Inverse Monodromy Research Articles

Classification of Rank 2 Cluster Varieties

We classify rank $2$ cluster varieties (those for which the span of the rows of the exchange matrix is $2$-dimensional) according to the deformation type of a generic fiber $U$ of their ${\mathcal X}$-spaces, as defined by Fock and Goncharov [Ann. Sci. \'Ec. Norm. Sup\'er. (4) 42 (2009), 865-930]. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. Call $U$ positive if $\dim[\Gamma(U,{\mathcal O}_U)] = \dim(U)$ (which equals 2 in these rank 2 cases). This is the condition for the Gross-Hacking-Keel construction [Publ. Math. Inst. Hautes \'Etudes Sci. 122 (2015), 65-168] to produce an additive basis of theta functions on $\Gamma(U,{\mathcal O}_U)$. We find that $U$ is positive and either finite-type or non-acyclic (in the usual cluster sense) if and only if the inverse monodromy of the tropicalization $U^{\rm trop}$ of $U$ is one of Kodaira's monodromies. In these cases we prove uniqueness results about the log Calabi-Yau surfaces whose tropicalization is $U^{\rm trop}$. We also describe the action of the cluster modular group on $U^{\rm trop}$ in the positive cases.

Poles of Painlevé IV Rationals and their Distribution

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlev\'e IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when $m$ is large and $n$ fixed.

Fuchsian Riemann-Hilbert problem for “real doubles” of Hurwitz Frobenius manifolds

A solution to the matrix Fuchsian Riemann-Hilbert problem (inverse monodromy problem) corresponding to “real doubles” of Dubrovin's Frobenius structures on Hurwitz spaces is constructed. The solution is given in terms of certain meromorphic differentials integrated over a basis of an appropriate relative homology space of the Riemann surface. The relationship with the solution of Fuchsian Riemann-Hilbert problem for Dubrovin's Hurwitz Frobenius manifolds is established. A solution of the Riemann-Hilbert problem corresponding to deformations of the “real doubles” is also given.

Riemann–Hilbert problem for Hurwitz Frobenius manifolds: Regular singularities

In this paper we study the Fuchsian RiemannHilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this RiemannHilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space over a Riemann surface, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.

Apparent singularities of Fuchsian equations and the Painlevé property for Garnier systems

We study movable singularities of Garnier systems using the connection of the latter with Schlesinger isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.

On the solvability of the discrete second Painlevé equation

The inverse monodromy method for studying the Riemann–Hilbert problem associated with classical Painlevé equations is applied to the discrete second Painlevé equation.

On the Riemann–Hilbert–Birkhoff Inverse Monodromy Problem Associated with the Third Painlevé Equation

A linear system of ordinary differential equations corresponding by the isomonodromy deformation method to the third Painleve equation is considered. The surjectivity of the monodromy map generated by this system is proven using the Riemann–Hilbert factorization method.

Riemann–Hilbert Problems for Hurwitz Frobenius Manifolds

In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.

The Cauchy problem for the second member of a hierarchy

A rigorous method, the inverse monodromy transform, for studying the Riemann–Hilbert (RH) problem associated with the classical Painlevé equations, , is applied to the second member of a fourth Painlevé hierarchy. We show that the Cauchy problem for the second member of this hierarchy admits, in general, a global meromorphic solution in x. Moreover, for a particular choice of the monodromy data the associated RH problem can be reduced to a set of scalar RH problems and a special solution which can be written in terms of the Airy function is obtained.

Inverse monodromy problems in mathematical physics

After a short introduction devoted to the origin of the problem considered from the physical problems of quantum field theory and statistical mechanics, the paper considers the mathematical aspects of the Knizhnik-Zamolodchikov equation theory associated with various root systems and the results connected with the inverse monodromy problem for them.

Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with irreducible quasi-permutation monodromy representation outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of P1 . The solution is given in terms of a generalization of Szego kernel on the Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. The isomonodromy tau-function of these solutions is computed up to a nowhere vanishing factor independent of the elements of monodromy matrices. Results of this work generalize the results of papers [13] and [14] where the 2× 2 case was solved.

Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painleve equations. These hierarchies contain the Painleve equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P 1 and P 2 equations and special cases of the P 3 and P 5 equations. Four of the six Painleve equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.

Amalgamations of the Painlevé equations

New hierarchies of nonlinear ordinary differential equations that are generalizations of the Painlevé equations are presented. These hierarchies are shown to contain the Painlevé equations as special cases. Emphasis is on the sixth order ordinary differential equations. Special solutions for one of them are expressed via the general solutions of the P1, P2 equations and special cases of the P3 and P5 equations. Special solutions of another sixth order ODE are determined by the general solutions of the P2, P4, P3, and P5 equations. Five of six Painlevé equations can be considered as special cases of these sixth order ODEs. Isomonodromic linear problems to solve the Cauchy problems of hierarchies by the inverse monodromy transform are given.

On the Fourth Painlevé Hierarchy

We use the inverse monodromy transform to find the fourth Painleve hierarchy. The second and third members of this hierarchy are given. Special and rational solutions of the second and third members for the P4 hierarchy are discussed. We apply the Painleve test to the second member of the fourth Painleve hierarchy.

A special case of de Branges' theorem on the inverse monodromy problem

We give a new proof of a special case of de Branges' theorem on the inverse monodromy problem: when an associated Riemann surface is of Widom type with Direct Cauchy Theorem. The proof is based on our previous result (with M.Sodin) on infinite dimensional Jacobi inversion and on Levin's uniqueness theorem for conformal maps onto comb-like domains. Although in this way we can not prove de Branges' Theorem in full generality, our proof is rather constructive and may lead to a multi-dimensional generalization. It could also shed light on the structure of invariant subspaces of Hardy spaces on Riemann surfaces of infinite genus.

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds

We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painleve VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations.

J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem

J-inner matrix functions, interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem

J-Inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem

This is the second of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted to the inverse monodromy problem for canonical integral and differential systems. In this part, which focuses on the case of a diagonal signature matrixJ, a parametrization is obtained of the set of all solutionsM (t) for the inverse problem for integral systems in terms of two chains of entire matrix valued inner functions. Special classes of solutions correspond to special choices of these chains. This theme will be elaborated upon further in a third part of this paper which will be published in a subsequent issue of this journal. There the emphasis will be on symmetries and growth conditions all of which serve to specify or restrict the chains alluded to above, from the outside, so to speak.

Equations prescribed curvature and deformations of the monodromy group

Ordinary differential equations having prescribed curvature representation, especially the second Painlevé equation with some arbitrary function as its parameter, are studied by the inverse monodromy problem method. The equations of the dynamics of monodromy data which are first integrals for the corresponding zero curvature equations are derived by means of the differentiation of the suitable Riemann-Hilbert problem.

Integrable systems and isomonodromy deformations

We analyze in detail three classes of isomondromy deformation problems associated with integrable systems. The first two are related to the scaling invariance of the n × n AKNS hierarchies and the Gel'fand-Dikii hierarchies. The third arises in string theory as the representation of the Heisenberg group by [(L k n ) +, L] = I where L is an nth order scalar differential operator. The monodromy data is constructed in each case; the inverse monodromy problem is solved as a Riemann-Hilbert problem; and a simple proof of the Painlevé property is given for the general case.