arXiv subjects

Marco Bertola

Publications and source records attributed to Marco Bertola.

At least 55 records · Page 3Linked to original sources

arXiv2018

Noncommutative Painlevé equations and systems of Calogero type

All Painlevé equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of $β$-models. An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlevé correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlevé equation.

math-ph↗

arXiv2018

The Kontsevich-Penner matrix integral, isomonodromic tau functions and open intersection numbers

We identify the Kontsevich-Penner matrix integral, for finite size $n$, with the isomonodromic tau function of a $3\times 3$ rational connection on the Riemann sphere with $n$ Fuchsian singularities placed in correspondence with the eigenvalues of the external field of the matrix integral. By formulating the isomonodromic system in terms of an appropriate Riemann-Hilbert boundary value problem, we can pass to the limit $n\to\infty$ (at a formal level) and identify an isomonodromic system in terms of the Miwa variables, which play the role of times of the KP hierarchy. This allows to derive the String and Dilaton equations via a purely Riemann-Hilbert approach. The expression of the formal limit of the partition function as an isomonodromic tau function allows us to derive explicit closed formulæ for the correlators of this matrix model in terms of the solution of the Riemann-Hilbert problem with all times set to zero. These correlators have been conjectured to describe the intersection numbers for Riemann surfaces with boundaries, or open intersection numbers.

math-ph↗

arXiv2018

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlevé IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlevé transcendent is pole-free on a semiaxis.

math-ph↗

arXiv2018

Inversion formula and range conditions for a vector multi-interval finite Hilbert transform in $L^2$

Given $n$ disjoint intervals $I_j$, on $\mathbb R$ together with $n$ functions $ψ_j\in L^2(I_j)$, $j=1,\dots n$, and an $n\times n$ matrix $Θ$, the problem is to find an $L^2$ solution $\vec φ= {\rm Col} (φ_1,\dots, φ_n)$, $φ_j \in L^2(I_j)$ to the linear system $χΘ\mathcal H \vec φ= \vecψ$, where $\mathcal H = {\rm diag} (\mathcal H_1 ,\dots, \mathcal H_n)$ is a matrix of finite Hilbert transforms and $χ=\text{diag}(χ_1,\dots,χ_n)$ is a matrix of the corresponding characteristic functions on $I_j$, and $\vec ψ={\rm Col} (ψ_1,\dots,ψ_n)$. Since we can interpret $χΘ\mathcal H \vec φ$ as a generalized vector multi-interval finite Hilbert transform, we call the formula for the solution as "the inversion formula" and the necessary and sufficient conditions for the existence of a solution as the "range conditions". In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix $Θ$ is symmetric and positive definite, and; b) all the entries of $Θ$ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. When the matrix $Θ$ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann-Hilbert Problem. We also discuss other cases of the matrix $Θ$.

math.FA↗

arXiv2017

Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data

We consider the compressive wave for the modified Korteweg--de Vries equation with background constants $c>0$ for $x\to-\infty$ and $0$ for $x\to+\infty.$ We study the asymptotics of solutions in the transition zone $4c^2t-\varepsilon t 0,$ $σ\in(0,1),$ $β>0.$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $\frac{\varepsilon t}{\ln t}.$ Also we show how to obtain Khruslov--Kotlyarov's asymptotics in the domain $4c^2t-ρ\ln t<x<4c^2t$ with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann-Hilbert problem.

math-ph↗

arXiv2017

The Malgrange Form and Fredholm Determinants

We consider the factorization problem of matrix symbols relative to a closed contour, i.e., a Riemann-Hilbert problem, where the symbol depends analytically on parameters. We show how to define a function $τ$ which is locally analytic on the space of deformations and that is expressed as a Fredholm determinant of an operator of "integrable" type in the sense of Its-Izergin-Korepin-Slavnov. The construction is not unique and the non-uniqueness highlights the fact that the tau function is really the section of a line bundle.

math-ph↗

arXiv2017

Discriminant circle bundles over local models of Strebel graphs and Boutroux curves

We study special circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$. The space $\mathcal Q_0^{\mathbb R}(-7)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order 7 with real periods; it appears naturally in the study of a neighbourhood of the Witten's cycle $W_1$ in the combinatorial model based on Jenkins-Strebel quadratic differentials of $\mathcal M_{g,n}$. The space $\mathcal Q^{\mathbb R}_0([-3]^2)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most 3 with real periods; it appears in description of a neighbourhood of Kontsevich's boundary $W_{-1,-1}$ of the combinatorial model. The application of the formalism of the Bergman tau-function to the combinatorial model (with the goal of computing analytically Poincare dual cycles to certain combinations of tautological classes) requires the study of special sections of circle bundles over $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$; in the case of the space $\mathcal Q_0^{\mathbb R}(-7)$ a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$, also called the spaces of Boutroux curves, in detail, together with corresponding circle bundles.

math.AG↗

arXiv2017

The Kontsevich matrix integral: convergence to the Painlevé hierarchy and Stokes' phenomenon

We show that the Kontsevich integral on $n\times n$ matrices ($n< \infty$) is the isomonodromic tau function associated to a $2\times 2$ Riemann--Hilbert problem. The approach allows us to gain control of the analysis of the convergence as $n\to\infty$. By an appropriate choice of the external source matrix in Kontsevich's integral, we show that the limit produces the isomonodromic tau function of a special tronquée solution of the first Painlevé hierarchy, and we identify the solution in terms of the Stokes' data of the associated linear problem. We also show that there are several tau functions that are analytic in appropriate sectors of the space of parameters and that the formal Witten-Kontsevich tau function is the asymptotic expansion of each of them in their respective sectors, thus providing an analytic tool to analyze its nonlinear Stokes' phenomenon.

math-ph↗

arXiv2016

On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials $π_n(z)$ with the quartic exponential weight $\exp[-N(\frac 12 z^2+\frac 14 tz^4)]$, where $t\in {\mathbb C}$ and $N\in{\mathbb N}$, $N\to\infty$. Our goal is to describe these asymptotic behaviors globally for $t\in {\mathbb C}$ in different regions. We also describe the "breaking" curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic "phase portrait" of the recurrence coefficients of $π_n(z)$, which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the $g$-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex $t$-plane to all noncritical values of $t$. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation].

nlin.SI↗

arXiv2016

Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues' density, and write the universal limit in function of the Kontsevich matrix model ("matrix Airy function", as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral ("matrix Bessel function") known in the literature as generalized Kontsevich matrix model.

math-ph↗

arXiv2016

Rogue waves in multiphase solutions of the focusing NLS equation

Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalised rogue wave notion then naturally enters as a large-amplitude localised coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalised rogue waves.

nlin.PS↗

arXiv2016

The dependence on the monodromy data of the isomonodromic tau function

[Note: important Corrigendum now available at arXiv:1601.04790] The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno did not derive the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes' parameters). We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing and monodromy-preserving) are dealt with at the same level. We thus provide variational formulae for the isomonodromic tau function with respect to the (generalized) monodromy data. The construction applies more generally: given any (sufficiently well-behaved) family of Riemann-Hilbert problems (RHP) where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form Omega (not necessarily closed) on the deformation space (Malgrange's differential), defined off Malgrange's divisor. We then introduce the notion of discrete Schlesinger transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Even if Omega is not closed, its difference evaluated along the original solution and the transformed one, is shown to be the logarithmic differential (on the deformation space) of a function. As a function of the position of the points of the Schlesinger transformation, yields a natural generalization of Sato formula for the Baker-Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector. Some exemples (Painleve' II and finite Toplitz/Hankel determinants) are provided.

nlin.SI↗

arXiv2016

Maximum amplitudes of finite-gap solutions for the focusing Nonlinear Schrödinger Equation

In this paper we prove that the maximum amplitude of a finite-gap solution to the focusing Nonlinear Schrödinger equation with given spectral bands does not exceed half of the sum of the length of all the bands. This maximum will be attained for certain choices of the initial phases. A similar result is also true for the defocusing Nonlinear Schrödinger equation.

math-ph↗

arXiv2015

Simple Lie algebras and topological ODEs

For a simple Lie algebra $\mathfrak g$ we define a system of linear ODEs with polynomial coefficients, which we call the topological equation of $\mathfrak g$-type. The dimension of the space of solutions regular at infinity is equal to the rank of the Lie algebra. For the simplest example $\mathfrak g=sl_2(\mathbb C)$ the regular solution can be expressed via products of Airy functions and their derivatives; this matrix valued function was used in our previous work for computing logarithmic derivatives of the Witten - Kontsevich tau-function. For an arbitrary simple Lie algebra we construct a basis in the space of regular solutions to the topological equation called generalized Airy resolvents. We also outline applications of the generalized Airy resolvents to computing the Witten and Fan - Jarvis - Ruan invariants of the Deligne - Mumford moduli spaces of stable algebraic curves.

math-ph↗

arXiv2015

Symplectic geometry of the moduli space of projective structures in homological coordinates

We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure coincides with the canonical Poisson structure on the cotangent bundle of the moduli space of Riemann surfaces, and therefore the homological coordinates provide a new system of Darboux coordinates. We define a natural family of commuting "homological flows" on the moduli space of quadratic differentials and find the corresponding action-angle variables. The space of projective structures over the moduli space can be identified with the cotangent bundle upon selection of a reference projective connection that varies holomorphically and thus can be naturally endowed with a symplectic structure. Different choices of projective connections of this kind (Bergman, Schottky, Wirtinger) give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations: the corresponding generating functions are found. We also study the monodromy representation of the Schwarzian equation associated with a projective connection, and we show that the natural symplectic structure on the the space of projective connections induces the Goldman Poisson structure on the character variety. Combined with results of Kawai, this result shows the symplectic equivalence between the embeddings of the cotangent bundle into the space of projective structures given by the Bers and Bergman projective connections.

math.SG↗

arXiv2015

Hankel Determinant Approach to Generalized Vorob'ev-Yablonski Polynomials and their Roots

Generalized Vorob'ev-Yablonski polynomials have been introduced by Clarkson and Mansfield in their study of rational solutions of the second Painlevé hierarchy. We present new Hankel determinant identities for the squares of these special polynomials in terms of Schur polynomials. As an application of the identities, we analyze the roots of generalized Vorob'ev-Yablonski polynomials and provide formulæ\, for the boundary curves of the highly regular patterns observed numerically in \cite{CM}.

math-ph↗

arXiv2015

Universality conjecture and results for a model of several coupled positive-definite matrices

The paper contains two main parts: in the first part, we analyze the general case of $p\geq 2$ matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a $(p+1)\times (p+1)$ matrix valued solution of a Riemann-Hilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multi-level random determinantal point fields which include the Bessel$_ν$ universality for $1$-level and the Meijer-$G$ universality for $2$-level. Our analysis uses the Deift-Zhou nonlinear steepest descent method and the explicit construction of a $(p+1)\times (p+1)$ origin parametrix in terms of Meijer G-functions. The solution of the full Riemann-Hilbert problem is derived rigorously only for $p=3$ but the general framework of the proof can be extended to the Cauchy chain of arbitrary length $p$.

math-ph↗

arXiv2014

A degeneration of two-phase solutions of focusing NLS via Riemann-Hilbert problems

Two-phase solutions of focusing NLS equation are classically constructed out of an appropriate Riemann surface of genus two, and expressed in terms of the corresponding theta-function. We show here that in a certain limiting regime such solutions reduce to some elementary ones called "Solitons on unstable condensate". This degeneration turns out to be conveniently studied by means of basic tools from the theory of Riemann-Hilbert problems. In particular no acquaintance with Riemann surfaces and theta-function is required for such analysis.

math-ph↗