose the following conditions are satisfied for all i = 1, ⋯, n:

1. D_{i}(t_{0}, c_{1}, ⋯, c_{n}) ≠ 0;

2. g_{i}c_{i} ≠ 0.

Then, the leading exponents are Fuchsian.

Proof: Let d_{i} be the g_{∗}-weighted degree of F_{i}. The condition D_{i}(t_{0}, c_{1}, ⋯, c_{n}) ≠ 0 implies that, after substituting in the balance, the lowest-order term in F_{i}(t, u_{1}, ⋯, u_{n}) with a nonzero coefficient is D_{i}(t_{0}, c_{1}, ⋯, c_{n})(t − t_{0})^{−d_{i}}. On the other hand, the lowest-order term in u′_{i} with a nonzero coefficient is −g_{i}c_{i}(t − t_{0})^{−g_{i}−1}. These two lowest-order terms must be balanced against each other. Therefore, we conclude that

g_{i} + 1 = d_{i}, −g_{i}c_{i} = D_{i}(t_{0}, c_{1}, ⋯, c_{n}).

The equality g_{i}+1 = d_{i} shows that the leading exponents g_{1}, ⋯, g_{n} are formally balanced.

□

Note that under the condition of the proposition, we clearly have

D_{i} = Fi^{D}, E_{i} = Fi^{E}.

Moreover, the condition D_{i}(t_{0}, c_{1}, ⋯, c_{n}) ≠ 0 means that, in substituting the leading terms in the balance, there is no “internal cancellation” among the highest g_{∗}-weighted terms in F_{i}. Because of this, the derivative must be involved in the dominant balance.

2.3 Resonance and Principal Balance

Because the solutions of (11) and (12) may not be unique, there may be some free parameters allowed in the balance in addition to the free choice of t_{0}. In this section, we describe the variation of the balance in the direction of these free parameters. There are three cases.

The first case is the free choice of t_{0} in the balance. Taking the derivative of the balance (8) in t_{0}, we find the variation is

\frac{∂u_{i}}{∂t_{0}} = −g_{i}c_{i}(t − t_{0})^{−g_{i}−1} + (1 − g_{i})u_{i,1}(t − t_{0})^{−g_{i}} + ⋯ + (j − g_{i})u_{i,j}(t − t_{0})^{j−g_{i}−1} + · · · .

Therefore, the direction of the variation is characterized by the vector (−g_{1}c_{1}, ⋯, −g_{n}c_{n}). To relate the vector to the Kowalevskian matrix, we take the derivative of the ODE system (7) in t and get

u′′_{i} = \frac{∂F_{i}}{∂t} + \frac{∂F_{i}}{∂u_{1}} u′_{1} + ⋯ + \frac{∂F_{i}}{∂u_{n}} u′_{n}.

We then substitute in the balance and compare the coefficients of (t − t_{0})^{−g_{i}−2} to get

(−g_{i})(−g_{i} − 1)c_{i} = \frac{∂Fi^{D}}{∂u_{1}} (t_{0}, c_{1}, ⋯, c_{n})(−g_{1})c_{1} + ⋯ + \frac{∂Fi^{D}}{∂u_{n}} (t_{0}, c_{1}, ⋯, c_{n})(−g_{n})c_{n}.

The equalities can be rewritten as (K + I)(−g_{1}c_{1}, ⋯, −g_{n}c_{n}) = 0.

Proposition 2 Suppose the leading exponents of a balance (8) are Fuchsian and (9) is satisfied. Then (−g_{1}c_{1}, ⋯, −g_{n}c_{n}) is an eigenvector of K with eigenvalue −1.

Because of the proposition, we call t_{0} a resonance parameter, with j = −1 as the corresponding resonance and (−g_{1}c_{1}, ⋯, −g_{n}c_{n}) as the corresponding resonance vector.

The second case is the free parameters in the leading coefficients, which are obtained by solving (11). The variations of the balance (8) in the direction of these parameters can be described by the tangents to the variety determined by (11). Thus, we take the derivative of the equations (11) with respect to one such free parameter ρ and get

−g_{i} \frac{dc_{i}}{dρ} = \frac{∂Fi^{D}}{∂u_{1}} \frac{dc_{1}}{dρ} + ⋯ + \frac{∂Fi^{D}}{∂u_{n}} \frac{dc_{n}}{dρ} .

The equalities can be rewritten as K \frac{dc_{∗}}{dρ} = 0.

Proposition 3 Suppose the leading exponents of a balance (8) are Fuchsian. Then the tangent vectors to the subvariety of allowable leading coefficients are the eigenvectors of K with eigenvalue 0.

Because of the proposition, we call a free parameter ρ in the leading coefficients a resonance parameter, with j = 0 as the corresponding resonance and the tangent vector

\frac{dc_{∗}}{dρ} as the corresponding resonance vector.

The third case is the free parameters in the subsequent coefficients, which are obtained by solving the recursive relations (12). One has to worry about the compatibility between K −jI and the right side of the recursive relation, which affects the existence of solutions. Leaving aside the existence issue for a moment, we may conclude the following from the recursive relation.

Proposition 4 Suppose the leading exponents of a balance (8) are Fuchsian. Then for j > 0, the j-th coefficient vectors form an affine space parallel to the eigenspace of K with eigenvalue j.

The affine space is trivial (i.e., of zero dimension) unless j is an eigenvalue of K. If j is indeed an eigenvalue, then as the solutions of (12), the j-th coefficient vectors are of the form

u_{∗,j} = U + ρ_{1}V_{1} + ⋯ + ρ_{s}V_{s},  (13)

where

1. U (analytically) depends only on c_{∗}, u_{∗,1}, ⋯, u_{∗,j−1};

2. V_{1}, ⋯, V_{s} is a basis of the eigenspace of K with eigenvalue j; and

3. ρ_{1}, ⋯, ρ_{s} can be any number.

We call ρ_{i} a resonance parameter, with j as the corresponding resonance and V_{i} as the corresponding resonance vector.

Definition 3 A balance (8) with Fuchsian leading exponents is a principal balance if for all allowable leading coefficients (i.e., solutions of (11)), the Kowalevskian matrix K satisfies the following conditions

1. K is diagonalizable;

2. Eigenvalues of K are −1 and non-negative numbers;

3. The eigenspace of −1 is of dimension 1,

and the number of resonance parameters, including t_{0}, is n.

Our main theorem is concerned with what happens in a neighborhood of a particular Laurent series solution in a principal balance (i.e., resonance parameters near a special choice). The local nature of the theorem makes it sufficient for us to give only a local interpretation of the conditions in the definition of the principal balance. The conditions on K imply that the eigenvalues and their multiplicities are constants, which we usually label in ascending order (with repetition to account for the multiplicity)

−1 = j_{1} < j_{2} ≤ j_{3} ≤ ⋯ ≤ j_{n}.

Since the rank of K is constant, it follows from (a modified version of) the implicit function theorem that the collection of the solutions of the dominant balance equation (11) is locally a manifold. Whatever parametrizes the manifold, it can be considered as the resonance parameters, with resonance j = 0 and the tangent vectors in the direction of the parameters as the resonance vectors. Moreover, we can find a matrix R(t_{0}, c_{∗}), analytically depending on its variables, such that

K = RΛR^{−1}, Λ = ( \begin{matrix}j_{1} & 0 & ⋯ & 0 \\ 0 & j_{2} & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & j_{n} \end{matrix} ) .  (14)

In other words, we may diagonalize K with the eigenvectors (i.e., the resonance vectors) chosen to be analytically dependent on t_{0} and the leading coefficients. We call the matrix R(t_{0}, c_{∗}) the resonance matrix. Finally, the condition on the number of resonance parameters implies that for any i and any solution c_{∗}, u_{∗,1}, ⋯, u_{∗,i−1} of (11) and (12) for 0 ≤ j < i, equation (12) for j = i is always consistent.

With (10), the principal balance (5) of system (4) has Fuchsian leading exponents. The balance then has resonance parameters t_{0}, r_{2}, r_{3}, r_{4}, with corresponding resonances −1, 2, 5, 8. The resonance matrix is

( \begin{matrix}2 & 1/3 & −2/3 & −1/3 \\ 0 & 1 & −1 & 3/2 \\ −5 & 2/3 & 1/6 & −11/3 \\ 3 & 0 & 1 & 1 \end{matrix} ) .  (15)

2.4 Resonance in Hamiltonian Systems

We consider a Hamiltonian system

u′ = J∇H,  ∇H = (\frac{∂H}{∂u_{1}} , ⋯, \frac{∂H}{∂u_{n}} ) ,  J = ( \begin{matrix}O & I_{m} \\ −I_{m} & O \end{matrix} )  (16)

in which n = 2m and H(t, u) is a polynomial in u and analytic in t. We denote

u = (q, p), \{ \begin{matrix}q = (q_{1}, ⋯, q_{m}), \\ p = (p_{1}, ⋯, p_{m}). \end{matrix}

Definition 4 The Hamiltonian system (16) is almost weighted homogeneous relative to the leading exponents g_{1}, ⋯, g_{n} of a balance (8) if

g_{i} + g_{i+m} = d − 1,

for all i = 1, ⋯, m, where

d = g_{∗}-weighted degree of H.

Denote h_{i} = g_{i+m} for i = 1, ⋯, m. Then under the almost weighted homogeneous

assumption, the g_{∗}-weighted degree of \frac{∂H}{∂p_{i}} is ≤ d−h_{i} = g_{i} +1, and the g_{∗}-weighted degree

of \frac{∂H}{∂q_{i}} is ≤ d − g_{i} = h_{i} + 1. Consequently, the leading exponents are Fuchsian. Moreover,

for

D = ∑{terms in H with g_{∗}-weighted degree = d},
E = ∑{terms in H with g_{∗}-weighted degree < d},

we have

H = D + E,  (J∇H)^{D} = J∇D,  (J∇H)^{E} = J∇E,

and the Kowalevskian matrix is

K = \frac{∂(J∇H)^{D}}{∂u} + Γ = JH_{D} + Γ,

where H_{D} is the Hessian of D. The special shape of the Kowalevskian matrix implies certain symplectic structures among the resonance vectors.

Proposition 5 Suppose the Hamiltonian system (16) is almost weighted homogeneous relative to the leading exponents of a balance. Suppose v and w are eigenvectors of the Kowalevskian matrix with eigenvalues j and k. If j + k ≠ d − 1, then ⟨v, Jw⟩ = 0.

Proof: From Kv = jv, Kw = kw, we have

(j + k)⟨v, Jw⟩ = ⟨Kv, Jw⟩ + ⟨v, JKw⟩
= −⟨J(JH_{H} + Γ)v, w⟩ + ⟨v, J(JH_{H} + Γ)w⟩
= ⟨(H_{H} − JΓ)u, v⟩ + ⟨v, (−H_{H} + JΓ)w⟩
= ⟨v, ΓJw⟩ + ⟨v, JΓw⟩
= (d − 1)⟨v, Jw⟩,

where the assumption g_{i} + h_{i} = d − 1 is used in (and only in) ΓJ + JΓ = d − 1.

□

Now suppose the balance is a principal one, with the resonances

−1 = j_{1} < j_{2} ≤ ⋯ ≤ j_{m} ≤ k_{m} ≤ k_{m−1} ≤ ⋯ ≤ k_{1}, k_{i} = j_{2m−i+1}.

Then K is diagonalizable, with the resonances as eigenvalues. Moreover, Proposition 5 implies that we can find

1. eigenvectors v_{1}, ⋯, v_{m} of K, with j_{1}, ⋯, j_{m} as eigenvalues;

2. eigenvectors w_{1}, ⋯, w_{m} of K, with k_{1}, ⋯, k_{m} as eigenvalues,

such that {v_{1}, ⋯, v_{m}, w_{1}, ⋯, w_{m}} is a symplectic basis of R^{n}, and the sum of the eigenvalues of v_{i} and w_{i} is

j_{i} + k_{i} = d − 1.  (17)

Note that v_{1} has to be a nonzero multiple of the vector

(−g_{1}c_{1}, −g_{2}c_{2}, ⋯, −g_{m}c_{m}, −h_{m}c_{n}, −h_{m−1}c_{n−1}, ⋯, −h_{1}c_{m+1}).

In fact, we can assume that v_{1} is identical to this vector, otherwise we can always rescale v_{1} and w_{1} in the opposite way to make it so.

In terms of the diagonalization (14) of K, the discussion gives us

K = S ( \begin{matrix}Λ′ & 0 \\ 0 & (d − 1)I − Λ′ \end{matrix} ) S^{−1}, Λ′ = ( \begin{matrix}j_{1} & 0 & ⋯ & 0 \\ 0 & j_{2} & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & j_{m} \end{matrix} ) ,

and S = [v_{1} ⋯ v_{m} w_{1} ⋯ w_{m}] is a symplectic matrix:

S^{T} JS = J.  (18)

As an example, we may rescale and reverse the last two columns of the resonance matrix (15) of the system (4) and get a symplectic matrix

( \begin{matrix}2 & 1/3 & 2/81 & −4/9 \\ 0 & 1 & −1/9 & −2/3 \\ −5 & 2/3 & 22/81 & 1/9 \\ 3 & 0 & −2/27 & 2/3 \end{matrix} ) .

For later use in the proof of Theorem 9, we discuss next the LU decomposition of the resonance matrix for Hamiltonian systems.

A lagrangian of C^{n} is a subspace L ⊂ C^{n} of dimension m, such that ⟨v_{1}, Jv_{2}⟩ = 0 for any v_{1}, v_{2} ∈ L. For any subset I ⊂ {1, ⋯, n}, we denote by π_{I} : C^{n} → C^{I} the projection onto the coordinates indexed by numbers in I. We also denote

I^{∗} = \{i^{∗} : i ∈ I\}, i^{∗} = \{ \begin{matrix}i + m, & for 1 ≤ i ≤ m, \\ i − m, & for m + 1 ≤ i ≤ n. \end{matrix}

The following is a basic result in symplectic linear algebra.

Proposition 6 For any lagrangian L ⊂ C^{n}, there is a subset I ⊂ {1, ⋯, n}, such that π_{I} : L → C^{I} is an isomorphism, and I ∩ I^{∗} = ∅.

By dividing S into four m × m matrices, the property (18) means

S = ( \begin{matrix}A & B \\ C & D \end{matrix} ) , A^{T} C = C^{T} A, B^{T} D = D^{T} B, A^{T} D − C^{T} B = I_{m}.  (19)

Applying Proposition 6 to the lagrangian spanned by v_{1}, ⋯, v_{m} (which are the columns

of \binom{A}{C}), we find the projection π_{I}. If, for any i + m ∈ I, 1 ≤ i ≤ m, we exchange q_{i}

with p_{i}, then correspondingly we exchange the i-th row and the (i + m)-th row of S. The fact that π_{I} is an isomorphism means that after some exchanges, the matrix A in the new (still symplectic) matrix S is invertible. Thus, up to exchanging some q_{i} with p_{i}, we may assume that the matrix A in (19) is already invertible.

Since A is invertible, we may permute some rows of A, so that A = LU for a lower triangular L and an upper triangular U. Now, if the i-th row of A is permuted to the i′-th row, then we also permute the i-th variable pair (q_{i}, p_{i}) to become the i′-th variable pair. This has a similar effect on the rows of S and the new matrix S is still symplectic. Thus, up to permuting the pairs (q_{i}, p_{i}), we may assume that the matrix A in (19) has an LU decomposition.

Now we assume that the exchanges and permutations of the variables are done, so that A in (19) is invertible and has an LU decomposition. Next, we consider the LU-

decomposition of the resonance matrix. Because the eigenvalues of the columns of \binom{B}{D} are decreasing, we get a resonance matrix only after reversing the last m columns of S. Moreover, for the proof of Theorem 9, we would like to rearrange further the order of the dependent variables to become q_{1}, ⋯, q_{m}, p_{m}, ⋯, p_{1}, which corresponds to reversing the last m rows of S. Thus, the resonance matrix under our consideration is

R = ( \begin{matrix}I_{m} & O \\ O & T \end{matrix} ) S ( \begin{matrix}I_{m} & O \\ O & T \end{matrix} ) , T = ( \begin{matrix}0 & ⋯ & 0 & 1 \\ 0 & ⋯ & 1 & 0 \\ ⋮ & & ⋮ & ⋮ \\ 1 & ⋯ & 0 & 0 \end{matrix} )_{m×m} .

Let A = LU be an LU decomposition. Then (19) implies that

R = ( \begin{matrix}L & O \\ T CA^{−1}L & T (L^{T})^{−1}T \end{matrix} ) ( \begin{matrix}U & L^{−1}BT \\ O & T (U^{T})^{−1}T \end{matrix} ) .

We note that the right side is an LU decomposition.

3 The Mirror Transform for ODE Systems

3.1 Main Theorem

Theorem 7 Suppose (8) is a principal balance of the ODE system (7), with Fuchsian leading exponents. Suppose t_{0}, r_{2}, r_{3}, ⋯, r_{n} are all the resonance parameters, with corresponding resonances −1 = j_{1} < j_{2} ≤ j_{3} ≤ ⋯ ≤ j_{n}. Then up to rearranging the order of the variables u_{1}, ⋯, u_{n}, there is a change of variables of the form

{ u_{1} = θ^{−g_{1}}, ; u_{2} = θ^{−g_{2}}(a_{2,0} + a_{2,1}θ + ⋯ + a_{2,j_{2}−1}θ^{j_{2}−1} + ξ_{2}θ^{j_{2}}), ; ⋮ ; u_{n} = θ^{−g_{n}}(a_{n,0} + a_{n,1}θ + ⋯ + a_{n,j_{n}−1}θ^{j_{n}−1} + ξ_{n}θ^{j_{n}}),

such that

1. a_{i,j} is an analytic function of t and ξ_{s} with j_{s} ≤ j;

2. the ODE system (7) is converted to a regular ODE system for the new variables (θ, ξ_{2}, ⋯, ξ_{n});

3. the principal balance is converted to a solution of the new ODE system with the

initial data

θ(t_{0}) = 0,

and

ξ_{i}(t_{0}) = b_{i,i}(t_{0}, c_{∗})r_{i} + ⋯ + b_{i,s}(t_{0}, c_{∗})r_{s} + b_{i}(t_{0}, c_{∗}, r_{∗}with resonances < j_{i}),

where b_{∗,∗}, b_{∗} are analytic, b_{i,i}(t_{0}, c_{∗}) ≠ 0, and r_{i}, r_{i+1}, ⋯, r_{s} are all the resonance parameters indexed ≥ i and with resonance j = j_{i}.

Proof: The proof begins with some formal manipulation of the principal balance (8), which gives rise to the new variables θ, ξ_{2}, ⋯, ξ_{n}.

The Laurent series (8) has functions of t_{0}, r_{2}, ⋯, r_{n} as its coefficients. We convert the coefficients into functions of t, r_{2}, ⋯, r_{n} by substituting t_{0} = t − (t − t_{0}) to get

u_{i} = c_{i}^{(1)}(t − t_{0})^{−g_{i}} + u_{i,1}^{(1)}(t − t_{0})^{1−g_{i}} + ⋯ + u_{i,j}^{(1)}(t − t_{0})^{j−g_{i}} + ⋯, i = 1, ⋯, n, (20)

where c_{i}^{(1)} is the same function as c_{i}, except that the variable t_{0} is changed to t. Moreover, it is easy to see that the j-th coefficient vector becomes

u_{∗,j}^{(1)} = U^{(1)} + ρ_{1}V_{1}^{(1)} + ⋯ + ρ_{s}V_{s}^{(1)},  (21)

where V_{1}^{(1)}, ⋯, V_{s}^{(1)} are the same as V_{1}, ⋯, V_{s} in (13), except that the variable t_{0} is changed to t. In particular, all the resonance vectors remain the same, except for the change from t_{0} to t. Denote the resonance matrix

R^{(1)} = (v_{∗,∗}^{(1)})_{n×n} = ( \begin{matrix}−g_{1}c_{1}^{(1)} & v_{1,2}^{(1)} & v_{1,3}^{(1)} & ⋯ \\ −g_{2}c_{2}^{(1)} & v_{2,2}^{(1)} & v_{2,3}^{(1)} & ⋯ \\ ⋮ & ⋮ & ⋮ \\ −g_{n}c_{n}^{(1)} & v_{n,2}^{(1)} & v_{n,3}^{(1)} & ⋯ \end{matrix} )

in which all the entries are functions of t and the leading coefficients c_{∗}.

Next, we choose a dependent variable, say u_{1}, such that g_{1}c_{1} ≠ 0. Then, we introduce the indicial normalization

u_{1} = θ^{−g_{1}}.

We fix some g_{1}-th root a_{0} of c_{1}^{(1)} and expect the leading behavior of θ to be a_{0}^{−1}(t − t_{0}). Therefore we postulate a power series

t − t_{0} = a_{0}θ + a_{1}θ^{2} + a_{2}θ^{3} + ⋯,  (22)

where a_{∗} are functions of t, r_{2}, ⋯, r_{n}. To determine the coefficients, we substitute (22) into (20) for i = 1 and get

θ^{−g_{1}} = c_{1}^{(1)}a_{0}^{−g_{1}}θ^{−g_{1}}(1 + a_{0}^{−1}a_{1}θ + a_{0}^{−1}a_{2}θ^{3} + ⋯)^{−g_{1}}
+u_{1,1}^{(1)}a_{0}^{1−g_{1}}θ^{1−g_{1}}(1 + a_{0}^{−1}a_{1}θ + a_{0}^{−1}a_{2}θ^{3} + ⋯)^{1−g_{1}}
+ ⋯
+u_{1,j}^{(1)}a_{0}^{j−g_{1}}θ^{j−g_{1}}(1 + a_{0}^{−1}a_{1}θ + a_{0}^{−1}a_{2}θ^{3} + ⋯)^{j−g_{1}}
+ · · · .

By expanding the right side into a Laurent θ-series and comparing the coefficients, we get a recursive relation for a_{∗}. Solving the recursive relation, we get

{ a_{0}^{g_{1}} = c_{1}^{(1)}, ; a_{1} = −(g_{1}c_{1}^{(1)})^{−1}a_{0}^{2}u_{1,1}^{(1)}, ; a_{2} = −(g_{1}c_{1}^{(1)})^{−1}a_{0}^{3}u_{1,2}^{(1)} + (g_{1}c_{1}^{(1)})^{−2}(1 − g_{1})a_{0}^{3}(u_{1,1}^{(1)})^{2}, ; ⋮ ; a_{j} = −(g_{1}c_{1}^{(1)})^{−1}a_{0}^{j+1}u_{1,j}^{(1)} + ⋯, ; ⋮  (23)

We substitute (22) into (20) for i = 2, ⋯, n and find the following θ-series

u_{i} = c_{i}^{(2)}θ^{−g_{i}} + u_{i,1}^{(2)}θ^{1−g_{i}} + ⋯ + u_{i,j}^{(2)}θ^{j−g_{i}} + ⋯, i = 2, ⋯, n,  (24)

where the coefficients are still functions of t, r_{2}, ⋯, r_{n}. The coefficients can be determined more precisely through the following computation

u_{i} = c_{i}^{(1)}a_{0}^{−g_{i}}θ^{−g_{i}}(1 + a_{0}^{−1}a_{1}θ + a_{0}^{−1}a_{2}θ^{2} + a_{0}^{−1}a_{3}θ^{3} + ⋯)^{−g_{i}}

+u_{i,1}^{(1)}a_{0}^{1−g_{i}}θ^{1−g_{i}}(1 + a_{0}^{−1}a_{1}θ + a_{0}^{−1}a_{2}θ^{2} + a_{0}^{−1}a_{3}θ^{3} + ⋯)^{1−g_{i}} + ⋯ +u_{i,j}^{(1)}a_{0}^{j−g_{i}}θ^{j−g_{i}}(1 + a_{0}^{−1}a_{1}θ + a_{0}^{−1}a_{2}θ^{2} + a_{0}^{−1}a_{3}θ^{3} + ⋯)^{j−g_{i}} + ⋯

= c_{i}^{(1)}a_{0}^{−g_{i}}θ^{−g_{i}}[1 − g_{i}a_{0}^{−1}a_{1}θ − (g_{i}a_{0}^{−1}a_{2} + ⋯)θ^{2} − (g_{i}a_{0}^{−1}a_{3} + ⋯)θ^{3} − ⋯]

+(u_{i,1}^{(1)}a_{0}^{1−g_{i}} + ⋯)θ^{1−g_{i}} + ⋯ + (u_{i,j}^{(1)}a_{0}^{j−g_{i}} + ⋯)θ^{j−g_{i}} + ⋯

= c_{i}^{(1)}a_{0}^{−g_{i}}θ^{−g_{i}} + (u_{i,1}^{(1)}a_{0}^{1−g_{i}} − g_{i}c_{i}^{(1)}a_{0}^{−g_{i}−1}a_{1} + ⋯)θ^{1−g_{i}} + ⋯

+(u_{i,j}^{(1)}a_{0}^{j−g_{i}} − g_{i}c_{i}^{(1)}a_{0}^{−g_{i}−1}a_{j} + ⋯)θ^{j−g_{i}} + ⋯

=^{(23)} c_{i}^{(1)}a_{0}^{−g_{i}}θ^{−g_{i}} + [u_{i,1}^{(1)} − \frac{g_{i}c_{i}^{(1)}}{g_{1}c_{1}^{(1)}} u_{1,1}^{(1)} + ⋯] a_{0}^{1−g_{i}}θ^{1−g_{i}} + ⋯

+ [u_{i,j}^{(1)} − \frac{g_{i}c_{i}^{(1)}}{g_{1}c_{1}^{(1)}} u_{1,j}^{(1)} + ⋯] a_{0}^{j−g_{i}}θ^{j−g_{i}} + · · · .

Therefore, we conclude that

c_{i}^{(2)} = c_{i}^{(1)}a_{0}^{−g_{i}} = c_{i}^{(1)}(c_{1}^{(1)})^{−g_{i}/g_{1}},  (25)

and

u_{i,j}^{(2)} = [u_{i,j}^{(1)} − \frac{g_{i}c_{i}^{(1)}}{g_{1}c_{1}^{(1)}} u_{1,j}^{(1)}] a_{0}^{j−g_{i}} + [terms involving c_{∗}^{(1)}, u_{∗,1}^{(1)}, ⋯, u_{∗,j−1}^{(1)}].  (26)

Formula (25) can be interpreted as a map (c_{1}^{(1)}, ⋯, c_{n}^{(1)}) → (c_{1}^{(2)}, ⋯, c_{n}^{(2)}). The Jacobian of the map is

J^{(1)} = ( \begin{matrix}−\frac{g_{2}c_{2}^{(1)}}{g_{1}c_{1}^{(1)}} a_{0}^{−g_{2}} & a_{0}^{−g_{2}} & 0 & ⋯ & 0 \\ −\frac{g_{3}c_{3}^{(1)}}{g_{1}c_{1}^{(1)}} a_{0}^{−g_{3}} & 0 & a_{0}^{−g_{3}} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ −\frac{g_{n}c_{n}^{(1)}}{g_{1}c_{1}^{(1)}} a_{0}^{−g_{n}} & 0 & 0 & ⋯ & a_{0}^{−g_{n}} \end{matrix} ) .

Thus, the tangent vectors to the variety of allowable new leading coefficient vectors (c_{2}^{(2)}, ⋯, c_{n}^{(2)}) are J^{(1)}v, with v being the tangent vectors to the variety of allowable old leading coefficient vectors (see Proposition 3). Moreover, comparing (21) and (26) gives us

u_{∗,j}^{(2)} = U^{(2)} + ρ_{1}V_{1}^{(2)} + ⋯ + ρ_{s}V_{s}^{(2)},  V_{k}^{(2)} = J^{(1)}V_{k}^{(1)}.  (27)

We may still talk about the resonance matrix R^{(2)} for (24), which characterizes the variations of the coefficients in the direction of the resonance parameters. From the discussion above, we see that R^{(2)} is obtained by multiplying J^{(1)} to the left of R^{(1)} and then taking the minor opposite the (1, 1)-entry. Equivalently, we use the entry v_{1,1}^{(1)} = −g_{1}c_{1}^{(1)} to eliminate the other terms in the first column of R^{(1)} and get

( \begin{matrix}−g_{1}c_{1}^{(1)} & v_{1,2}^{(1)} & v_{1,3}^{(1)} & ⋯ \\ 0 & v_{2,2}^{(1)} − \frac{g_{2}c_{2}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,2}^{(1)} & v_{2,3}^{(1)} − \frac{g_{2}c_{2}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,3}^{(1)} & ⋯ \\ ⋮ & ⋮ & ⋮ \\ 0 & v_{n,2}^{(1)} − \frac{g_{n}c_{n}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,2}^{(1)} & v_{n,3}^{(1)} − \frac{g_{n}c_{n}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,3}^{(1)} & ⋯ \end{matrix} ) .

Then we multiply a_{0}^{−g_{∗}} by the rows and a_{0}^{j_{∗}} by the columns, and the (n − 1) × (n − 1) minor opposite of v_{1,1}^{(1)} is the resonance matrix for (24)

R^{(2)} = (v_{∗,∗}^{(2)})_{(n−1)×(n−1)} = ( \begin{matrix}(v_{2,2}^{(1)} − \frac{g_{2}c_{2}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,2}^{(1)}) a_{0}^{j_{2}−g_{2}} & (v_{2,3} − \frac{g_{2}c_{2}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,3}^{(1)}) a_{0}^{j_{3}−g_{2}} & ⋯ \\ ⋮ & ⋮ \\ (v_{n,2}^{(1)} − \frac{g_{n}c_{n}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,2}) a_{0}^{j_{2}−g_{n}} & (v_{n,3}^{(1)} − \frac{g_{n}c_{n}^{(1)}}{g_{1}c_{1}^{(1)}} v_{1,3}) a_{0}^{j_{3}−g_{n}} & ⋯ \end{matrix} ) .

Next we assume v_{2,2}^{(2)} ≠ 0, which may be achieved by rearranging u_{2}, ⋯, u_{n}. We cut the θ-series of u_{2} at the first resonance parameter r_{2} by introducing ξ_{2}:

u_{2} = c_{2}^{(2)}θ^{−g_{2}} + u_{2,1}^{(2)}θ^{1−g_{2}} + ⋯ + u_{2,j_{2}−1}^{(2)}θ^{j_{2}−g_{2}−1} + ξ_{2}θ^{j_{2}−g_{2}}.

Then, we have the following θ-series for ξ_{2}:

ξ_{2} = u_{2,j_{2}}^{(2)} + u_{2,j_{2}+1}^{(2)}θ + u_{2,j_{2}+2}^{(2)}θ^{2} + · · · .  (28)

By the definition of the resonance vectors, we also have (compare with (27))

u_{2,j_{2}}^{(2)} = u_{2,2}^{(2)} + r_{2}v_{2,2}^{(2)} + r_{3}v_{2,3}^{(2)} + ⋯ + r_{s}v_{2,s}^{(2)},  (29)

where r_{2}, r_{3}, ⋯, r_{s} are all the resonance parameters with resonance j = j_{2}, and u_{2,2}^{(2)} depends only on t and c_{∗}. Note that in case j_{2} = 0, the formula (29) is true only up to the leading term. This means that we can find a θ-series for u_{2,j_{2}}^{(2)} with (29) as the leading term. But this modification does not essentially change the subsequent discussion.

By substituting (29) into (28), we get

ξ_{2} = (u_{2,2}^{(2)} + r_{2}v_{2,2}^{(2)} + r_{3}v_{2,3}^{(2)} + ⋯ + r_{s}v_{2,s}^{(2)}) + u_{2,j_{2}+1}^{(2)}θ + u_{2,j_{2}+2}^{(2)}θ^{2} + · · · .  (30)

Solving for r_{2}, we get a θ-series,

r_{2} = (ξ_{2} − u_{2,2}^{(2)} − r_{3}v_{2,3}^{(2)} − ⋯ − r_{s}v_{2,s}^{(2)})(v_{2,2}^{(2)})^{−1} + r_{2,1}θ + r_{2,2}θ^{2} + ⋯,

in which the coefficients are functions of t, ξ_{2}, r_{3}, ⋯, r_{n}. By substituting this into (24) for i = 3, ⋯, n, we get a new θ-series,

u_{i} = c_{i}^{(3)}θ^{−g_{i}} + u_{i,1}^{(3)}θ^{1−g_{i}} + ⋯ + u_{i,j}^{(3)}θ^{j−g_{i}} + ⋯, i = 3, ⋯, n,  (31)

with functions of t, ξ_{2}, r_{3}, ⋯, r_{n} as the coefficients.

We may analyze the coefficients in (31) in a way similar to the analysis of the coefficients of (24). The result is that the variations of the coefficients in the direction of the resonance parameters r_{3}, ⋯, r_{n} are characterized by another resonance matrix R^{(3)} for (31). Moreover, R^{(3)} is obtained in the following way. By using the first entry v_{2,2}^{(2)} to eliminate the other terms in the first column of R^{(2)}, we end up with a row operation

R^{(2)}  ⇒  ( \begin{matrix}v_{2,2}^{(2)} & ∗ \\ 0 & R^{(3)} \end{matrix} ) ,

in which R^{(3)} is the minor opposite of the first entry.

The analysis of the variations of the coefficients in (31) allows us, after making v_{3,3}^{(3)} ≠ 0 by rearranging u_{3}, ⋯, u_{n}, to introduce a new variable ξ_{3}:

u_{3} = c_{3}^{(3)}θ^{−g_{3}} + u_{3,1}^{(3)}θ^{1−g_{3}} + ⋯ + u_{3,j_{3}−1}^{(3)}θ^{j_{3}−g_{3}−1} + ξ_{3}θ^{j_{3}−g_{3}}.

Moreover, we have a θ-series

ξ_{3} = (u_{3,3}^{(3)} + r_{3}v_{3,3}^{(3)} + r_{4}v_{3,4}^{(3)} + ⋯ + r_{s}v_{3,s}^{(3)}) + u_{3,j_{3}+1}^{(3)}θ + u_{3,j_{3}+3}^{(3)}θ^{3} + ⋯

similar to (30), in which r_{3}, r_{4}, ⋯, r_{s} are all the resonance parameters with index ≥ 3 and resonance j = j_{3}. Moreover, u_{3,3}^{(3)} depends only on t and the resonance parameters with resonance < j_{3}.

Clearly, we have established a pattern. Up to rearranging u_{∗}, we have successive row operations

R^{(i)}  ⇒  ( \begin{matrix}v_{i,i}^{(i)} & ∗ \\ 0 & R^{(i+1)} \end{matrix} ) ,  i = 1, ⋯, n − 1.

Then, it is possible to cut successively u_{i+1} at r_{i+1} by introducing ξ_{i+1}. The end result is a transform given by u_{1} = θ^{−g_{1}} and

u_{i} = c_{i}^{(i)}θ^{−g_{i}} + u_{i,1}^{(i)}θ^{1−g_{i}} + ⋯ + u_{i,j_{i}−1}^{(i)}θ^{j_{i}−g_{i}−1} + ξ_{i}θ^{j_{i}−g_{i}}, i = 2, ⋯, n.

This is the transform described in the theorem. Moreover, we also have a θ-series

ξ_{i} = (u_{i,i}^{(i)} + r_{i}v_{i,i}^{(i)} + r_{i+1}v_{i,i+1}^{(i)} + ⋯ + r_{s}v_{i,s}^{(i)}) + u_{i,j_{i}+1}^{(i)}θ + u_{i,j_{i}+i}^{(i)}θ^{i} + ⋯, i = 2, ⋯, n, (32)

similar to (30), in which r_{i}, r_{i+1}, ⋯, r_{s} are all the resonance parameters with index ≥ i and resonance j = j_{i}. Moreover, u_{i,i}^{(i)} depends only on t and the resonance parameters with resonance < j_{i}.

Now let us find out how the Laurent series (8) is converted under the transform. By taking the (−g_{1})-th root of the Laurent series for u_{1} and assuming the leading behavior for θ to be a_{0}^{−1}(t − t_{0}), we get a power series for θ:

θ = a_{0}^{−1}(t − t_{0}) − \frac{u_{1,1}}{g_{1}a_{0}^{g_{1}+1}} (t − t_{0})^{2} + · · · .  (33)

Substituting this series into (32) and expanding the coefficients in (32) into a Taylor series at t_{0}, we get a power series for ξ_{i}:

ξ_{i} = (b_{i} + r_{i}b_{i,i} + r_{i+1}b_{i,i+1} + ⋯ + r_{s}b_{i,s}) + ξ_{i,1}(t − t_{0}) + ξ_{i,2}(t − t_{0})^{2} + ⋯,  (34)

where b_{i}, b_{i,i}, b_{i,i+1}, ⋯, b_{i,s} are the same as u_{i,i}^{(i)}, v_{i,i}^{(i)}, v_{i,i+1}^{(i)}, ⋯, v_{i,s}^{(i)}, except that the variable t is replaced by t_{0}. Thus, we conclude that the transform converts the Laurent series solution (8) into the power series solution (33) and (34). The power series solution is characterized by the initial condition described in the theorem.

Finally, we prove the regularity of the new system for θ, ξ_{2}, ⋯, ξ_{n}. From the shape of the transform, the new system must be of the form (ξ_{1} = θ)

ξ′_{i} = ϕ_{i}(t, θ, ξ_{2}, ⋯, ξ_{n})
= ψ_{i}(t, θ, ξ_{2}, ⋯, ξ_{n}) + ϕ_{i,1}(t, ξ_{2}, ⋯, ξ_{n})θ^{−1} + ⋯ + ϕ_{i,N_{i}}(t, ξ_{2}, ⋯, ξ_{n})θ^{−N_{i}}, (35)

where ϕ_{∗,∗} and ψ_{∗} are analytic in their variables. Since the Laurent series (8) is a formal solution of the ODE system (7), the power series (33) and (34) is a formal solution of the new system (35). Substituting the power series into the i-th equation, we find that the left side is a power series, while the right side is a Laurent series, with a_{0}^{N_{i}}ϕ_{i,N_{i}}(t_{0}, ξ_{2}(t_{0}), ⋯, ξ_{n}(t_{0})) as the coefficient of (t − t_{0})^{−N_{i}}. Since a_{0} ≠ 0, we thus conclude that

ϕ_{i,N_{i}}(t_{0}, ξ_{2}(t_{0}), ⋯, ξ_{n}(t_{0})) = 0

for all choices of t_{0} and the resonance parameters. From the formulas for the initial data ξ_{2}(t_{0}), ⋯, ξ_{n}(t_{0}) (see (34) or the third part of the theorem), we see that t_{0}, ξ_{2}(t_{0}), ⋯, ξ_{n}(t_{0}) may be independently chosen in a small region. Since ϕ_{i,N_{i}} is analytic, we conclude that ϕ_{i,N_{i}} is constantly zero.

By induction, we may further prove that ϕ_{i,N_{i}} = ϕ_{i,N_{i}−1} = ⋯ = ϕ_{i,1} = 0. This completes the proof of the regularity of (35).

□

3.2 The Essence of the Mirror Transform

A closer look at the proof of Theorem 7 reveals that the mirror transform is essentially defined for a collection of Laurent series with enough numbers of “independent” free parameters in the coefficients. We illustrate this point by considering the following three Laurent series involving two free parameters

u_{1} = (t − t_{0})^{−2} + 2(t − t_{0})^{−1} + r_{2} + r_{2}r_{3}^{2}(t − t_{0}) + ⋯,
u_{2} = (t − t_{0})^{−1} + r_{2} + r_{2}^{2}r_{3}(t − t_{0}) + ⋯,
u_{3} = (t − t_{0})^{−3} + 2r_{2}(t − t_{0})^{−1} + r_{3} + · · · .

The example has the following features: first, we will think of u_{2} as having the formal leading exponent −2, so that the resonance parameters r_{2} and r_{3} will have corresponding resonances j = 2 and 3. Second, r_{3} first appears in the coefficients in a non-linear way and with functions of r_{2} as coefficients (such phenomena cannot happen under the conditions of Theorem 7). Because of such non-linearity, we should consider the matrix

( \begin{matrix}2 & r_{2} & r_{2}r_{3}^{2} \\ 0 & r_{2} & r_{2}^{2}r_{3} \\ 3 & 2r_{2} & r_{3} \end{matrix} )  (36)

as playing a role similar to the resonance matrix defined before (the true resonance matrix is some kind of derivative of this; see later discussions).

We introduce the indicial normalization u_{1} = θ^{−2} and compute the following (with the choice of positive sign in θ = ±(t − t_{0}) + ⋯):

t − t_{0} = θ + θ^{2} + (\frac{1}{2}r_{2} + \frac{1}{2}) θ^{3} + (\frac{r_{2}}{2} r_{3}^{2} + r_{2}) θ^{4} + · · · .

Substituting this into (36), we get

u_{1} = θ^{−2},
u_{2} = θ^{−1} + r_{2} − 1 + (r_{2}^{2}r_{3} − \frac{r_{2}}{2} + \frac{1}{2}) θ + ⋯,  (37)
u_{3} = θ^{−3} − 3θ^{−2} + (\frac{1}{2}r_{2} + \frac{9}{2}) θ^{−1} + (−\frac{3r_{2}}{2} r_{3}^{2} + r_{3} + r_{2} − 4) + · · · .

We truncate u_{2} at the constant term by introducing

ξ_{2} = r_{2} − 1 + (r_{2}^{2}r_{3} − \frac{r_{2}}{2} + \frac{1}{2}) θ + · · · .  (38)

We convert this into

r_{2} = ξ_{2} + 1 + (−ξ_{2}^{2}r_{3} − 2ξ_{2}r_{3} − r_{3} + \frac{ξ_{2}}{2} ) θ + · · · .  (39)

Substituting this into u_{3} in (37) yields

u_{1} = θ^{−2},
u_{2} = θ^{−1} + ξ_{2},  (40)
u_{3} = θ^{−3} − 3θ^{−2} + (5 + \frac{ξ_{2}}{2} ) θ^{−1} + (−\frac{3(ξ_{2} + 1)}{2}r_{3}^{2} + \frac{(1 − 2ξ_{2} − ξ_{2}^{2})}{2}r_{3} − 3 + \frac{5ξ_{2}}{4} ) + · · · .

We truncate u_{3} at the constant term by introducing

ξ_{3} = (−\frac{3(ξ_{2} + 1)}{2}r_{3}^{2} + \frac{(1 − 2ξ_{2} − ξ_{2}^{2})}{2}r_{3} − 3 + \frac{5ξ_{2}}{4} ) + ⋯
= (−\frac{3r_{2}}{2} r_{3}^{2} + (1 − \frac{r_{2}^{2}}{2} ) r_{3} + \frac{5r_{2}}{4} − \frac{17}{4}) + · · · .  (41)

Then we get the mirror transform,

u_{1} = θ^{−2},
u_{2} = θ^{−1} + ξ_{2},
u_{3} = θ^{−3} − 3θ^{−2} + (5 + \frac{ξ_{2}}{2} ) θ^{−1} + ξ_{3},

associated with (36).

In terms of the resonance matrix, consider the Gaussian elimination of (36)

( \begin{matrix}2 & r_{2} & r_{2}r_{3}^{2} \\ 0 & r_{2} & r_{2}^{2}r_{3} \\ 3 & 2r_{2} & r_{3} \end{matrix} ) ⟶ ( \begin{matrix}2 & r_{2} & r_{2}s^{2} \\ 0 & r_{2} & r_{2}^{2}r_{3} \\ 0 & \frac{r_{2}}{2} & −\frac{3r_{2}}{2} r_{3}^{2} + r_{3} \end{matrix} ) ⟶ ( \begin{matrix}2 & r_{2} & r_{2}r_{3}^{2} \\ 0 & r_{2} & r_{2}^{2}r_{3} \\ 0 & 0 & −\frac{3r_{2}}{2} r_{3}^{2} + (1 − \frac{r_{2}^{2}}{2} ) r_{3} \end{matrix} ) .

Observe that in the second matrix, the 2×2 minor opposite the first entry is the “resonance matrix” for u_{2} and u_{3} in (37), up to adding constants in the r_{2}-column and adding functions of r_{2} in the r_{3}-column. Moreover, in the third matrix, the 1 × 1 minor formed by the last entry is the “resonance matrix” for u_{3} in (40), up to adding a function of r_{2}. Finally, the second and the third diagonal elements in the third matrix are the leading terms for the power series of ξ_{2} and ξ_{3}, again up to adding a constant and a function of r_{2}, respectively (see (38) and (41)).

The key to deriving the mirror transform is the invertibility of the leading coefficients of the leading terms of the new variables as functions of the corresponding resonance parameters. In the example above, this means that the second and the third diagonal terms r_{2} and −\frac{3r_{2}}{2} r_{3}^{2} + (1 − \frac{r_{2}^{2}}{2} ) r_{3} are respectively invertible functions of r_{2} and r_{3}. Such invertibility enables us to convert (38) to (39) (and similarly for (41)). The invertibility will also be crucial in the final proof of the regularity of the mirror system.

The invertibility can be detected by taking the partial derivatives with respect to the resonance parameters. Thus, we take the derivative of the second column of the matrix (36) in r_{2} and the derivative of the third column in r_{3}. This commutes with the Gaussian elimination process, such that we have

( \begin{matrix}2 & 1 & 2r_{2}r_{3} \\ 0 & 1 & r_{2}^{2} \\ 3 & 2 & 1 \end{matrix} ) ⟶ ( \begin{matrix}2 & 1 & 2r_{2}r_{3} \\ 0 & 1 & r_{2}^{2} \\ 0 & \frac{1}{2} & −3r_{2}r_{3} + 1 \end{matrix} ) ⟶ ( \begin{matrix}2 & 1 & 2r_{2}r_{3} \\ 0 & 1 & r_{2}^{2} \\ 0 & 0 & −3r_{2}r_{3} + 1 − \frac{r_{2}^{2}}{2} \end{matrix} ) .

Thus, strictly speaking, the mirror transform is valid for all r_{2} and r_{3} satisfying −3r_{2}r_{3} + 1 − \frac{r_{2}^{2}}{2} ≠ 0.

In conclusion, the following is the exact set-up needed for constructing the mirror transform (such that the new variables have power series with leading coefficients equivalent to the resonance parameters). Consider the Laurent series (8) satisfying (9). Assume that the coefficients of the Laurent series contain some resonance parameters. Each resonance parameter r is associated with an integer j ≥ 0 (the resonance), such that u_{i,k} (u_{i,0} = c_{i}) is independent of r for k < j. Then, we define the corresponding resonance

vector to be (\frac{∂u_{1,j}}{∂r} , ⋯, \frac{∂u_{n,j}}{∂r} ). Moreover, we also consider t_{0} to be a resonance parame-

ter with associated resonance j = −1 and the resonance vector (−g_{1}c_{1}, ⋯, −g_{n}c_{n}). Now, the exact condition for constructing the mirror transform is that the resonance vectors form a basis of C^{n}.

Now, back to the proof of Theorem 7. We note that the proof consists of three parts:

1. construct the mirror transform;

2. find the power series (especially for the initial data) for the new variables;

3. prove the regularity of the mirror system.

From the discussion on the essence of the mirror transform, we see that the first two steps make no use of any explicit information about the differential equation, such as the kind of functions on the right side, the Fuchsian condition, and the Kowalevskian matrix. It is in the third part that the explicit information on the differential equation is used. More precisely, the polynomial assumption is used for showing that the right side of the mirror system consists of functions meromorphic in θ and holomorphic in all other variables. The regularity of the mirror system then follows from this and the fact that the resonance parameters are equivalent to the initial values of the new variables.

The discussion indicates that Theorem 7 can be extended to more general differential equations. In fact, we will do so in discussing the mirror system for a single, high-order ODE in Section 4.1.

Finally, we remark that the rearrangement of the order of the variables u_{1}, ⋯, u_{n} in Theorem 7 is needed for the precise reason that, after the same rearrangement of the rows, the resonance matrix has an LU decomposition. We will see that in the case of single, highorder equation, the resonance matrix already has an LU decomposition with the natural order of rows. Therefore, the mirror system in this case is in a form compatible with the natural order. We will also see that for Hamiltonian systems, the resonance matrix has an LU decomposition up to a symplectic type of rearrangement. Consequently, the mirror system is in a certain special form up to such kind of rearrangement.

3.3 Localness and Local Invertibility of the Mirror Transform

Theorem 7 is a local statement. The following are some technical considerations that restrict the theorem to be a local one. First, we need to choose a non-zero analytic function a_{0} satisfying a_{0}^{g_{1}} = c_{1}. For a choice of t_{0} and the resonance parameters with resonance j = 0, a_{0} generally exists only in the neighborhood of the choice. Second, we need to choose a rearrangement of the dependent variables so that the resonance matrix has an LU decomposition. For different choices of t_{0} and resonance parameters with resonance j = 0, we may have to use different rearrangements. Third, we need to formally solve several equations of the θ-series. This depends on expanding the Taylor series of the coefficients in the series. The radii of convergence of the series depend on t_{0} and the resonance parameters.

Thus, for a given choice of t_{0} and the resonance parameters, the claims of the theorem hold for those Laurent series solutions in which t_{0} and the resonance parameters are sufficiently close to the given choice. In particular, the radius of convergence of the power series solutions of the new system (as well as the radius of convergence of the Laurent series in the original balance) depends on the choice. On the other hand, the mirror transform is often (but not always) given by a globally valid formula. This could be useful in patching the local understandings of the solutions to gain global results. For an example of the extra work needed to extract global results from the mirror system, see [9].

One potential application of the mirror transform is in the completion of the phase space of the ODE system, which is needed for the study of algebraic complete integrable systems [3] [4]. The completion is constructed by adding the singularities to the regular part of the phase space. And the mirror transform may be used as the patching map for this addition. However, we need to make sure that the mirror transform is locally invertible in order to do this.

Consider the example of mirror transform (6). It is two-to-one as a global transform. On the other hand, the transform is one-to-one when restricted to the domain

|Q_{1}|^{2} < \frac{2}{3|Q_{2}|}, |Q_{1}|^{4} < \frac{8}{9|Q_{2}|^{2}} , |Q_{1}|^{5} < \frac{1}{3|P_{2}|}  (42)

around Q_{1} = 0. To see this, assume that both (Q_{1}, Q_{2}, P_{1}, P_{2}) and (Q̂_{1}, Q̂_{2}, P̂_{1}, P̂_{2}) correspond to the same (q_{1}, q_{2}, p_{1}, p_{2}). Then we have Q̂_{1} = −Q_{1} from the formula for q_{1}. Substituting this into the formula for p_{2}, we have

2Q_{1}^{−3} − \frac{1}{2}Q_{1}^{−1}(Q_{2} + Q̂_{2}) + \frac{3}{8}Q_{1}(Q_{2}^{2} + Q̂_{2}^{2}) + Q_{1}^{2}(P_{2} − P̂_{2}) = 0.

Now, if both points were in the domain (42), then we would have

|\frac{1}{2}Q_{1}^{−1}(Q_{2} + Q̂_{2})| < \frac{1}{3}|2Q_{1}^{−3}|, |\frac{3}{8}Q_{1}(Q_{2}^{2} + Q̂_{2}^{2})| < \frac{1}{3}|2Q_{1}^{−3}|, |Q_{1}^{2}(P_{2} − P̂_{2})| < \frac{1}{3}|2Q_{1}^{−3}|,

which would lead to a contradiction.

In general, we may consider the situation as described in Theorem 7 and look for the natural leading exponents in a principal balance. For example, we consider the leading exponents of the principal balance (5) to be 2, 2, 5, 3, instead of the later choice 2, 4, 5, 3 (see (10)) that fits into the Fuchsian condition. If these natural leading exponents are coprime, then a similar argument as above gives us a neighborhood

|θ| < ϵ(ξ_{2}, ⋯, ξ_{n}), (ϵ is a positive continuous function)

around θ = 0, such that the restriction of the mirror transform to the neighborhood is one-to-one.

In general, it is easy to construct collections of Laurent series so that the associated mirror transform is not locally one-to-one. On the other hand, the mirror transforms constructed for the single, high-order ODEs (see Theorem 8) are always locally one-to-one. At the moment, we are not sure whether mirror transforms constructed under the context of Theorem 7 are always locally one-to-one or not, although many examples suggest this is the case.

4 The Mirror Transform for Other Equations

4.1 For Single, High-Order ODEs

Consider a differential equation

u^{(n)} = F (t, u, u′, ⋯, u^{(n−1)}),  (43)

where F is a polynomial in u, u′, ⋯, u^{(n−1)} and analytic in t. We may convert the equation to a system of first-order ODEs in the usual way and get a version of Theorem 7 for single, high-order ODEs. However, the observations made in Section 3.2 allow us to refine the theorem.

Definition 5 A principal balance for equation (43) is a formal Laurent series solution

u = c(t_{0})(t − t_{0})^{−k} + u_{1}(t_{0})(t − t_{0})^{1−k} + ⋯  (44)
+r_{2}(t − t_{0})^{j_{2}−k} + u_{j_{2}+1}(t_{0}, r_{2})(t − t_{0})^{j_{2}+1−k} + ⋯
+r_{3}(t − t_{0})^{j_{3}−k} + u_{j_{3}+1}(t_{0}, r_{2}, r_{3})(t − t_{0})^{j_{3}+1−k} + ⋯
+ · · · · · ·
+r_{n}(t − t_{0})^{j_{n}−k} + u_{j_{n}+1}(t_{0}, r_{2}, ⋯, r_{n})(t − t_{0})^{j_{n}+1−k} + ⋯
+ · · · · · · ,

where

1. the singularity t_{0} can be any point; 2. the coefficient functions u_{p} are analytic;

3. the leading exponent k > 0 is an integer;

4. the leading coefficient c(t_{0}) ≠ 0 near t_{0}. In case j_{2} = 0, this means r_{2} can be any non-zero number;

5. the resonances satisfy j_{n} > ⋯ > j_{2} > j_{1} = −1, and the resonance parameters r_{2}, ⋯, r_{n} can be arbitrary constants.

Note that the Definition 3 for principal balances of ODE systems makes use of the Kowalevskian matrix, which further depends on the fact that the leading exponents is Fuchsian. The definition of principal balances of single, high-order ODEs does not explicitly rely on the Fuchsian type condition. Although it may happen that after converting single, high-order ODEs to systems of ODEs, the two definitions coincide, we are not concerned with the technical issue here. The bottom line is that Definition 5 is more suitable for practical applications.

Theorem 8 Suppose (44) is a principal balance of the ODE (43). Then there is a change of variables of the form

{ u = θ^{−k}, ; u′ = θ^{−k−1}(a_{2,0} + a_{2,1}θ + ⋯ + a_{2,j_{2}−1}θ^{j_{2}−1} + ξ_{2}θ^{j_{2}}), ; ⋮ ; u^{(n−1)} = θ^{−k−n+1}(a_{n,0} + a_{n,1}θ + ⋯ + a_{n,j_{n}−1}θ^{j_{n}−1} + ξ_{n}θ^{j_{n}}),

such that

1. a_{i,j} is an analytic function of t and ξ_{s} with j_{s} ≤ j;

2. the ODE (43) is converted to a regular ODE system for the new variables (θ, ξ_{2}, ⋯, ξ_{n});

3. if j_{2} > 0, then the principal balance is converted to a solution of the new ODE

system with the initial data

θ(t_{0}) = 0,

and

ξ_{i}(t_{0}) = b_{i}(t_{0})r_{i} + c_{i}(t_{0}, r_{2}, ⋯, r_{i−1}),

where b_{∗} and c_{∗} are analytic, b_{i}(t_{0}) ≠ 0. If j_{2} = 0, then everything is still true except that r_{2} should be replaced by r̄_{2} satisfying r̄_{2}^{−k} = r_{2}, and b_{∗} depends only on r̄_{2}.

Proof: The theorem may be proved in a similar way as Theorem 7. As indicated in Section 3.2, the proof involves three steps, and the key for the first two steps is the invertibility and the LU decomposition of the resonance matrix. Now, in terms of the dependent variables u, u′, ⋯, u^{(n−1)}, the resonance matrix for the principal balance (44) is

R = ( \begin{matrix}−kcP_{0}(j_{1} − k) & P_{0}(j_{2} − k) & ⋯ & P_{0}(j_{n} − k) \\ −kcP_{1}(j_{1} − k) & P_{1}(j_{2} − k) & ⋯ & P_{1}(j_{n} − k) \\ −kcP_{2}(j_{1} − k) & P_{2}(j_{2} − k) & ⋯ & P_{2}(j_{n} − k) \\ ⋮ & ⋮ & & ⋮ \\ −kcP_{n−1}(j_{1} − k) & P_{n−1}(j_{2} − k) & ⋯ & P_{n−1}(j_{n} − k) \end{matrix} ) ,

where

P_{i}(x) = x(x − 1) ⋯ (x − i + 1).

Without permuting rows of R, we have the following LU decomposition

R = ( \begin{matrix}1 & 0 & ⋯ & 0 \\ ∗ & 1 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ ∗ & ∗ & ⋯ & 1 \end{matrix} ) ( \begin{matrix}−kc & ∗ & ∗ & ⋯ & ∗ \\ 0 & (j_{2} − j_{1}) & ∗ & ⋯ & ∗ \\ 0 & 0 & (j_{3} − j_{1})(j_{3} − j_{2}) & ⋯ & ∗ \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & ⋯ & ∏_{i=1}^{n−1}(j_{n} − j_{i}) \end{matrix} ) .

By the same process as in the proof of Theorem 7, we may introduce the new variables by u = θ^{−k} and successively truncating the θ-series of u′, u′′, ⋯, u^{(n−1)} at θ^{j_{2}−k−1}, θ^{j_{3}−k−2}, ⋯, θ^{j_{n}−k−n+1}. The result is a change of variable in the shape indicated in the theorem.

The transform converts the Laurent series (44) in to power series for the new variables. More precisely, from the LU decomposition above and (32), we have

θ(t_{0}) = 0,

and (a_{0}(t_{0}) is replaced by r̄_{2} in case j_{2} = 0)

ξ_{i}(t_{0}) = [a_{0}(t_{0})^{j_{i}−k−i+1} ∏_{l=1}^{i−1}(j_{i} − j_{l})] r_{i} + c_{i}(t_{0}, r_{2}, ⋯, r_{i−1}),

where the coefficient of r_{i} is clearly nonzero. Therefore, the initial data is as described in the theorem.

Finally, the discussion in Section 3.2 indicates that we have all the key ingredients for proving the regularity of the new system.

□

4.2 Equations with Rational Right Sides

For the mirror systems to be regular, some kind of rationality assumption on the functions on the right side seems to be necessary. The reason is that the mirror transform involves θ^{−1}. Therefore, for the mirror system to be analytic near θ = 0, the right side of the original equation should be analytic or, at most, have poles at infinity.

Let us consider the single, high-order ODE (43), with function F rational in u and a polynomial in the derivatives. Specifically, we consider

F (t, u, u′, ⋯, u^{(n−1)}) = \frac{G(t, u, u′, ⋯, u^{(n−1)})}{a(t)(u^{m} + b_{1}(t)u^{m−1} + ⋯ + b_{m}(t))},

where G, a, b_{i} are analytic in t, and G is a polynomial in u, u′, ⋯, u^{(n−1)}. For t_{0} away from some isolated points, we have

u^{m} + b_{1}(t)u^{m−1} + ⋯ + b_{m}(t) = (u − U_{1}(t))^{m_{1}} ⋯ (u − U_{s}(t))^{m_{s}},  (45)

such that

1. a(t_{0}) ≠ 0;

2. U_{1}, ⋯, U_{s} are analytic near t_{0};

3. U_{1}(t_{0}), ⋯, U_{s}(t_{0}) are distinct.

We call those isolated points where the above is not satisfied fixed singularities. In particular, if some of U_{1}(t_{0}), ⋯, U_{s}(t_{0}) coincide, then t_{0} is also considered as a fixed singularity.

Away from fixed singularities, equation (43) has movable singularities in case (45) vanishes. This leads us to consider regular solutions u satisfying u(t_{0}) = U_{p}(t_{0}) for some U_{p} and movable singularity t_{0}. In this case, we define a principal balance to be a formal power series solution

u = U_{p}(t) + c(t_{0})(t − t_{0})^{k} + u_{1}(t_{0})(t − t_{0})^{1+k} + ⋯
+r_{2}(t − t_{0})^{j_{2}+k} + u_{j_{2}+1}(t_{0}, r_{2})(t − t_{0})^{j_{2}+1+k} + ⋯
+r_{3}(t − t_{0})^{j_{3}+k} + u_{j_{3}+1}(t_{0}, r_{2}, r_{3})(t − t_{0})^{j_{3}+1+k}
+ · · · · · ·
+r_{n}(t − t_{0})^{j_{n}+k} + u_{j_{n}+1}(t_{0}, r_{2}, ⋯, r_{n})(t − t_{0})^{j_{n}+1+k}
+ · · · · · · ,

where k > 0 is a positive integer and all the other conditions in the definition of principal Laurent balances are satisfied. For such a principal balance, we can still construct the mirror transform. Specifically, we introduce the indicial normalization u = U_{p} + θ^{k}. Then, by fixing a k-th root of c(t) near t_{0}, we may find the θ-series of t − t_{0}. We may then manipulate the series for the derivatives of u as before and introduce the new variables ξ_{2}, ⋯, ξ_{n} by successively truncating at the resonance parameters. The resulting mirror transform is a polynomial in the new variables. Moreover, the power series of u is converted to the power series of the new variables, with the leading terms for ξ_{2}, ⋯, ξ_{n} essentially equivalent to the resonance parameters.

The only new ingredient is the regularity of the mirror system. By the assumption that a(t_{0}) ≠ 0 and U_{1}(t_{0}), ⋯, U_{s}(t_{0}) are distinct, substituting the mirror transform changes the right side of (43) into the product of θ^{−m_{i}k} and a function analytic in t, θ, and a polynomial in ξ_{2}, ⋯, ξ_{n} near t = t_{0} and θ = 0. Then, it is easy to see that the right side of the mirror system consists of functions analytic in t, meromorphic in θ, and a polynomial in ξ_{2}, ⋯, ξ_{n} near t = t_{0} and θ = 0. By making use of the fact that the leading terms for ξ_{2}, ⋯, ξ_{n} are essentially equivalent to the resonance parameters (so that they can be arbitrarily chosen), the same proof as before shows that the right side of the mirror system is regular near t = t_{0} and θ = 0.

The discussion above also applies to ODE systems. As pointed out in Section 3.2, the derivation of the mirror transform does not make use of much explicit information about the equation (except U_{p} for the case of a rational right side). Therefore, we still have the mirror transform in the expected shape. The only problem is to make sure the right side of the mirror system consists of functions meromorphic in θ and analytic in all others. For this to happen, it is sufficient to assume that the right side of the original system consists of functions rational in one u_{i} and a polynomial in all other dependent variables, such that g_{i}c_{i} ≠ 0 (this condition allows us to introduce indicial normalization u_{i} = U_{p} + θ^{g_{i}}).

Suppose the right side of (43) is rational (instead of being a polynomial) in all the derivatives instead of just u. In this case, we can still derive the mirror transform from the principal balance. The key problem is that we are no longer sure of the regularity of the mirror system. On the other hand, as long as we know that the right side of the mirror system consists of functions meromorphic in θ and analytic in all others, the proof of regularity still applies, and we may conclude that the mirror system is regular. The same remark also applies to the case of ODE systems with right sides consisting of functions rational in all dependent variables.

4.3 For Hamiltonian Systems

Theorem 9 Suppose the Hamiltonian system (16) is almost weighted homogeneous relative to the leading exponents of a principal balance. Then, up to exchanging some p_{i} with q_{i} and rearranging the order among the pairs (q_{1}, p_{1}), ⋯, (q_{m}, p_{m}), there is a change of variables of the form

{ q_{1} = Q_{1}^{−g_{1}}, ; q_{2} = Q_{1}^{−g_{2}}(a_{0,2} + a_{1,2}Q_{1} + ⋯ + a_{j_{2}−1,2}Q_{1}^{j_{2}−1} + Q_{2}Q_{1}^{j_{2}}), ; ⋮ ; q_{m} = Q_{1}^{−g_{m}}(a_{0,m} + a_{1,m}Q_{1} + ⋯ + a_{j_{m}−1,m}Q_{1}^{j_{m}−1} + Q_{m}Q_{1}^{j_{m}}), ; p_{1} = Q_{1}^{−h_{1}}(b_{0,1} + b_{1,1}Q_{1} + ⋯ + b_{k_{1}−1,1}Q_{1}^{k_{1}−1} − g_{1}^{−1}P_{1}Q_{1}^{k_{1}}), ; p_{2} = Q_{1}^{−h_{2}}(b_{0,2} + b_{1,2}Q_{1} + ⋯ + b_{k_{2}−1,2}Q_{1}^{k_{2}−1} + P_{2}Q_{1}^{k_{2}}), ; ⋮ ; p_{m} = Q_{1}^{−h_{m}}(b_{0,m} + b_{1,m}Q_{1} + ⋯ + b_{k_{m}−1,m}Q_{1}^{k_{m}−1} + P_{m}Q_{1}^{k_{m}}),

such that 1. a_{j,i} is an analytic function of t and Q_{s} with j_{s} ≤ j;

2. b_{j,i} is an analytic function of t, Q_{2}, ⋯, Q_{m}, and P_{s} with k_{s} ≤ j;

3. the symplectic form is preserved

dq_{1} ∧ dp_{1} + ⋯ + dq_{m} ∧ dp_{m} = dQ_{1} ∧ dP_{1} + ⋯ + dQ_{m} ∧ dP_{m};

4. the Hamiltonian system is converted to another Hamiltonian system with a polynomial of (Q_{∗}, P_{∗}) as the Hamiltonian function. In the autonomous case, the new Hamiltonian is obtained by substituting the change of variable to the original Hamiltonian. In the nonautonomous case, the new Hamiltonian is obtained by substituting the change of variable to the original Hamiltonian, and then deleting the terms with negative powers of Q_{1};

5. the principal balance is converted to a regular solution of the new Hamiltonian system with the initial data equivalent to the resonances.

The precise meaning of “terms with negative powers of Q_{1}” in the fourth item can be found in Section 4.4.

The initial data in the fifth item are actually given by the formulas similar to those in the third item of Theorem 7.

Proof: In the proof of Theorem 7, we have already shown how to construct a change of variables that converts the Laurent series solution into a power series solution. For the Hamiltonian system, we need to do the construction in a more careful way so that the third and the fourth items are satisfied.

We have seen that the construction of a change of variables corresponds to the LU decomposition of the resonance matrix. The construction is always possible because, up to rearranging the rows, any invertible matrix has an LU decomposition. As pointed out in Section 2.4, at least for almost weighted homogeneous Hamiltonian systems, we may exchange some p_{i} with q_{i} and rearrange the order among the pairs (q_{1}, p_{1}), ⋯, (q_{m}, p_{m}), so that when the dependent variables are presented in the order of q_{1}, ⋯, q_{m}, p_{m}, ⋯, p_{1}, the resonance matrix has an LU decomposition. Consequently, we have a change of variable in the form as described in the theorem, such that the new system for Q_{∗}, P_{∗} is regular, and the first, second and fifth items are satisfied.

Now we prove the third item. Note that dq_{i} ∧ dp_{i} contains the term (keep in mind Q_{1} is the variable θ in Theorem 7)

Q_{1}^{j_{i}−g_{i}}dQ_{i} ∧ Q_{1}^{k_{i}−h_{i}}dP_{i} = Q_{1}^{j_{i}+k_{i}−g_{i}−h_{i}}dQ_{i} ∧ dP_{i} = dQ_{i} ∧ dP_{i},

where we use the almost weighted homogeneous assumption g_{i} + h_{i} = d − 1 and equality (17) in the second equality. Since this term clearly has the highest power in Q_{1}, we have

dq_{i} ∧ dp_{i} = dQ_{i} ∧ dP_{i} + Q_{1}^{−1}(⋯) + Q_{1}^{−2}(⋯) + ⋯,

where (⋯) means 2-forms in Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}, with functions of Q_{2}, ⋯, Q_{m}, P_{1}, ⋯, P_{m} as coefficients. Thus, we conclude that

dq_{1} ∧ dp_{1} + ⋯ + dq_{m} ∧ dp_{m}
= dQ_{1} ∧ dP_{1} + ⋯ + dQ_{m} ∧ dP_{m} + Q_{1}^{−1}ω_{1} + Q_{1}^{−2}ω_{2} + ⋯ + Q_{1}^{−N}ω_{N}.  (46)

It remains for us to show that all the 2-forms ω_{N} vanish.

We may think of the Laurent series for q_{1}, ⋯, q_{m}, p_{1}, ⋯, p_{m} as a transform between (q_{∗}, p_{∗}) and (t_{0}, r_{∗}) and substitute into the left side of (46). The result is a 2-form in t_{0} and r_{∗}, with regular functions of t_{0} and r_{∗} as coefficients. We emphasize that since the symplectic form is left invariant by Hamiltonian flows, the coefficients are independent of t.

On the other hand, from Theorem 7, we have the power series solution (33) for Q_{1} and (34) for Q_{2}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}. If we think of these power series as a transform between (Q_{∗}, P_{∗}) and (t_{0}, r_{∗}), and substitute into the right side of (46), the result should be the same as what we obtained by substituting the Laurent series for q_{1}, ⋯, q_{m}, p_{1}, ⋯, p_{m} into the left side of (46). In particular, we should get a 2-form in t_{0} and r_{∗}, with coefficients independent of t.

Now let us discuss in detail what we get by substituting the power series for Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m} into the right side of (46). Let

ω_{N} = dQ_{1} ∧ ρ + ϕ,

where ρ and ϕ are 1- and 2-forms in Q_{2}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}, with functions of Q_{2}, ⋯, Q_{m}, P_{1}, ⋯, P_{m} as coefficients. After substituting the power series into ω_{N}, we get

ω_{N} = dt_{0} ∧ ρ̄ + ϕ̄ + (t − t_{0})ψ̄,

where

1. ρ̄ is a 1-form in t_{0} and r_{∗};

2. ϕ̄ is obtained as follows: Think of the initial data formulas as a transform between (Q_{2}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}) and (r_{2}, ⋯, r_{n}), with t_{0} as a constant. Substituting the transform into ϕ gives us ϕ̄;

3. ψ̄ is a regular 2-form in t_{0} and r_{∗}.

Thus, substituting the power series for Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m} into the right of (46) will give us a 2-form

a_{0}^{N}(t − t_{0})^{−N} ϕ̄ + (⋯),  (47)

in t_{0} and r_{∗}, with (⋯) consisting of terms that are either of order > −N or have dt_{0} as a factor. On the other hand, we have already argued that (47) should be a 2-form in t_{0} and r_{∗}, with regular functions of t_{0} and r_{∗} (and no t) as coefficients. Therefore, we conclude that ϕ̄ = 0. Since ϕ̄ is obtained from ϕ by an invertible transform, this implies ϕ = 0.

Once we know ω_{N} = dQ_{1} ∧ ρ, substituting the power series for Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m} into ω_{N} will give us

ω_{N} = −a_{0}^{−1}dt_{0} ∧ ρ̄ + (t − t_{0})ψ̄,

where ρ̄ is obtained from ρ in the similar way as ϕ̄ being obtained from ϕ above, and ψ̄ is a regular 2-form in t_{0} and r_{∗}. Thus, substituting the power series for Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m} into the right of (46) will give us a 2-form

−a_{0}^{N−1}(t − t_{0})^{−N}dt_{0} ∧ ρ̄ + (⋯),

in t_{0} and r_{∗}, with (⋯) consisting of terms of order > −N. Again, since (47) should be a 2-form in t_{0} and r_{∗}, with regular functions of t_{0} and r_{∗} (and no t) as coefficients, we must conclude ρ̄ = 0, which further implies ρ = 0.

We have proved that ω_{N} = 0. By induction, we further have ω_{N} = ω_{N−1} = ⋯ = ω_{1} = 0. This completes the proof of the third item.

Now, we consider the fourth item. Suppose the original Hamiltonian system is autonomous. Since we have shown in the third item that the transform is canonical, the new system for the new variables is also an autonomous Hamiltonian system, and the new Hamiltonian function is simply obtained by substituting the transform into the old Hamiltonian function. To see that the new Hamiltonian function is a polynomial, we first note that the new Hamiltonian function is of the form (from the shape of the transform)

H̄(Q_{∗}, P_{∗}) = k + h_{1}Q_{1}^{−1} + ⋯ + h_{N}Q_{1}^{−N},

where h_{1}, ⋯, h_{N} are polynomials of Q_{2}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}, and k is a polynomial of Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}. Since the Hamiltonian is a first integral of the Hamiltonian system, substituting the power series solution for Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m} into H̄ should give us an expression independent of t. Then, as in the proof of the regularity of the new system in Theorem 7, we may consider the most singular term after the substitution and conclude that h_{N} = 0. By induction, we find that H̄(Q_{∗}, P_{∗}) = k is regular. This completes the proof of the fourth item in the autonomous case.

The proof of the claim in the fourth item about the nonautonomous case can be found in the discussion following the proof of Proposition 10 in Section 4.4.

□

Note that a close inspection of the proof reveals that the key to the extra Hamiltonian properties (the third and fourth items) is the following: after exchanging some p_{i} with q_{i} and rearranging the order among the pairs (q_{1}, p_{1}), ⋯, (q_{m}, p_{m}), we have

1. j_{i} + k_{i} = g_{i} + h_{i};

2. with row coordinates given in the order (q_{1}, ⋯, q_{m}, p_{m}, ⋯, p_{1}), the resonance matrix has an LU decomposition.

While the almost weighted homogeneous condition immediately implies both, we believe that, as long as the leading exponents of a principal balance are Fuchsian for the Hamiltonian system, the two key requirements can always be satisfied. We also believe that the reason is due to some hidden symplectic structure in the principal balance that we do not yet fully understand.

4.4 On Non-autonomous Systems

For any given non-autonomous Hamiltonian system, it is obvious that the Hamiltonian function under the canonical mirror transform is no longer the Hamiltonian function of the mirror (Hamiltonian) system. In fact, the Hamiltonian function under the mirror transform is often not regular in θ. In this section, we show that the Hamiltonian function of the mirror system is still a regular function. Actually, it is the “regular part” of the original Hamiltonian function under the mirror transform. Some examples on this can be found in [8].

Indeed, for a non-autonomous system (7), we think of t as a parametrization in the mirror transform. Denote the mirror transform by u = μ(t, ξ), where ξ = (θ, ξ_{2}, ⋯, ξ_{n}).

Then the Jacobian from the new variable ξ to the old variable u is J_{μ} = \frac{∂μ}{∂ξ}, and we have

u′ = \frac{∂μ}{∂t} + J_{μ}ξ′.

Consequently, the mirror system is

ξ′ = J_{μ}^{−1}F (t, μ(t, ξ)) − J_{μ}^{−1} \frac{∂μ}{∂t}.  (48)

Note that we have

J_{μ} = ( \begin{matrix}−g_{1}θ^{−g_{1}−1} & 0 & 0 & ⋯ & 0 \\ \{θ^{<j_{2}−g_{2}}\} & θ^{j_{2}−g_{2}} & 0 & ⋯ & 0 \\ \{θ^{<j_{3}−g_{3}}\} & \{θ^{<j_{2}−g_{3}}\} & θ^{j_{3}−g_{3}} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ \{θ^{<j_{n}−g_{n}}\} & \{θ^{<j_{2}−g_{n}}\} & \{θ^{<j_{3}−g_{n}}\} & ⋯ & θ^{j_{n}−g_{n}} \end{matrix} ) ,  (49)

where {θ^{<k}} denotes a series in θ^{i} with powers i < k and coefficients given by analytic functions of t and ξ. Moreover, F (t, μ(t, ξ)) is also a series in powers of θ with coefficients

given by analytic functions of t and ξ. Therefore both J_{μ}^{−1}F (t, μ(t, ξ)) and J_{μ}^{−1} \frac{∂μ}{∂t} are series in powers of θ.

Proposition 10 J_{μ}^{−1} \frac{∂μ}{∂t} is a series with purely negative powers of θ. In particular, if we denote by [?]_{reg} the non-negative power terms in a θ-series, then the mirror system is

ξ′ = [J_{μ}^{−1}F (t, μ(t, ξ))]_{reg} .

Proof: Let J_{μ}^{−1} \frac{∂μ}{∂t} = σ(t, ξ) = (σ_{1}(t, ξ), ⋯, σ_{n}(t, ξ)). Then we have \frac{∂μ}{∂t} = J_{μ}σ. From (49), we have

0 = −g_{1}θ^{−g_{1}−1}σ_{1},

{θ^{<j_{2}−g_{2}}} = {θ^{<j_{2}−g_{2}}}σ_{1} + θ^{j_{2}−g_{2}}σ_{2},
{θ^{<j_{3}−g_{3}}} = {θ^{<j_{3}−g_{3}}}σ_{1} + {θ^{<j_{2}−g_{3}}}σ_{2} + θ^{j_{3}−g_{3}}σ_{3},
⋮
{θ^{<j_{n}−g_{n}}} = {θ^{<j_{n}−g_{n}}}σ_{1} + {θ^{<j_{2}−g_{n}}}σ_{2} + θ^{j_{3}−g_{3}}σ_{3} + ⋯ + θ^{j_{n}−g_{n}}σ_{n}.

From the first equation, we get σ_{1} = 0. Substituting into the second equation, we get σ_{2} = {θ^{<0}}. Substituting σ_{1} = 0 and σ_{2} = {θ^{<0}} into the third equation, we get σ_{3} = {θ^{<0}}. In general, we get σ_{k} = {θ^{<0}} by induction.

Thus, we see that all the terms in J_{μ}^{−1} \frac{∂μ}{∂t} have θ in negative power. On the other hand, the right side of (48) should be a polynomial in ξ. Therefore, we conclude that the right side of (48) is the regular part of J_{μ}^{−1}F (t, μ(t, ξ)).

□

Now we focus on a non-autonomous Hamiltonian system (16). By the third item of Theorem 9, the mirror transform is canonical. This implies that

J_{μ}^{−1} [J∇_{q_{∗},p_{∗}}H(t, q_{∗}, p_{∗})]_{(q_{∗},p_{∗})=μ(t,Q_{∗},P_{∗})} = J∇_{Q_{∗},P_{∗}}H(t, μ(t, Q_{∗}, P_{∗})).

Note that H(t, μ(t, Q_{∗}, P_{∗})) is a polynomial in Q_{1}^{−1}, Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}. Denote

H(t, μ(t, Q_{∗}, P_{∗})) = [H(t, μ(t, Q_{∗}, P_{∗}))]_{reg} + [H(t, μ(t, Q_{∗}, P_{∗}))]_{sing} ,

where [H(t, μ(t, Q_{∗}, P_{∗}))]_{reg} is a polynomial in Q_{1}, ⋯, Q_{m}, P_{1}, ⋯, P_{m}, and all terms in [H(t, μ(t, Q_{∗}, P_{∗}))]_{sing} have negative powers in Q_{1}. Then J∇_{Q_{∗},P_{∗}} [H(t, μ(t, Q_{∗}, P_{∗}))]_{reg} is regular, and all terms in J∇_{Q_{∗},P_{∗}} [H(t, μ(t, Q_{∗}, P_{∗}))]_{sing} have negative powers in Q_{1}. Thus, we conclude that

[J_{μ}^{−1} [J∇_{q_{∗},p_{∗}}H(t, q_{∗}, p_{∗})]_{(q_{∗},p_{∗})=μ(t,Q_{∗},P_{∗})}]_{reg} = J∇_{Q_{∗},P_{∗}} [H(t, μ(t, Q_{∗}, P_{∗}))]_{reg} .

It then follows from Proposition 10 that the mirror system is a Hamiltonian system with the regular part of H(t, μ(t, Q_{∗}, P_{∗})) as the Hamiltonian function. This is the claim made in the fourth item of Theorem 9 for the non-autonomous case.

4.5 For PDEs

For partial differential equations, we still have the analogue of Theorems 7 and 8. However, the mirror system is in general not in the form to which we can apply the Cauchy-Kowalevski theorem.

Instead of presenting the general theory, we will illustrate the main points by considering third-order partial differential equations of the form

u_{x^{3}} = f(t, x, u, u_{x}, u_{t}, u_{x^{2}}, u_{xt}, u_{t^{2}}),  (50)

where f is analytic in t, x, and is a polynomial in all the other variables. A principal balance for (50) is a formal solution of the form

u = c(t, ψ)(x − ψ)^{−k} + u_{1}(t, ψ, ψ_{t})(x − ψ)^{1−k} + u_{2}(t, ψ, ψ_{t}, ψ_{t^{2}})(x − ψ)^{2−k}
+ · · · · · ·
+r(t)(x − ψ)^{i−k} + u_{i+1}(t, ψ, ⋯, ψ_{t^{i+1}}; r)(x − ψ)^{i+1−k}
+ · · · · · ·
+s(t)(x − ψ)^{j−k} + u_{j+1}(t, ψ, ⋯, ψ_{t^{j+1}}; r, ⋯, r_{t^{j−i}}; s)(x − ψ)^{j+1−k}
+ · · · · · · ,

where

1. the singularity x = ψ = ψ(t) can be any analytic function;

2. the coefficient functions u_{p} are analytic;

3. the leading exponent k > 0 is an integer;

4. the leading coefficient c(t, x) is nonzero everywhere along the singularity x = ψ(t). In case i = 0, this means r(t) can be any nowhere zero analytic function;

5. the resonances satisfy j > i > −1, and the resonance parameters r = r(t), s = s(t) can be any analytic function.

Note that in our definition of the principal balance, we imposed some restrictions on the orders of the derivatives of ψ, r, s appearing in u_{p}. Such restrictions are satisfied by many important examples.

We introduce the indicial normalization u = θ^{−k}. We fix a branch a_{0}(t, x) = c(t, x)^{\frac{1}{k}} and postulate a θ-series

x − ψ = a_{0}(t, x)θ + a_{1}(t, x; θ_{t})θ^{2} + a_{2}(t, x; θ_{t}, θ_{t^{2}})θ^{3}
+ · · · · · ·
+[α(t, x)r(t) + â_{i}(t, x; θ_{t}, ⋯, θ_{t^{i}})]θ^{i+1} + a_{i+1}(t, x; θ_{t}, ⋯, θ_{t^{i+1}}; r)θ^{i+2}
+ · · · · · ·
+[β(t, x)s(t) + â_{j}(t, x; θ_{t}, ⋯, θ_{t^{j}}; r, ⋯, r_{t^{j−i−1}})]θ^{j+1}
+a_{j+1}(t, x; θ_{t}, ⋯, θ_{t^{j+1}}; r, ⋯, r_{t^{j−i}}; s)θ^{j+2}
+ · · · · · · .

The postulation gives us

−ψ_{t} = θ_{t}a_{0} + (∂_{1}a_{0} + 2θ_{t}a_{1})θ + (∂_{1}a_{1} + θ_{t^{2}}∂_{3}a_{1} + 3θ_{t}a_{2})θ^{2}
+ · · · · · ·

+[∂_{1}a_{i−1} + θ_{t^{2}}∂_{3}a_{i−1} + θ_{t^{3}}∂_{4}a_{i−1} + ⋯ + θ_{t^{i}}∂_{i+1}a_{i−1} + (i + 1)θ_{t}(αr + â_{i})]θ^{i}

+[α_{t}r + αr_{t} + ∂_{1}â_{i} + θ_{t^{2}}∂_{3}â_{i} + ⋯ + θ_{t^{i+1}}∂_{i+2}â_{i} + (i + 2)θ_{t}a_{i+1}]θ^{i+1}
+ · · · · · ·
+[∂_{1}a_{j−1} + θ_{t^{2}}∂_{3}a_{j−1} + ⋯ + θ_{t^{j}}∂_{j+1}a_{j−1}
+r_{t}∂_{j+2}a_{j−1} + ⋯ + r_{t^{j−i−1}}∂_{2j−i}a_{j−1} + (j + 1)θ_{t}(βr + â_{j})]θ^{j}
+[β_{t}s + βs_{t} + ∂_{1}â_{j} + θ_{t^{2}}∂_{3}â_{j} + ⋯ + θ_{t^{j+1}}∂_{j+2}â_{j}
+r_{t}∂_{j+3}â_{j} + ⋯ + r_{t^{j−i}}∂_{2j−i+2}â_{j} + (j + 2)θ_{t}a_{j+1}]θ^{j+1}
+ · · · · · ·

and similar θ-series for higher-order derivatives of ψ in t. Substituting into the Laurent series for u, we get

θ^{−k} = c(t, x − a_{0}θ − a_{1}θ^{2} − ⋯)(a_{0} + a_{1}θ + ⋯)^{−k}θ^{−k}

+u_{1}(t, x − a_{0}θ − ⋯, −θ_{t}a_{0} − (∂_{1}a_{0} + 2θ_{t}a_{1})θ − ⋯)(a_{0} + a_{1}θ + ⋯)^{1−k}θ^{1−k} + · · · · · ·

= a_{0}^{−k}cθ^{−k}

+ [−ka_{0}^{−k−1}a_{1}c + a_{0}^{1−k}u_{1} − a_{0}^{1−k}∂_{2}c] θ^{1−k}

+[−ka_{0}^{−k−1}a_{2}c + a_{0}^{2−k}u_{2}

+\frac{k(k + 1)}{2}a_{0}^{−k−2}a_{1}^{2}c + (k − 1)a_{0}^{−k}a_{1}∂_{2}c + \frac{1}{2}a_{0}^{2−k}∂_{22}c

+a_{0}^{−k}((1 − k)a_{1}u_{1} − ∂_{2}u_{1}) − a_{0}^{1−k}(∂_{1}a_{0} + 2θ_{t}a_{1})∂_{3}u_{1}]θ^{2−k} + · · · · · · ,

where c, u_{1}, u_{2}, ⋯, and their partial derivatives are evaluated at (t, x), (t, x, −a_{0}θ_{t}), (t, x, −a_{0}θ_{t}, −θ_{t^{2}}a_{0} − 2θ_{t}∂_{1}a_{0} − 2θ_{t}^{2}a_{1}), · · ·. By comparing the coefficients of both sides, we get

a_{0}^{−k}c  = 1,
ka_{0}^{−k−1}ca_{1} = a_{0}^{1−k}u_{1} − a_{0}^{1−k}∂_{2}c,
ka_{0}^{−k−1}ca_{2} = a_{0}^{2−k}u_{2} + \frac{k(k + 1)}{2}a_{0}^{−k−2}a_{1}^{2}c + (k − 1)a_{0}^{−k}a_{1}∂_{2}c + \frac{1}{2}a_{0}^{2−k}∂_{22}c
+a_{0}^{−k}((1 − k)a_{1}u_{1} − ∂_{2}u_{1}) − a_{0}^{1−k}(∂_{1}a_{0} + 2θ_{t}a_{1})∂_{3}u_{1},
⋮

ka_{0}^{−k−1}ca_{p} = a_{0}^{p−k}u_{p} + [ \begin{matrix}c, a_{0}, ⋯, a_{p−1}, u_{1}, ⋯, u_{p−1}, and their derivatives; \\ θ_{t}, ⋯, θ_{t^{p−1}}; \\ r, ⋯, r_{t^{p−i−1}}, if p > i; \\ s, ⋯, s_{t^{p−j−1}}, if p > j. \end{matrix} ] ,

⋮

where a_{i} = αr + â_{i}, a_{j} = βs + â_{j}, u_{i} = r, u_{j} = s, and [A, B, ⋯] denotes an expression involving A, B, · · ·. Note that since u_{1}, ⋯, u_{p} and their derivatives are evaluated at places determined by 1. a_{0}, ⋯, a_{p−1} and there partial derivatives;

2. t, x, θ_{t}, ⋯, θ_{t^{p−1}};

3. r, ⋯, r_{t^{p−i−1}} in case p ≥ i − 1;

4. s, ⋯, s_{t^{p−j−1}} in case p ≥ j − 1,

the recursive relation gives us the coefficients a_{∗} with variables as described in our postulation.

We emphasize that the essential condition we used to find the θ-series of x − ψ is our assumption on the orders of the derivatives of ψ, r and s appearing the in coefficients in the principal balance.

Now, we may take the derivative of the Laurent series for u and substitute in the θ-series of x − ψ, ψ_{t}, ψ_{t^{2}}, ⋯ to get following θ-series for u_{x}

u_{x} = −kc(x − ψ)^{−k−1} + (1 − k)u_{1}(x − ψ)^{−k} + (2 − k)u_{2}(x − ψ)^{1−k} + ⋯
= −ka_{0}^{−k−1}cθ^{−k−1}

+ [k(k + 1)a_{0}^{−k−2}a_{1}c + (1 − k)a_{0}^{−k}u_{1} + ka_{0}^{−k}∂_{2}c] θ^{−k}

+[k(k + 1)a_{0}^{−k−2}a_{2}c + (2 − k)a_{0}^{1−k}u_{2}

−\frac{k(k + 1)(k + 2)}{2}a_{0}^{−k−3}a_{1}^{2}c − k^{2}a_{0}^{−k−1}a_{1}∂_{2}c + \frac{k}{2}a_{0}^{1−k}∂_{22}c

−a_{0}^{−k−1}(k(1 − k)a_{1}u_{1} + (1 − k)∂_{2}u_{1}) − (1 − k)a_{0}^{−k}(∂_{1}a_{0} + 2θ_{t}a_{1})∂_{3}u_{1}]θ^{1−k} + ⋯,

and the θ-series for u_{x^{2}}. Moreover, up to adding an expression involving c, a_{0}, ⋯, a_{i−1}, u_{1}, ⋯, u_{i−1}, their derivatives, and θ_{t}, ⋯, θ_{t^{i−1}}, the coefficient of θ^{i−k−1} in the θ-series of u_{x} is of the form

k(k + 1)a_{0}^{−k−2}a_{i}c + (i − k)a_{0}^{i−k−1}u_{i} ∼ (k + 1)a_{0}^{−1}a_{0}^{i−k}r + (i − k)a_{0}^{i−k−1}r
= (i + 1)a_{0}^{i−k−1}r,

and the coefficient of θ^{i−k−2} in the θ-series of u_{x^{2}} is of the form

−k(k + 1)(k + 2)a_{0}^{−k−3}a_{i}c + (k − i)(k − i + 1)a_{0}^{i−k−2}u_{i}
∼ −(k + 1)(k + 2)a_{0}^{−2}a_{0}^{i−k}r + (k − i)(k − i + 1)a_{0}^{i−k−2}r
= (i + 1)(i − 2k − 2)a_{0}^{i−k−2}r.

We also have the similar formula for the coefficient of θ^{j−k−1} and θ^{j−k−2} in the θ-series for u_{x} and u_{x^{2}}.

In the coefficients of the θ-series for u_{x} and u_{x^{2}}, we may substitute the formulas for a_{1}, a_{2}, ⋯, and evaluate c, u_{1}, u_{2}, ⋯, and their partial derivatives at (t, x), (t, x, −a_{0}θ_{t}), (t, x, −a_{0}θ_{t}, −θ_{t^{2}}a_{0} − 2θ_{t}∂_{1}a_{0} − 2θ_{t}^{2}a_{1}), · · ·. The result is

u_{x} = −ka_{0}(t, x)^{−1}θ^{−k−1} + v_{1}(t, x, θ_{t})θ^{−k} + v_{2}(t, x, θ_{t}, θ_{t^{2}})θ^{1−k}
+ · · · · · ·
+[(i + 1)a_{0}(t, x)^{i−k−1}r(t) + v̂_{i}(t, x, θ_{t}, ⋯, θ_{t^{i}})]θ^{i−k−1}
+v_{i+1}(t, x, θ_{t}, ⋯, θ_{t^{i+1}}; r(t))θ^{i−k}
+ · · · · · ·
+[(j + 1)a_{0}(t, x)^{j−k−1}s(t) + v̂_{j}(t, x, θ_{t}, ⋯, θ_{t^{j}}; r(t), ⋯, r_{t^{j−i−1}}(t))]θ^{j−k−1}
+v_{j+1}(t, x, θ_{t}, ⋯, θ_{t^{j+1}}; r(t), ⋯, r_{t^{j−i}}(t); s(t))θ^{j−k}
+ · · · · · ·

and

u_{x^{2}} = k(k + 1)a_{0}(t, x)^{−2}θ^{−k−2} + w_{1}(t, x, θ_{t})θ^{−k−1} + w_{2}(t, x, θ_{t}, θ_{t^{2}})θ^{−k}
+ · · · · · ·
+[(i + 1)(i − 2k − 2)a_{0}(t, x)^{i−k−2}r(t) + ŵ_{i}(t, x, θ_{t}, ⋯, θ_{t^{i}})]θ^{i−k−2}
+w_{i+1}(t, x, θ_{t}, ⋯, θ_{t^{i+1}}; r(t))θ^{i−k−1}
+ · · · · · ·
+[(j + 1)(j − 2k − 2)a_{0}(t, x)^{j−k−2}s(t) + ŵ_{j}(t, x, θ_{t}, ⋯, θ_{t^{j}}; r(t), ⋯, r_{t^{j−i−1}}(t))]θ^{j−k−2}
+w_{j+1}(t, x, θ_{t}, ⋯, θ_{t^{j+1}}; r(t), ⋯, r_{t^{j−i}}(t); s(t))θ^{j−k−1}
+ · · · · · · .

We then truncate the θ-series of u_{x} at θ^{i−k−1} (where the resonance parameter r(t) first appears) by introducing a new variable ξ

u_{x} = −ka_{0}(t, x)^{−1}θ^{−k−1}+v_{1}(t, x, θ_{t})θ^{−k}+⋯+v_{i−1}(t, x, θ_{t}, ⋯, θ_{t^{i−1}})θ^{i−k−2}+ξθ^{i−k−1}. (51)

The new variable has the following θ-series

ξ = [(i + 1)a_{0}^{i−k−1}r + v̂_{i}] + v_{i+1}θ + ⋯ + [(j + 1)a_{0}^{j−k−1}s + v̂_{j}]θ^{j−i} + v_{j+1}θ^{j−i+1} + · · · .

By appropriate postulation and solving the subsequent recursive relation, we may convert the θ-series for ξ into a θ-series for r

r = (i + 1)^{−1}a_{0}^{k+1−i}[ξ − v̂_{i}(t, x, θ_{t}, ⋯, θ_{t^{i}})] + r_{1}(t, x, θ_{t}, ⋯, θ_{t^{i+1}}; ξ)θ
+ · · · · · ·
+[−(i + 1)^{−1}(j + 1)a_{0}^{j−i}s + r_{j−i}(t, x, θ_{t}, ⋯, θ_{t^{j}}; ξ, ⋯, ξ_{t^{j−i−1}})]θ^{j−i}
+r_{j−i+1}(t, x, θ_{t}, ⋯, θ_{t^{j+1}}; ξ, ⋯, ξ_{t^{j−i}}; s)θ^{j−i+1}
+ · · · · · · .

We update the θ-series for u_{x^{2}} by substituting the θ-series for r into the θ-series for u_{x^{2}}:

u_{x^{2}} = k(k + 1)a_{0}(t, x)^{−2}θ^{−k−2} + w_{1}(t, x, θ_{t})θ^{−k−1} + w_{2}(t, x, θ_{t}, θ_{t^{2}})θ^{−k}

+ · · · · · ·
+[(i − 2k − 2)a_{0}(t, x)^{−1}ξ + ˆ w̄_{i}(t, x, θ_{t}, ⋯, θ_{t^{i}})]θ^{i−k−2}
+w̄_{i+1}(t, x, θ_{t}, ⋯, θ_{t^{i+1}}; ξ)θ^{i−k−1}
+ · · · · · ·
+[(j + 1)(j − i)a_{0}(t, x)^{j−k−2}s + ˆ w̄_{j}(t, x, θ_{t}, ⋯, θ_{t^{j}}; ξ, ⋯, ξ_{t^{j−i−1}})]θ^{j−k−2}
+w̄_{j+1}(t, x, θ_{t}, ⋯, θ_{t^{j+1}}; ξ, ⋯, ξ_{t^{j−i}}; s)θ^{j−k−1}
+ · · · · · · .

Now, we truncate the θ-series of u_{x^{2}} at θ^{j−k−i} (where the resonance parameter s(t) first appears) by introducing a new variable η

u_{x^{2}} = k(k + 1)a_{0}^{−2}θ^{−k−2} + w_{1}θ^{−k−1} + ⋯ + w̄_{j−1}θ^{j−k−3} + ηθ^{j−k−2}.  (52)

The new variable has the following θ-series:

η = [(j + 1)(j − i)a_{0}^{j−k−2}s + ˆ w̄_{j}] + w̄_{j+1}θ + · · · .

The mirror transform (u, u_{x}, u_{x^{2}}) ↔ (θ, ξ, η) consists of u = θ^{−k}, (51), and (52). The transform converts the original Laurent series for u to a power series for θ, ξ, and η:

θ = a_{0}(t, ψ)^{−1}(x − ψ) + θ_{2}(t, ψ, ψ_{t})(x − ψ)^{2} + ⋯,
ξ = [(i + 1)a_{0}(t, ψ)^{i−k−1}r + ξ_{0}(t, ψ, ⋯, ψ_{t^{i}})] + ξ_{1}(t, ψ, ⋯, ψ_{t^{i+1}}; r)(x − ψ) + ⋯,
η = [(j + 1)(j − i)a_{0}(t, ψ)^{j−k−2}s(t) + η_{0}(t, ψ, ⋯, ψ_{t^{j}}; r, ⋯, r_{t^{j−i−1}})]
+ξ_{2}(t, ψ, ⋯, ψ_{t^{j+1}}; r, ⋯, r_{t^{j−i}}; s)(x − ψ) + · · · .

It is also easy to see that the mirror system is of the form

\{ \begin{matrix}θ_{x} & = & A(t, x, θ_{t^{p}}, ξ_{t^{p}}, η_{t^{p}}, p ≥ 0) \\ ξ_{x} & = & B(t, x, θ_{t^{p}}, ξ_{t^{p}}, η_{t^{p}}, p ≥ 0) \\ η_{x} & = & C(t, x, θ_{t^{p}}, ξ_{t^{p}}, η_{t^{p}}, p ≥ 0) \end{matrix}

where A, B and C are Laurent polynomials (i.e., polynomials in θ and θ^{−1}), with analytic functions of the other variables as coefficients.

As in the case of ODE, A, B and C are in fact polynomials in θ, so that they are analytic in all their variables. The proof is similar to the ODE case. The only new concern is to show that the arbitrariness of ψ(t), r(t), and s(t) implies the arbitrariness of the values of θ_{t}, ξ, and η and their derivatives in t along x = ψ(t).

Denote

X = a_{0}(t, ψ)^{−1},
Y = (i + 1)a_{0}(t, ψ)^{i−k−1},
Z = (j + 1)(j − i)a_{0}(t, ψ)^{j−k−2},

which we already know are nonzero. Then we find the following initial values at x = ψ(t)

θ(t, ψ) = 0,
θ_{t}(t, ψ) = −Xψ_{t},
⋮
θ_{t^{p}}(t, ψ) = −Xψ_{t^{p}} + [ψ, ⋯, ψ_{t^{p−1}}],
⋮
ξ(t, ψ) = Y r + [ψ, ⋯, ψ_{t^{i}}],
θ_{t^{i+1}}(t, ψ) = −Xψ_{t^{i+1}} + [ψ, ⋯, ψ_{t^{i}}; r],
⋮
ξ_{t^{p−i−1}}(t, ψ) = Y r_{t^{p−i−1}} + [ψ, ⋯, ψ_{t^{p−1}}; r, ⋯, r_{t^{p−i−2}}],
θ_{t^{p}}(t, ψ) = −Xψ_{t^{p}} + [ψ, ⋯, ψ_{t^{p−1}}; r, ⋯, r_{t^{p−i−1}}],
⋮
η(t, ψ) = Zs + [ψ, ⋯, ψ_{t^{j}}; r, ⋯, r_{t^{j−i−1}}],
ξ_{t^{j−i}}(t, ψ) = Y r_{t^{j−i}} + [ψ, ⋯, ψ_{t^{j}}; r, ⋯, r_{t^{j−i−1}}; s],
θ_{t^{j+1}}(t, ψ) = −Xψ_{t^{j+1}} + [ψ, ⋯, ψ_{t^{j}}; r, ⋯, r_{t^{j−i}}; s],
⋮

Since X, Y, Z are nonzero, the values of θ_{t}, ξ, η and their derivatives in t along x = ψ(t) are bi-holomorphically equivalent to the values of ψ_{t}, r(t), and s(t) and their derivatives in t. This is sufficient for us to conclude the analyticity of the functions on the right side of the mirror system.

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