This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type
This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type A n , n ∈ N with ϵ sign: 2 (w i ) tt = ϵ e 2(wi+1-wi) -e 2(wi-wi-1) , ϵ = ±1, i ∈ Z,
(1.0.1)
where w i : C \ {0} → R and subject to the conditions w i = w i+n+1 (periodicity) , w i = w i (|t|) (radial condition) , w i + w n-i = 0 (anti-symmetry) .
Since Mikhailov-Olshanetsky-Perelomov [Mik81,MOP81] proposed the two dimensional generalization of the classical Toda lattice equation on the affine root systems, the 2D Toda equations have been an important integrable system with many aspects. Among various boundary conditions and underlying root systems, our focus is on the 2D periodic Toda equation of the affine root system A n , which has been studied extensively from both mathematical and physical perspectives. When ϵ = 1 in (1.0.1), for example, it can be interpreted as the equation for primitive harmonic maps taking values in a compact flag manifold (see [BP94,BPW95]). Such maps are closely related to harmonic maps into symmetric spaces. These, in turn, have numerous geometrical interpretations, such as surfaces in R 3 of constant mean curvature (see [Dor08]) or special Lagrangian cones in C 3 (see [McI03,Joy08]). In contrast, when ϵ = -1 in (1.0.1), it is an example of the tt * (topological-anti-topological fusion) equations introduced by Cecotti and Vafa [CV91,CV92,CV93] to describe certain deformations of super-symmetric quantum field theories. In their context, imposing radial symmetry and anti-symmetry for the solution of (1.0.1) is natural. Due to its Toda structure, (1.0.1) with ϵ = -1 is known as the tt * -Toda equation.
The project is a continuation of the works [GL12, GL14, GIL15a, GIL15b, GIL20, GIL23] devoted to the case ϵ = -1. Of special importance are the global solutions of the tt * -Toda equation first predicted by Cecotti and Vafa, i.e. the solutions which are smooth for all 0 < |t| < ∞. A comprehensive analysis of these solutions was performed in [GL12,GIL23]. Earlier, a class of the solutions of the tt * -Toda equation was introduced and their behavior at t = 0 and t = ∞ had been evaluated by Tracy and Widom in [TW98]. As it follows from the comparison of the results of [TW98, GL12, GIL23], the Tracy-Widom solutions turned out to be exactly the Cecotti-Vafa global solutions of (1.0.1).
The ultimate goal of the current project is to study the general radial solution of (1.0.1) with ϵ = -1. The principal issue is the connection formulae between the asymptotic parameters describing the behavior of the solution at t = 0 and t = ∞. In [GL12,GIL23] the problem was solved for the global solutions. The technique used was a fusion of the Iwasawa factorization in the loop group theory and the Riemann-Hilbert nonlinear steepest descent method of Deift and Zhou [DZ93] which is applicable to (1.0.1) in view of its Lax-integrability (see more details in the next section).
The extension of the methods of [GL12,GIL23] to the general families of solutions is not trivial. The main problem is that the generic solution of (1.0.1) with ϵ = -1 is singular in the neighborhood of both t = 0 and t = ∞. This fact creates very serious technical challenges for the Riemann-Hilbert approach. Hence the idea is to try to obtain first the connection formulae for the general solution of (1.0.1) with ϵ = 1. The obvious reason is that for this sign of ϵ, every solution is smooth for all 0 < |t| < ∞ (again, the details are given in the next section).
In this paper, we shall consider the case ϵ = 1 and n = 2. The last restriction is not of principal importance. Indeed, the associated Riemann-Hilbert problem, as we will see, is of matrix rank 3. That is, we are already beyond the usual and well-developed rank 2 Riemann-Hilbert setting, and the n = 2 case actually captures the principal features of the general n case. At the same time, we think that it makes sense to first present all the details assuming n = 2. Indeed, the transition to n > 2, which we will do in a subsequent paper of the series, will be then much easier to follow. This is the same strategy as the one used in [GL12,GIL23]: first to consider n = 3 [GL12, GIL23, GIL20] and then extend the analysis to general n [GIL23].
Introduce the variable,
Then, equation (1.0.1) becomes (w i ) xx + 1 x (w i ) x = ϵ 2e 2(wi+1-wi) -2e 2(wi-wi-1) , i ∈ Z.
(1.0.2)
When n = 1, the periodicity condition implies that there are only two functions involved: w 0 and w 1 . Moreover, the anti-symmetry yields w 1 = -w 0 , and hence the system (1.0.2) reduces to a single equation on w 0 ,
or
This is a particular case of the third Painlevé equation whose connection problem for a special global solution, in the case ϵ = -1, was first solved by McCoy et al in [MTW77] and was used there to describe in detail the transition regime in the Ising model. A complete solution of the connection problem for the general solution of (1.0.3) was obtained in the series of works [Nov84, Nov85, Kit87a, Nov07] (see the monograph [FIKN06] for more on the history of the question).
In the case n = 2, which is the subject of this paper, one has w 1 = 0, w 2 = -w 0 , and hence the system (1.0.2) again becomes a single equation for w 0 ,
This equation is known as the radial version of the Bullough-Dodd equation or Tzitzeica equation (see [LM16] and references therein), and it is yet another particular case of the Painlevé III equation. The connection formulae for this equation have been studied by Kitaev in [Kit87b]. (At the end of this introduction, we will say more about Kitaev's results.) When ϵ = 1 in (1.0.4), we obtain
(1.0.5) which we call the radial Toda equation (with n = 2) in our context. Our main result is the following theorem:
Theorem 1.1. For every γ ∈ (-1/2, 1) and every ρ ∈ R, there exists a unique real-valued, smooth for all x > 0, solution of equation (1.0.5) such that w 0 (x) = γ ln x + ρ + o (1) , x → 0.
(1.0.6)
The large x behavior of this solution is described by the asymptotic formula,
(1.0.7)
The connection formulae, i.e. the expression of σ and ψ in terms of γ and ρ, are given by the equations
(1.0.9) where X = 1 2π ln 1 8 cos π (1-γ) 3 sin 2 π (1-γ)
Moreover, every solution of (1.0.5) can be characterized by the asymptotic behavior at x = 0 given by (1.0.6) for some γ ∈ (-1/2, 1) and ρ ∈ R (and, consequently, behaving at infinity according to (1.0.7) for some real σ and ψ).
As already mentioned above, the 2D periodic Toda equation is Lax-integrable due to Mikhailov [Mik81]. In particular, this means that its radial version (1.0.2) describes the isomonodromic deformations of a certain (n + 1) × (n + 1) system of linear differential equations. In the case n = 2, which we study in this paper, this is a 3 × 3 linear system (2.1.1) presented in the next section. The monodromy data of this linear system constitute the first integrals of the nonlinear equation (1.0.5). The connection problem is solved as soon as the asymptotic behaviors of w 0 (x) at x = ∞ and x = 0 are explicitly described in terms of these integrals. We solve the first problem by obtaining the explicit formulae for the parameters X and α in terms of the monodromy data of the linear system (2.1.1), in section 3, using the extension of the Deift-Zhou steepest descent method to the associated 3 × 3 matrix Riemann-Hilbert problem. The second problem is the derivation of the explicit formulae for the parameters γ and ρ in terms of the same monodromy data that is done in section 4 with the help of Iwasawa Factorization. Combining these yields the direct connection formulae between the asymptotic parameters (σ, ψ) at infinity and the asymptotic parameters (γ, ρ) at zero which are given in theorem 1.1. Section 4 essentially borrows the technique already developed in [GIL20]. Section 3 represents our main technical development-a higher-rank version of the Deift-Zhou nonlinear steepest descent method which we will use in the following paper of this series where the case of general n will be studied.
The fact that we have accounted for all solutions of (1.0.5), i.e. the proof of the last statement of theorem 1.1, follows from the above computations-this is explained at the end of section 4.
The 'first half' of theorem 1.1, i.e. the asymptotic formulae (1.0.7), (1.0.8), (1.0.9) at x = ∞ with the parameters X and α given in terms of the monodromy data of the linear system (2.1.1) had already been obtained in [Kit87b]. Also, in [Kit87b] a complete connection formula for the one-parameter family of the global solution in the case ϵ = -1 was presented. The important difference between our approach and the one by [Kit87b] is that the latter is based on the WKB asymptotic solution of the associated direct monodromy problem. Unlike [Kit87b], our paper gives an alternative derivation and proof of the connection formulae based on the asymptotic solution of the inverse monodromy problem via the Deift-Zhou nonlinear steepest descent method. As already stressed, a key methodological point of our work is that in order to prove theorem 1.1, we need to develop an extension of the Deift-Zhou method in the case of a Riemann-Hilbert problem whose matrix rank is higher than 2.
In the smooth case, with ϵ = 1, a complete description of the connection formulae had been obtained in [KV04]. In fact, in this work a much broader class of the complex and generally singular solutions of (1.0.5) is studiedfoot_0 . The authors of [KV04] use an alternative, 2 × 2 Lax pair for (1.0.5) and they apply again the WKB-based technique of solution of the associated direct monodromy problem. The 2 × 2 Lax pair used in [KV04] has no analog for the general n case. Hence the methods of [KV04] can not be immediately extended to the radial Toda equation for general n, while our 3 × 3 based approach is easily, at least in principle, generalized to the n × n case. It should also be mentioned that matching our theorem 1.1 with the formulae of [KV04] is an outstanding issue since in [KV04] a different monodromy parametrization is used, and the authors do not extract from their formulae the exact equations directly relating the asymptotic parameters at 0 and ∞. Surely, this, though necessarily somewhat cumbersome, would be possible to carry out if needed.
Our paper is, of course, not the first one where the nonlinear steepest descent method has been extended to the higher-rank Riemann-Hilbert setting. In particular, close to our Riemann-Hilbert problem (but not coinciding with it), some 3 × 3 Riemann-Hilbert problems have been analyzed in the papers [BdMS13, BdMLS19, CL21, CLW23].
In this section, we will show that every solution of the radial Toda equation (1.0.5) has the corresponding monodromy data. To this end, we give the Lax pair representation of (1.0.5) and then describe all the monodromic properties.
Before getting into this point, we shall prove that (1.0.5) is equivalent to a certain Painlevé III equation. Indeed, if we introduce the change of variables
then one can readily verify that (1.0.5) is transformed into
which is a special case of the Painlevé III (D 7 ) equation, and vice versa.
Proposition 2.1. Every solution w 0 (x) of (1.0.5) is smooth on the positive real line.
Proof. From the Painlevé property, a solution of the equation (2.0.1) could have only a pole on R >0 , i.e. w(s) is of the form,
where s 0 ∈ R >0 . This implies that w 0 (x) might have a singularity only of the form,
where x 0 ∈ R >0 , but one can show that the ansatz (2.0.2) is not compatible with equation (1.0.5) for real a, x 0 , and b.
The radial Toda equation (1.0.5) admits a Lax pair representation:
where
The compatibility condition of (2.1.1) gives (1.0.5).
If we take any solution w 0 (x) of the radial Toda equation (1.0.5) and the corresponding solution Ψ(x, ζ) of (2.1.1) and put
then Ψ will satisfy Mikhailov's Lax pair (see [Mik81]) for the 2D periodic Toda equation (1.0.1) of type A 2 with ϵ = 1:
where w 0 (x) in (2.1.2) is replaced by w 0 (t, t) := w 0 (|t|) to obtain w and W in (2.1.3). Then, the compatibility condition of (2.1.3) is equivalent to the PDE 2 (w 0 ) tt = e -2w0 -e 4w0 .
We shall say more in section 4 about the relationship between these two Lax pairs.
Following the general theory of ODEs with singular points (see for instance chapter 1 of [FIKN06]), we will obtain the monodromy data for the first equation of system (2.1.1).
We start with the formal solutions of the equation
This equation has two irregular singularities at ζ = 0 and at ζ = ∞. The leading term of (2.2.1) at ζ = 0 is -W, and it can be diagonalized as follows:
-W = P 0 (-d 3 ) P -1 0 where
Moreover, matrix P 0 has the following decomposition
where we defined
By proposition 1.1 of [FIKN06], one can obtain a unique formal solution of (2.2.1) at ζ = 0 of the form
(2.2.2)
Similarly, we can obtain the formal solution at ζ = ∞. After the transformation ζ → 1 ζ , equation (2.2.1) becomes
The leading term is x 2 W T , and W T can be diagonalized as follows:
Again, by proposition 1.1 of [FIKN06], one can obtain a unique formal solution of (2.2.1) at ζ = ∞ of the form
(2.2.3)
Remark. Observe that ω 2 + ω + 1 = 0, thus
where
Since C 2 = I, we have
(2.2.4)
Next, we describe the Stokes rays and Stokes sectors of (2.2.1). We start with ζ = 0. Recall that Stokes sectors are defined so that the fundamental solution is uniquely determined there by the asymptotic solution Ψ (0) f . To begin with, suppose we have two fundamental solutions Ψ (0) and Ψ(0) with the same asymptotics at ζ = 0. Then their ratio
To obtain the unique fundamental solution in some sector Ω (0) n , we need to have Since Ψ (0) and Ψ(0) have the same asymptotics at ζ = 0, it holds that
Thus, to get
n to contain at least one Stokes ray which is defined by
for each pair (i, j), i < j. More precisely, Stokes rays are
for n ∈ Z. Now we define Stokes sectors by those which contain exactly one Stokes ray for each superscript (i, j) with i < j, i.e. Ω (0)
(2,3) n for each n ∈ Z. Thus, one can take the following Stokes sectors (see figure 1):
Similarly, we can define the Stokes sectors at ζ = ∞. First, we find the Stokes rays at ζ = ∞:
Then, the Stokes sectors at ∞ are
Moreover, we have
(2.2.5)
Strictly speaking, sectors Ω (∞,0) n should be considered as lying on the universal covering C * .
Next, we define the canonical solutions. According to the general theory (see e.g. theorem 1.4 in [FIKN06]), in each sector Ω (∞,0) n where n ∈ Z, there exist unique solutions Ψ (∞,0) n of (2.2.1) satisfying the asymptotic condition,
It also should be noticed that
(2.2.6)
Remark. Another fundamental fact following from the general theory is that every solution of (2.2.1), and, in particular, all our canonical solutions Ψ (∞,0) n admit analytical continuation on the whole universal covering C * of C * = C \ {0}. Relations (2.2.6) should be then understood as equations on C * where we accept a somewhat 'colloquial' convention to denote the point on C * and its projection on C * by the same letter ζ. We also note that, in the case of C * , we can write that
so that the canonical 'altering sheets' involution β on C * can be indicated in short writing as
With this convention, the rigorous version of relations (2.2.6) reads,
In what follows we shall however use a 'colloquial' form (2.2.6) of this relation, but when talking about Ψ (∞,0) n (ζ) will always recognize that arg ζ matters.
The Stokes matrices are defined by
where n ∈ Z. By the analytic continuation property, it holds that
n-2 . Thus, for our purposes to find monodromy data, it is enough to focus on choices n = 1, 2.
Lemma 2.2. Stokes matrices have the following structures,
Proof. We only give proof for S (∞) 1
. The same approach can be applied to determine the structure of other Stokes matrices.
By the definition of the Stokes matrices (2.2.7) we have
n+1 , it can be further written as
And it follows that
Thus equality (2.2.8) holds if S (∞) 1
have the following structure
Lemma 2.2 describes the structure of S (∞,0) n . In the next subsections, we will discuss some symmetries of these matrices that will help us to parameterize each S (∞,0) n .
In this section, we consider the antisymmetry relations of the formal and canonical solutions and the Stokes matrices at ζ = ∞, and the inversion symmetry relations of those at ζ = 0.
Let A(ζ) be the coefficient matrix of (2.2.1):
Then, one can observe the following symmetry:
where
Moreover, (2.2.9) implies the following lemma:
Taking a transposition and changing ζ → -ζ, we have
) and definition of Ψ.
One can observe that
Together with lemma 2.3 this leads to the anti-symmetry relation for the formal solution:
Moreover, considering the Stokes sectors of each canonical solution (see figure 2), we have that (2.2.11) implies the anti-symmetry of canonical solutions at ζ = ∞:
(2.2.12)
In terms of Stokes matrices, this can be interpreted as follows.
Lemma 2.4. One has
(2.2.13) Proof.
T-1 d 3 3 by (2.2.12)
The matrix A(ζ) admits another symmetry, namely
For this, we have the following lemma.
Lemma 2.5. If Ψ(ζ) is a solution of (2.2.1), then so is Ψ(ζ) := ∆Ψ -1 x 2 ζ . On the level of Stokes matrices, it gives the following result.
Lemma 2.6. One has
(2.2.14)
Proof. From lemma 2.5 one can obtain that the corresponding formulae for the formal solutions are
and for the canonical solutions:
where n = 1, 2. In terms of Ψ (0) k it can be read as
(2.2.17) which gives the corresponding formulae for the Stokes matrices.
The matrix A(ζ) has what is called cyclic symmetry, namely
Using this symmetry, we obtain the next lemma:
Let us introduce
Then lemma 2.7 together with equalities
gives the following formula for the formal solution:
(2.2.18)
Similarly, for ζ → 0, we have
which follows from the equalities
.
In order to formulate the cyclic symmetry of the fundamental solutions we need a collection of Stokes sectors compatible with this symmetry.
First, we consider the case ζ = ∞. For n ∈ 1 3 Z we introduce additional sectors
.
In these sectors we consider fundamental solutions
1+n .
Then one can show that the cyclic symmetry relations of the formal solutions (2.2.18) translates to
for n ∈ 1 3 Z. Similarly, near ζ = 0, for n ∈ 1 3 Z we introduce additional sectors
which is consistent with the relations (2.2.5) we had before. In these sectors we consider fundamental solutions
where n ∈ 1 3 Z. Then the cyclic symmetry relations (2.2.19) translates to
for n ∈ 1 3 Z.
It is not straightforward to formulate the cyclic symmetries for the Stokes matrices. We will obtain them in several steps; first, we define jump matrices for π/3 rotated fundamental solutions.
Lemma 2.8. For n ∈ 1 3 Z, consider constant matrices
Again, from the analytic continuation property, it holds that Q
n-2 and it is enough for us to consider n = 1, 2 only. These jump matrices have the following structure,
Proof. We will prove the structure for
only. The same approach can be applied to determine the structure of other jump matrices.
For
, equation (2.2.22) can be written as
and
and
has the following structure
Similarly for n ∈ 1 3 Z, we consider constant matrices
Using the same reasoning as for (2.2.13) and (2.2.14), one can obtain the following symmetry relations for Q
where n ∈ 1 3 Z. Observe that from (2.2.24) it follows that
n has the same structure as
Lemma 2.9. For n ∈ 1 3 Z, we have the following cyclic symmetry relations of jump matrices Q
(2.2.25)
(2.2.26)
Proof. From (2.2.20) we have
. Other cases can be proved similarly.
Finally, the Stokes matrices
(2.2.28)
, we have
. Other cases can be proved similarly.
In the past two subsections, we formulated many symmetry equations that will help us to parameterize the monodromy data. Let us list them here.
• Symmetry relations for formal solutions (2.2.11), (2.2.15), (2.2.18) and (2.2.19):
• Symmetry relations for canonical solutions (2.2.12), (2.2.16), (2.2.17), (2.2.20) and (2.2.21):
• Symmetry relations for Jump matrices (2.2.24)-(2.2.26):
Then, using the anti-symmetry and inversion symmetry relations (2.2.24) and the cyclic symmetry of Jump matrices (2.2.25) and (2.2.26), we have
(2.2.29) and
(2.2.30)
Using (2.2.24) once more, we have
(2.2.31) Therefore, each Stokes matrix is parameterized by a single parameter a-Stokes data.
Next we define the connection matrices E k which connects solutions at ζ = ∞ and solutions at ζ = 0 by
(2.2.32) for all ζ in the universal covering C * .
Proposition 2.11. One has
.2.35) As a corollary of (2.2.34) and (2.2.35), we have E 1 = E T 1 . Proof. For the first equality of (2.2.33): By definitions of the Stokes matrices and connection matrices,
for all ζ ∈ C * . Similarly, using the definition of the Stokes matrices, we have
By (2.2.32), it becomes
For the second equality of (2.2.33): By definitions of the Stokes matrices and connection matrices,
Thus, it follows that
For the anti-symmetry relation (2.2.34): From (2.2.12), we have
(2.2.38) Now, using (2.2.17):
(2.2.38) can be proceeded as follows,
Comparing it with the definition of the connection matrices, we get
For the inversion symmetry relation (2.2.35): From (2.2.16) and definition of E 1 , we have
(2.2.39)
If we use (2.2.16) once more with (2.2.6), we could obtain
.2.40) From (2.2.39) and (2.2.40), it follows that
Thus, we have
Remark. Using the anti-symmetry relation of E 1 ,
we can calculate the determinant of E 1 . By (2.2.14), we have
Taking a determinant of previous equality, we get det
where we used det
We can also determine the sign of det E 1 . Taking a determinant of (2.2.32), we have
Using the Jacobi's formula
and the fact that tr
By definition of the jump matrices Q
Using the cyclic symmetry relation, (2.2.26) and (2.2.21), it follows that
and
Substituting (2.2.43) and (2.2.44) into (2.2.32), we have
Thus, we obtain that
Moreover, using the symmetry relation (2.2.24), equation (2.2.45) becomes
One can verify that d -1
This is what we call the cyclic symmetry relation of E 1 .
In summary,
• Anti-symmetry and inversion relation of E 1 (2.2.41) and (2.2.35):
• Cyclic symmetry relation of E 1 (2.2.46):
Recall that we considered only real solutions w 0 (x) ∈ R of the radial Toda equation (1.0.5). This implies that the coefficient matrix A(ζ) of (2.2.1):
, which gives us the following Lemma.
Lemma 2.12. If Ψ(ζ) is a solution of (2.2.1), then so is Ψ( ζ).
Recall that the formal solution at ζ = ∞ is given by (2.2.3), so we have
where we used the property C 2 = I, the definition P ∞ = e w Ω -1 , and equality
Observe that by (2.2.4), we have
Thus, we get
Therefore, we obtain the reality condition of the formal solutions:
(2.3.1)
Similarly, one can show that the formal solution at ζ = 0 has the same property:
Next, we will describe the reality condition on the level of the canonical solutions and the Stokes matrices. Recall that the Stokes sectors for the canonical solution at ζ = ∞ are given by:
By taking complex conjugate of ζ, i.e. arg ζ = 2π -arg ζ, the Stokes sectors for the solutions
Thus, we have
,
, and the relations for the canonical solutions are
(2.3.3) Finally, the reality conditions for the Stokes matrices are
(2.3.4)
Recall that we have parametrization of S (∞)
given by (2.2.29) and of S (∞) 2
, given by (2.2.30). Using these parametrizations, (2.3.4) becomes
Thus, introducing a real parameter s R , one can write a as a = ω 2 s R .
(2.3.5)
In this section, we will describe the reality condition for the connection matrix E 1 . Since it does not contain ζ, we have freedom in the definition of the complex conjugate of ζ. Let arg ζ = -arg ζ this time. Then, the Stokes sectors have the property,
and
which implies the following reality conditions of the canonical solutions,
(2.3.6) By (2.2.32) and (2.3.6), it follows that
(2.3.7)
In summary,
• reality condition of the formal solutions (2.3.1) and (2.3.2):
• reality condition of the canonical solutions (2.3.3):
• reality condition of the Stokes matrices (2.3.4):
• reality condition of the connection matrices (2.3.7):
In this section, we will use symmetry relations to parameterize the connection matrix E 1 and will show that the monodromy data consists only of two real parameters.
Proposition 2.13. The connection matrix E 1 can be parameterized by two real numbers A R and s R , and one complex number B:
where the parameter s R was introduced in (2.3.5) and parameters A R , s R , B satisfy
The proof of this proposition is given in appendix A.
Remark. In the following paper [GIK + ] in this series we shall show that the case A R < 0 corresponds to having a complex solution to (1.0.5) of the form w 0 = v 0 + i π 2 where v 0 is real.
Hence we must have A R > 0 for any real solution of our equation (1.0.5). Together with (2.3.9) it gives A R ⩾ 1 3 .
Summarizing, we have that all Stokes matrices S (0,∞) n and connection matrices E n for n = 1, 2 depend on the parameters (A R , B, s R ).
Proposition 2.14. The data (A R , B, s R ) can be further parameterized by two real parameters {s R , y R }, where s R satisfies -3 < s R < 1 and y R is arbitrary. In fact,
where y R = Im(ωB).
The proof of this proposition will be given in appendix B.
Recall that we called the parameter s R Stokes data. Together with the new data y R coming from connection matrix, we introduce the set of monodromy data M by
(2.3.11)
Recall that Ψ (∞,0) n were originally defined on the Stokes sector Ω (∞,0) n and then were extended to the universal covering C * of C * as was explained in the section 2.2.3. In this section instead we consider Ψ Then, we will reconstruct the coefficient A(ζ) in the equation (2.2.1) (and so the solution w 0 (x) of (1.0.5)) from the given monodromy data.
This inverse monodromy problem forms a Riemann-Hilbert Problem. Ψ has jumps only on Γ 1 with the jump matrices shown in figure 3. For example • For ζ on the outer part of the arg ζ = π 2 ray,
At the self-intersection of Γ 1 , points ζ = i ρ and ζ = -i ρ in figure 3, by proposition 2.11, we have
In particular, we have no formal monodromy around these points. From (2.2.2) and (2.2.3) the asymptotic behaviour of Ψ(ζ) is
It is important to emphasize that Ψ(ζ) is defined as a piecewise analytic function on the complex plane C * and not on its universal covering; i.e. for Ψ(ζ) we do not need to indicate arg ζ.
In the next subsection, we will deform the original Ψ-problem by a series of transformations.
First, we define a Φ-problem by figure 4. The function Φ is a piecewise analytic function whose analytic pieces are obtained from the function Ψ by the proper right matrix multiplication as shown in figure 4. This transformation comes from the decomposition of the Stokes matrices proved in equations (2.2.27) and (2.2.28). Note that our new oriented jump contours are the unit circle and infinite rays with nπ 3 angles and we call it Γ 2 . One can check that the new jump matrices on Γ 2 for this problem are the ones written in figure 4 Next, we define a Φ-problem by
One can check that this transformation does not affect the contour but the jump matrices will be changed:
• Jumps on the ray outside of S 1 ρ do not change. • Jumps on the ray inside of
The jump matrices inside the circle can be further simplified. For example,
One can repeat a similar discussion for the others. Next, introducing Ẽ-1 n = 1 3 E -1 n C for n = 1, 2, the jump matrices on S 1 ρ can be also simplified, for instance,
. The updated jump matrices on Γ 2 are depicted in figure 5. By definition of Φ(ζ) and (2.2.3), we observe that this transformation does not change the asymptotics at ζ = ∞ : Φ (ζ) = P ∞ I + O ζ -1 e -x 2 ζd3 , (3.1.1) but at ζ = 0, definition of Φ(ζ) and (2.2.2) imply the following change:
3 .
(3.1.2)
Recall that ρ in the original RHP was arbitrary. We now set ρ = where
3) is just a scaling, the jump contour does not change from Γ 2 as in figure 5; the only difference is that previously the circle has a radius ρ > 0, but now it is a unit circle. However, jump matrices do change from G Φ to G Y :
For convenience, let us define
The normalization condition of this problem becomes the following:
by (3.1.2) and the fact P ∞ = e w Ω -1 .
Therefore we pose the following RHP.
RHP 1. Find a matrix-valued function Y satisfying the following conditions
• The jump conditions are
• The normalization condition is
Here the contour Γ 2 and jump matrices G Y are depicted in figure 6.
Our goal is to solve the Y-problem asymptotically as x → ∞. To this end, we will transform the Y-problem (if needed) to the one where jump matrices are close to the identity matrix with respect to the L 2 ∩ L ∞ norm so that one can apply the small norm theorem (see [FIKN06], theorem 8.1) and prove the asymptotic solvability of the RHP. Let us first analyze matrix G Y .
Let
(3.2.1) By (3.1.4), we have
e -xφ2 * e xφ1 * * e -xφ3 * e xφ2 * e xφ3 * , (3.2.2) where * do not depend on ζ. To understand the sign-change of the real part of each exponent in (3.2.2), we shall compute the stationary points. First, consider φ 1 . We have
The zero level curve and the sign of Re(φ 1 (ζ)) are shown in figure 7.
Next, consider φ 2 , we have and
The zero level curve and the sign of Re(φ 2 (ζ)) are shown in figure 8. Finally, consider φ 3 ; we have
and
The zero level curve and the sign of Re(φ 3 (ζ)) are shown in figure 9.
Using figures 7-9, one can verify that the jump matrices of the Y-problem on each ray approach the identity matrix in the L 2 ∩ L ∞ norm as x → ∞. For instance,
and from figure 7 we have that ||ae -xφ1(ζ) || → 0 when ζ belongs to the region described by figure 6. All other jumps on the rays can be checked similarly.
The remaining jump matrices to be considered are the ones on the unit circle, i.e. conjugated connection matrices. Since G Y has structure (3.2.2) and each exponent is purely imaginary on the unit circle by (3.2.1), we see that they are oscillating (no decay), so we cannot apply the small norm theorem immediately to the Y-problem. To overcome this issue, we need to make further modifications.
First, we decompose the jump matrices of the Y-problem as follows
so that Ľk and Řk have diagonal entries equal to 1 and non-diagonal entries exponentially decaying as x → ∞, and D k are some constant diagonal matrices. We discuss it in section 3.2.3. Then, one can transform the Y-problem to the following Y-problem defined in figure 10. Such procedure is called opening lenses. We call the new oriented contour Γ 3 and the jump matrices GY. After this decomposition all jump matrices GY except D k 's tend to the identity matrix as x → ∞.
RHP 2. Find a matrix-valued function Y satisfying the following conditions
• The jump conditions are
• The normalization condition is
Next, we solve a model RHP that has jumps only on the unit circle and the corresponding jump matrices are D k 's. We call its solution a global parametrix YD (ζ) and discuss it in section 3.2.5. Then the function Y(ζ) YD -1 (ζ) will have jump matrices which tend to the identity as x → ∞. However, the convergence of such jump matrices to the identity matrix is not in the L ∞ norm if ζ is near the stationary points. This suggests constructing other model RHPs near the stationary points. We call their solutions local parametrices P (p k ) (ζ) where p k 's are stationary points and discuss them in sections 3.2.6 and 3.2.7. Finally, in section 3.3.1 we consider a small norm RHP for an error function R(ζ) by comparing Y(ζ) with its parametrices.
Remark. In analogy with the classical steepest descent method, the main contribution to the asymptotics of Y(ζ) as x → ∞ comes from the unit circle and from the neighborhood of the stationary points.
The structures of the Ľk and Řk in (3.2.3) are dictated by the Re(φ 1 ), Re(φ 2 ), and Re(φ 3 ). For instance, Ľ1 and Ř1 in Ě-1 1 = Ľ1 D 1 Ř1 , need to be of the form
since the non-diagonal entries are exponentially decaying when ζ belongs to the corresponding regions in figure 10, as x → ∞. All the other structures of Ľk and Řk can be obtained similarly. One can verify that indeed such structured matrix decompositions exist by direct computations written in appendix C. Thus, we obtain the following proposition. (3.2.5)
Then, one has
Let us name the arcs of the unit circle in figure 11 by C k and the intersection points of the unit circle with the rays by p k where k = 1, 2, • • • , 6. Thus,
The following sections of model RHPs consist of three parts. First, we solve a model RHP 3 on the unit circle S 1 = 6 k=1 C k to get the global parametrix YD . Then, we consider a model RHP near p 1 to get a local parametrix P (1) (ζ). Lastly, we obtain the other local parametrices:
and symmetry relations.
Consider the following model RHP for the global parametrix.
RHP 3. Find a matrix-valued function YD (ζ) satisfying
• On the unit circle oriented counterclockwise, the following jump condition is valid:
where the jump matrices are
Theorem 3.2. The solution of the RHP 3 is
where
,
We check that YD (ζ) defined by (3.2.7) satisfies RHP 3.
(i) Observe that each f k (ζ) is a composition of a linear fractional transformation and a power function:
where p k and p k+1 are the endpoints of the arc C k (see figure 11), z k = -1 2π i ln D k . Such g k maps the arc C k to the ray from the origin with angle 5π/6. For each k, we select this ray as the branch cut of the complex logarithm to obtain
where L 5π/6 (z) := ln |z| + iA 5π/6 (z) and A 5π/6 (z) is the argument of z that belongs to (-7π/6, 5π/6] (we will use the fact that L 5π/6 1 2 -√ 3 2 i = -π i 3 later). This proves that each f k is holomorphic everywhere except the arc C k , and therefore, their product,
is holomorphic everywhere except S 1 . (ii) Then, one can readily check that L 5π/6,+ (z) = L 5π/6,-(z) -2π i on the ray {z ∈ C | arg(z) = 5π/6} oriented from the origin to infinity. Thus, it holds that
on each C k . This immediately shows that the jump condition is fulfilled since all matrices are diagonal and commute. (iii) Finally, as L 5π/6 (1) = 0, it holds that f k (∞) = I for each k, which shows that the normalization condition is satisfied and finishes the proof.
Remark. In the neighborhood of p 1 = 1, we consider a small disk U 1 centered at that point.
Introduce the following regions (see also figure 12):
Originally, we chose C \ C k as the domain for a branch f k . But here we let (ζ -1) ν with ν = -1 2π i ln 3A R be a branch with the cut {ζ ⩽ 1} where we take arg(ζ -1) ∈ (-π, π]. Then the global parametrix YD (ζ) admits the following representation in the neighborhood
where Θ(ζ) is locally analytic for ζ ∈ U 1 :
We are interested in finding a local parametrix P (1) (ζ), such that
• on the boundary ∂U 1 , we have the following matching relation
(3.2.9)
Recall that the contour and the jump matrices of the Y-problem near ζ = p 1 (Note that p 1 = 1 from (3.2.6); we might switch from one to another but they both mean the same point) are the ones in figure 13, where
and the matrices L k , R k are given in proposition 3.1.
Observe that, if we locally define Yp1 (ζ) by the transformation shown in figure 14(a), where
then we obtain a new contour and jump matrices shown in figure 14(b). There we introduced
and s 1 = ω 2 B A R . Note that A R cannot be zero (see (2.3.9) and (2.3.10)), so s 1 is always welldefined.
Out of these observations, notice that we see that the local problem can be reduced to a 2 × 2 setting because of the block structure of the jump matrices above. Moreover, near the stationary point p 1 (i.e. ζ = 1) the exponent φ 3 (ζ) -2 √ 3i has a second order zero:
All these facts suggest the parabolic cylinder function problem as the candidate for the model RHP in the neighborhood of the point p 1 . This intuitive idea can be implemented as follows. Consider the following 2 × 2 RHP for the function ϕ(z).
RHP 4. Find a matrix-valued function ϕ(z) satisfying the following conditions
where Γ 4 is the jump contour defined in figure 15.
, where the jump matrix G ϕ is given by (3.2.11).
where
One can solve this RHP using the parabolic cylinder function as demonstrated in section 1.5 in chapter 2 of [FIKN06]. The asymptotic behavior of ϕ as z → ∞ is
Γ (-ν) .
Next, we introduce Z(z) by
where we introduced
Remark. Observe that s 1 = 0 gives B = 0, A R = 1 3 , and s R = 0. This implies S
Let us define a local change of the variables by
where its asymptotics is given by
using the branch determined to obtain (3.2.8).
Since p 1 is a stationary point, i.e. φ 3 (p 1 ) = 0, equation (3.2.12) defines a holomorphic change of variables in some neighborhood U 1 :
Observe that Z(ζ) := Z(z(ζ)) has exactly the same jumps in U 1 as Yp1 does:
Thus, applying the inverse transformation described in figure 14 to Z(ζ), we obtain Ž(ζ), which has exactly the same jumps in U 1 as Y(ζ) does. Furthermore, Ž(ζ) has the following asymptotics:
Finally, to get the matching with the asymptotics (3.2.8) of YD (ζ), we define the local parametrix P (1) (ζ) near ζ = 1 by
One can observe that V(ζ) is holomorphic in U 1 and has the following Taylor expansion:
by (3.2.8) and (3.2.14). Moreover, on the boundary of U 1 , since Ž(ζ) has the asymptotic expansion (3.2.13), we have
(3.2.15)
In other words, the matching condition (3.2.9) is satisfied. Proposition 3.3. These local parametrices can be found using the following symmetries:
(3.2.16)
(3.2.17)
(3.2.18)
denote the jump matrices for the P (p k ) -problem.
Observe that we have the following sequence of equivalent statements
.2.21) Thus we need to check if (3.2.21) is satisfied for all jump matrices of the RHP near ζ = 1 and the ones near ζ = -1. Because of the anti-symmetry for Q (∞) n
we have that
and direct computation shows that indeed
Moreover, the matching condition with YD is automatically satisfied near ζ = -1, since by (3.2.15)
So, P (-1) (ζ) satisfies all local parametrix conditions. Similar reasoning can be applied to prove the other relations, (3.2.17)-(3.2.20).
Let us call the union of all neighborhoods U p k of the stationary points ζ = p k from the previous subsection U all :
Then, we define a piecewise holomorphic function Y(approx) (ζ) by
which is discontinuous on S 1 \ U all and on ∂U all .
Recall that in section 3.2.1 we discussed that we cannot apply the small norm theorem to the Y-problem directly. Instead, using Y(approx) (ζ) constructed in the previous subsections, we would be able to apply the small norm theorem to the
Observe that the error function R(ζ) solves the following RHP.
RHP 5. Find a matrix-valued function R(ζ) satisfying the following conditions:
Due to the structure of jump matrices of the Y-problem and the function Y(approx) (ζ) and the matching up conditions such as (3.2.15), we have that
Thus, by the small norm theorem, there is a unique solution R(ζ) of the RHP 5 for x large. Moreover, one can show the following fact:
Theorem 3.4. Given any point of M = {(s R , y R )} defined by (2.3.11) and for every x ∈ (0, ∞), all the above introduced Rieman-Hilbert Problems are solvable.
Proof. Let (s R , y R ) ∈ M be arbitrary. Then, for sufficiently large x ∈ R >0 , say x > x 0 , there exists a unique solution R(ζ) by the small norm theorem. Thus, all the other RH problems ( Y, Y, Φ, Φ, Ψ problems) are solvable for such x. It means that there exists a solution w 0 (x) of the radial Toda equation (1.0.5) for x > x 0 whose monodromy data (s R , y R ) is given. Since a solution w 0 (x) of (1.0.5) can be smoothly extended to the whole half line x > 0 (see proposition 2.1), the Riemann-Hilbert Problem for Ψ will be solvable for all x > 0 by the canonical solutions of the system (2.2.1) corresponding to the extended w 0 (x). Simultaneously, all the other RH problems will be solvable for all x > 0.
In this subsection, we will find the desired asymptotics of w 0 (x) as x → ∞. First, we note that by definition
From equation (3.2.7), one can see that YD (0) = I. Thus, by the definition of Y and (3.1.6), we have that
1 + e 2w0 + e -2w0 ω 2 + ωe 2w0 + e -2w0 ω 2 + ωe 2w0 + e -2w0 ω + ω 2 e 2w0 + e -2w0
1 + e 2w0 + e -2w0 .
(3.3.2)
On the other hand, we can write R(ζ) in a form of Cauchy integral,
Observe that inequality (3.3.1) allows us to apply the small norm theorem, which in turn implies
where for the last equality we used the following estimate
due to the Cauchy Schwarz inequality, (3.3.1) and (3.3.4). For convenience, let us denote the boundary of U p k 's as follows:
Let Γ 5 := Γ 5 \ ∪ 6 k=1 γ k . Since jump matrices on Γ 5 are exponentially close to the identity matrix, we can rewrite the last equation (3.3.5) as
Proposition 3.5. One has
The proof of this proposition is given in appendix D. Comparing (3.3.2) with (3.3.6), we have
where the leading term, as it is shown below, is of order O(x -1/2 ), thus the real solution should have the following asymptotics:
substituting α and β we obtain the following theorem.
Theorem 3.6. For every s R ∈ (-3, 1) and every y R ∈ R, the asymptotics of the corresponding solution w 0 (x) of the radial Toda equation (1.0.5) is given by the formula
where
The proof of this theorem is given in appendix E.
Corollary 1. For every s R ∈ (-3, 1) and every y R ∈ R, the asymptotics of the corresponding solution of the Painlevé III (D 7 ) equation (2.0.1) is given by the formula
where σ and ψ are the ones we introduced in theorem 3.6. where each function p i = p i (z) is holomorphic in a neighborhood of z 0 . To obtain radial solutions near zero, we need to take p i (z) = c i z ki for certain real k i and c i . Indeed, if we take p i (z) = c i z ki with c i > 0 and k i > -1, then radial solutions near zero satisfy
where -1 2 < γ < 1. All such solutions with this asymptotic behavior arise this way. As we have pointed out earlier, all of these solutions are in fact global solutions. We call the two real parameters {γ, ρ} asymptotic data and the two real parameters {c i , k i } holomorphic data. The relation between these two data will be given below.
As we obtained the asymptotic formula for the solution near x = ∞ as in theorem 3.6, our goal is now to find connection formulae between two kinds of asymptotic parameters. This will be done by making use of the monodromy data {s R , y R }, and a method based on the Iwasawa factorization.
The following discussion is parallel to the one in [GIL20], so we omit many details. Consider the (possibly multi-valued) holomorphic matrix function L(z, λ) that satisfies the following ODE,
where λ ∈ C * and
For convenience, we omit arguments of functions when no confusion is likely.
Consider the twisted loop group ΛSL t 3 C:
From (4.1.3) it follows that L ∈ ΛSL t 3 C. Theorem 4.1. Let L(z, λ) be as in (4.1.2). Then, there is a unique Iwasawa factorization of L(z, λ) :
where
where b 0 is smooth in a small neighbourhood of 0 and depends on |z|.
• The maps L R , L + are (possibly multi-valued) maps from a small neighborhood of 0 into
Proof. This is a special case of theorem 8.1.1 of [PS86]. See also [McI94].
Moreover, we have that
where w, W are as in (2.1.2). Next, we introduce a new variable t by
Note that dt dz = ν. Then, define
After this transformation, equation (4.1.4) becomes
which is exactly the Lax pair (2.1.3) for the 2D periodic Toda equation of type A 2 . This change of variables gives
Comparing it with (4.1.1), we have
As in [GIL20], the radial symmetry of w 0 implies that Ψ(t, t, λ)g(λ) T depends only on |t| and λ/t. We then write
and this transforms (4.1.5) into
(4.1.6) Simultaneously, let us introduce Φ (λ) := (gL) T Then, it transforms equation (4.1.2) into dΦ dλ = -3 N z λ 2 η T + 1 λ m Φ. (4.1.7)
It will be convenient to introduce
and from (4.1.7), using similar arguments to those in [GIL20], one can obtain
Note. The above Φ is not related to the Φ in section 3.1.2.
Summarizing, we have obtained the Lax pair (4.1.6) and the subsidiary ODE (4.1.8) from the Iwasawa factorization and the connection between the holomorphic data {c i , k i } and the asymptotic data {γ, ρ}:
4.1.2. Monodromy data for Φ.
We shall see that Ψ and Φ have very similar behavior at 0, in particular the same Stokes data there. Moreover, as we will see, the Stokes data of (4.1.8) can be calculated explicitly because of the very specific structure of (4.1.8). Recall from section 2.2 that we have • Canonical solutions near singularities:
• Connection matrix:
For Φ, one can derive similar formulae (see [GIL20]):
• Canonical solution near singularities:
• Stokes matrices:
• Connection matrices:
The proof for the Stokes matrices is exactly the same as corollary 4.3 in [GIL20]. We will give a proof for the connection matrices.
Since (gL R G) T and Ψ (0,∞) k are solutions of the first equation of (4.1.6), they differ by constant (in both ζ and x) matrices Y k , X k respectively:
(4.1.9)
Notice that
By the inversion symmetry relation (2.2.16) and reality condition (2.3.3),
Using the loop group reality L * R 1/λ -1 = L R (λ) for (4.1.9), we have
By (4.1.9), it also holds that
Since GG = I and g(e i π ζt)g e i π tζ = I, it follows that
where we used (2.2.17) for the last equality. By (4.1.10) and (4.1.11), we obtain
is positive definite. Therefore, the upper left 1 × 1 corner of E 1 C has a positive determinant, i.e. A R is positive in this situation, i.e. when the solution arises through the Iwasawa factorization.
It turns out that equation (4.1.8) can be reduced to 3rd order scalar differential equations which can be solved by Barnes-type integrals involving Gamma functions. This gives rise to an explicit formula for the connection matrix D 1 . As this is very similar to the discussion leading to theorem 3.13 in [GIL20], we just state the result.
Lemma 4.3. The connection matrix D 1 admits the decomposition,
Theorem 4.4. Let us introduce a real parameter q R by q R := 2 (γ -1)
Then,
where λ 0 = 4 3 sin 2 π 3 (1 -γ) and λ 1 = 2 cos π 3 (1 -γ). Proof. Substituting the decomposition (4.1.12) of D 1 into E 1 = D 1 D * 1 C, one can get (4.1.13) by direct computation. Moreover, (4.1.13) implies
Note that the LHS of (4.1.14) is parametrized only by the monodromy data
4.1.15) by (2.2.31) and (2.3.8), while the RHS of (4.1.14) is parametrized by the asymptotic data 1 λ 0
where
Comparing (4.1.15) with (4.1.16), we have
This provides us with the one-to-one correspondence between monodromy data {s R , y R } and asymptotic data {γ, ρ}.
Theorem 4.5. One has s
where λ 0 = 4 3 sin 2 π 3 (1 -γ), λ 1 = 2 cos π 3 (1 -γ), and q R = 2(γ-1) 2 e -2ρ 3 2(γ-1) Γ( γ-1
3 )Γ( 1-γ 3 )
.
Proof. Substitute (4.1.17) and (4.1.18) into (4.1.19) to have (4.1.20). Take the imaginary part of (4.1.18) multiplied by ω 2 to have (4.1.21).
Remark. Since γ satifies -1 2 < γ < 1, it follows that -3 < s R < 1 from (4.1.20).
In this subsection, we will describe different ways to write down the connection formulae in terms of q R , γ. Formulae (4.1.17)-(4.1.19) imply the following statement.
Proposition 4.6. Let ωB = x R + iy R and A R be the monodromy data, then one has
Direct computation shows that the relations between A R , B, s R described by (2.3.9) and (2.3.10) are automatically satisfied if we substitute formulae (4.2.1)-(4.2.3).
In order to describe the asymptotics of w 0 at infinity in terms of q R , γ we need the following corollary of proposition 4.6.
Corollary 2. For γ ∈ (-1/2, 1), one has
Proof. Follows from proposition 4.6 and the fact that λ 0 , λ 1 > 0 for γ ∈ (-1/2, 1).
Observe that the statement of theorem 3.6 can be written in the following way
where
Combining these observations we have the following connection formula between the asymptotics of w 0 (x) at zero and infinity.
Theorem 4.7. For every γ ∈ (-1/2, 1) and every ρ ∈ R, there exists a unique solution w 0 (x) of the radial Toda equation, i.e.
where
Every real solution of (1.0.5) corresponds to a point in the monodromy data set M (see (2.3.11)). Given any monodromy data (s R , y R ), with A R > 0, we can solve equations (4.1.20) and (4.1.21) for γ and ρ. As stated in the remark in section 2.3.4, we shall show in [GIK + ] that there are no solutions of (1.0.5) when A R < 0. These facts yield the last statement of theorem 1.1, i.e. the completeness of the description (1.0.6) of the behavior of solutions of (1.0.5) at x = 0.
It can be written as
then the LHS of (A.0.1) is
and the RHS of (
Therefore, we should have
So, the parametrization of E 1 can be reduced to 3 variables, A, B, and D:
which satisfies the symmetry E 1 = E T 1 observed in proposition 2.11. From the relation (2.2.42) with this symmetry, one has
where Under the conditions A = A, B = D the last equation is equivalent to (2.3.5): a = ω 2 s R . We will write A as A R to highlight that it is a real number. Equations (A.0.3) and (A.0.4) become
On the other hand, the LHS of (C.0.2) is
where A R , B were introduced in appendix A. Using identities (2.3.10) and (2.3.9) between A R and B we get
Comparing (C.0.4) and (C.0.5) we have
Finally, substituting (C.0.6) into (C.0.3), (C.0.2) and taking inverse, we obtain that the decomposition Ẽ-1
All the other decompositions can be obtained similarly.
where we set
Therefore, combining the residue calculations (D.0.2) and (D.0.3), we obtain
Appendix E. Proof of theorem 3.6
We have seen that
Observe that (E.0.1) Thus,
s 1 Γ (-ν)
s 1 Γ (ν)
Note that we have e -5π i 6 (-i)e -3π i 4 = e -π i 12 and e -5π i 6 (-ω)e -3π i 4 = e with ϵ = -1.
Comparing with our formulae, we observe that u (τ ) = u x 2 = -2w 0 (x) and
In [Kit87b], the following result was obtained:
) -1 ∼ -a √ 6 (3τ ) -1 4 cos 2 √ 3τ + a 2 ln √ 3τ + ϕ -π 4 , τ → ∞ or in our terms, e -2w0(x) -1 ∼ -a √ 6 3x 2 -1 4 cos 2 √ 3x + a 2 ln √ 3x + ϕ -π 4 , x → ∞, where a = | ln g 3 | 2π exp 1 2 arg ln g 3 ϕ = a 2 ln 24 -i 2 ln Γ -ia 2 Γ (ia 2 ) g 2 + sg 3 g 1 + 1 2 arg Γ -ia 2 Γ (ia 2 ) + 1 2 arg g 2 + sg 3 g 1
and the complex parameters g 1 , g 2 , g 3 satisfy g 1 + g 2 + g 3 = 1, g 1 g 2 + g 2 g 3 + g 1 g 3 (1 + s) = 0. (F.0.1)
It seems not clear how to make a correct correspondence between Kitaev's parameters (g 1 , g 2 , g 3 ) and our parameters (A R , B, s R ). But, one can make the following two observations:
(1) If we take
(F.0.2) then (F.0.1) becomes equivalent to (2.3.9) and (2.3.10), and it holds that
which implies our result differs by a minus sign. Although the correspondence (F.0.2) seems natural, one might give an alternative change of parameters as follows:
(2) If we take
then (F.0.1) is satisfied by (2.3.9) and (2.3.10), and we have
This asymptotics is exactly what we have.
Ryosuke Odoi https://orcid.org/0000-0002-6518-7828
The reader can find a detailed summary of the results of[Kit87b,KV04] in appendices B, C, and D of the recent paper[KV23]. In this paper, the authors have corrected some typos and small arithmetic mistakes contained in their earlier works, and they have made considerable efforts to check numerically and present in a more transparent and simplified way the asymptotic formulae of[Kit87b,KV23].