Monodromy Data Research Articles

Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

The third Painlevé equation in its generic form, often referred to as Painlevé-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $u_0(x)$ corresponding to parameters $\alpha$, $\beta$, denoted as the triple $(u_0(x),\alpha,\beta)$, we apply an explicit Bäcklund transformation to generate a family of solutions $(u_n(x),\alpha + 4n,\beta + 4n)$ indexed by $n \in \mathbb N$. We study the large $n$ behavior of the solutions $(u_n(x), \alpha + 4n, \beta + 4n)$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $u_n(z/n)$. Our main result is a proof that the limit of solutions $u_n(z/n)$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($D_8$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($D_6$), which are constructed using the seed solution $(1,4m,-4m)$ where $m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $D_6$ and $D_8$ at $z = 0$. We also deduce the large $n$ behavior of the Umemura polynomials in a neighborhood of $z = 0$.

Abelian tropical covers

AbstractLet $\mathfrak{A}$ be a finite abelian group. In this paper, we classify harmonic $\mathfrak{A}$ -covers of a tropical curve $\Gamma$ (which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on $\Gamma$ . We give a realisability criterion for harmonic $\mathfrak{A}$ -covers by patching local monodromy data in an extended homology group on $\Gamma$ . As an explicit example, we work out the case $\mathfrak{A}=\mathbb{Z}/p\mathbb{Z}$ and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.

On the Monodromy Manifold of q-Painlevé VI and Its Riemann–Hilbert Problem

We study the q-difference sixth Painlevé equation (qtext {P}_{text {VI}}) through its associated Riemann–Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of qtext {P}_{text {VI}} with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explicitly in terms of certain orthogonal polynomials yielding special function solutions of qtext {P}_{text {VI}}.

The sixth Painlevé equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

The sixth Painlevé equation PVI is both the isomonodromy deformation condition of a 2-dimensional isomonodromic Fuchsian system and of a 3-dimensional irregular system. Only the former has been used in the literature to solve the nonlinear connection problem for PVI, through the computation of invariant quantities . We prove a new simple formula expressing the invariants p jk in terms of the Stokes matrices of the irregular system, making the irregular system a concrete alternative for the nonlinear connection problem. We classify the transcendents such that the Stokes matrices and the p jk can be computed in terms of special functions, providing a full non-trivial class of 3-dim. examples such that the theory of non-generic isomonodromy deformations of Cotti et al (2019 Duke Math. J. 168 967–1108) applies. A sub-class of these transcendents realises the local structure of all the 3-dim Dubrovin–Frobenius manifolds with semisimple coalescence points of the type studied in Cotti et al (2020 SIGMA 16 105). We compute all the monodromy data for these manifolds (Stokes matrix, Levelt exponents and central connection matrix).

On the tau function of the hypergeometric equation

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss’ hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes G-function. In particular, if the Fuchsian singularities are placed to 0, 1 and ∞, this gives the structure constants of the asymptotical formula of Iorgov–Gamayun–Lisovyy for solutions of Painlevé VI equation.

Symplectic aspects of the tt*-Toda equations

We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt*-Toda equations. This constant problem for the sinh–Gordon equation, which is the case n = 1 of the tt*-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297–311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.

ELLIPTIC ASYMPTOTIC REPRESENTATION OF THE FIFTH PAINLEVÉ TRANSCENDENTS

For the fifth Painlevé transcendents, an asymptotic representation by the Jacobi sn-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part depends on a single integration constant, which is the phase shift and is parametrized by monodromy data for the associated isomonodromy deformation. In addition, under a certain supposition, the error term is also expressed by an explicit asymptotic formula, whose leading term is written in terms of integrals of the sn-function and the ϑ-function, and contains the other integration constant. Instead of the justification scheme for asymptotic solutions of Riemann-Hilbert problems by the Brouwer fixed point theorem, we begin with a boundedness property of a Lagrangian function, which enables us to determine the modulus of the sn-function satisfying the Boutroux equations and to construct deductively the elliptic representation.

Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori

Let $E_{\tau}:= \mathbb{C}/(\mathbb{Z}+ \mathbb{Z} \tau)$ be a flat torus and $G(z; \tau)$ be the Green function on $E_{\tau}$. Consider the multiple Green function $G_{n}$ on$(E_{\tau})^{n}$: \[ G_n (z_1, \cdots ,z_n ; \tau) := \sum_{i \lt j} G(z_i - z_j ; \tau) - n \sum_{i=1}^n G(z_i ; \tau). \] We prove that for $ \tau \in i \mathbb{R}_{\gt 0}$, i.e. $E_\tau$ is a rectangular torus, $G_n$ has exactly $2n + 1$ critical points modulo the permutation group $S_n$ and all critical points are non-degenerate. More precisely, there are exactly $n$ (resp. $n+1$) critical points $ \boldsymbol{a}$’s with the Hessian satisfying $(-1)^n \det D^2 G_n (\boldsymbol{a} ; \tau) \lt 0$ (resp. $\gt 0$). This confirms a conjecture in [4]. Our proof is based on the connection between $G_n$ and the classical Lame equation from [4, 19], and one key step is to establish a precise formula of the Hessian of critical points of $G_{n}$ in terms of the monodromy data of the Lame equation. As an application, we show that the mean field equation \[ \Delta_u + e^u = \rho \delta_0 \textrm{ on } E_\tau \] has exactly $n$ solutions for $8 \pi n - \rho \gt 0$ small, and exactly $n+1$ solutions for $\rho - 8 \pi n \gt 0$ small.

The space of monodromy data for the Jimbo–Sakai family ofq-difference equations

We formulate a geometric Riemann-Hilbert correspondence that applies to the derivation by Jimbo and Sakai of equation $q$-PVI from ``isomonodromy'' conditions. This is a step within work in progress towards the application of $q$-isomonodromy and $q$-isoStokes to $q$-Painleve.

Correction to: The Dependence on the Monodromy Data of the Isomonodromic Tau Function

The note corrects the aforementioned paper (Bertola in Commun Math Phys 294(2):539–579, 2010). The consequences of the correction are traced and the examples updated.

Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis: I

AbstractLeading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as are found. These asymptotics are parameterized by monodromy data of the associated linear ODE, urn:x-wiley:00222526:media:sapm12323:sapm12323-math-0002The parameterization allows one to derive connection formulae for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulae are also considered.

Opers for Higher States of Quantum KdV Models

We study the ODE/IM correspondence for all states of the quantum $\widehat{\mathfrak{g}}$-KdV model, where $\widehat{\mathfrak{g}}$ is the affinization of a simply-laced simple Lie algebra $\mathfrak{g}$. We construct quantum $\widehat{\mathfrak{g}}$-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum $\widehat{\mathfrak{g}}$-KdV opers satisfy the Bethe Ansatz equations of the quantum $\widehat{\mathfrak{g}}$-KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.

Local Moduli of Semisimple Frobenius Coalescent Structures

We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the A3-Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian G(2,4). In the latter case, we analyse in details the action of the braid group on the monodromy data. This proves that these data can be expressed in terms of characteristic classes of mutations of Kapranov's exceptional 5-block collection, as conjectured by one of the authors.

Vector perturbations of Kerr-AdS5 and the Painlev\xe9 VI transcendent

We analyze the Ansatz of separability for Maxwell equations in generically spinning, five-dimensional Kerr-AdS black holes. We find that the parameter μ introduced in [1] can be interpreted as apparent singularities of the resulting radial and angular equations. Using isomonodromy deformations, we describe a non-linear symmetry of the system, under which μ is tied to the Painlevé VI transcendent. By translating the boundary conditions imposed on the solutions of the equations for quasinormal modes in terms of monodromy data, we find a procedure to fix μ and study the behavior of the quasinormal modes in the limit of fast spinning small black holes.

Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa III: Iwasawa Factorization and Asymptotics

This paper, the third in a series, completes our description of all (radial) solutions on $${\mathbb {C}}^*$$ of the tt*-Toda equations $$2(w_i)_{ {t{\bar{t}}} }=-e^{2(w_{i+1}-w_{i})} + e^{2(w_{i}-w_{i-1})}$$, using a combination of methods from p.d.e., isomonodromic deformations (Riemann–Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth near 0. For such solutions, we compute explicitly the Stokes data and connection matrix of the associated meromorphic system, in the resonant cases as well as the non-resonant case. This allows us to give a complete picture of the monodromy data, holomorphic data, and asymptotic data of the global solutions.

The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potentialy″(z)=[∑k=03nk(nk+1)℘(z+ωk2|τ)+B]y(z),nk∈Z≥0 from the monodromy aspect. We prove the existence of a pre-modular form Zr,sn(τ) of weight 12∑nk(nk+1) such that the monodromy data (r,s) is characterized by Zr,sn(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n1=n2=n3=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equationsΔu+eu=16πδ0andΔu+eu=8π∑k=13δωk2 on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo–Miwa–Ueno differential

Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s [45]. Recent advances in the theory of tau functions [41], based on earlier works of B. Malgrange and M. Bertola, have allowed to extend the original Jimbo–Miwa–Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability 1−γ∈(0,1].

Three-Parameter Solutions of the PV Schlesinger-Type Equation near the Point at Infinity and the Monodromy Data

For the Schlesinger-type equation related to the fifth Painlev\'e equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the corresponding monodromy data. Two-parameter solutions of (V) with their monodromy data immediately follow from our results. Under certain conditions, these solutions of (V) admit sequences of zeros and of poles along the imaginary axis. The monodromy data are obtained by matching techniques for a perturbed linear system.

Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an [Formula: see text] linear system of ODEs with an irregular singularity of Poincaré rank 1 at [Formula: see text] and Fuchsian singularity at [Formula: see text], holomorphically depending on parameter [Formula: see text] within a polydisk in [Formula: see text] centered at [Formula: see text]. The eigenvalues of the leading matrix at [Formula: see text], which is diagonal, coalesce along a coalescence locus [Formula: see text] contained in the polydisk. Under minimal vanishing conditions on the residue matrix at [Formula: see text], we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808 .] that isomonodromic deformations can be extended to the whole polydisk, including [Formula: see text], in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of [Formula: see text], where it simplifies. Conversely, if the [Formula: see text]-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting [Formula: see text], and if suitable entries of the Stokes matrices vanish, then [Formula: see text] is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.