arXiv subjects

M. Bertola

Publications and source records attributed to M. Bertola.

At least 19 recordsLinked to original sources

arXiv2025

The $r$-matrix structure on the moduli space of framed Higgs pairs

On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson $r$-matrix structure. The known $r$-matrices are defined on the Riemann sphere (rational), the cylinder (trigonometric), or the torus (elliptic). We extend the formalism to the case of a Riemann surface $\mathcal C$ of higher genus $g$: we consider the moduli space of framed vector bundles of rank $n$ and degree $ng$, where the framing consists of a choice of basis of $n$ independent holomorphic sections that trivialize the fiber at a given point $\infty\in \mathcal C$. The cotangent space is known to be identified with the set of Higgs fields, i.e., one-forms on $\mathcal C$ with values in the endomorphisms of the vector bundle, with an additional simple pole at $\infty$. The natural symplectic structure on the cotangent bundle of the moduli space induces a Poisson structure on the Higgs fields. Building on Dolgushev's higher genus $r$-matrix construction we identify the kernel with an explicitly computable non-abelian Cauchy kernel and derive the complete dynamical Poisson algebra, including the mixed bracket between the Higgs field and the kernel. A detailed discussion of the elliptic case including a comparison with the literature, is also provided.

nlin.SI↗

arXiv2023

Hamiltonian structure of rational isomonodromic deformation systems

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space $L^*gl(r)$ of the loop algebra $Lgl(r)$. Nonautonomous isomonodromic counterparts of the isospectral systems generated by spectral invariants are obtained by identifying the deformation parameters as Casimir functions on the phase space. These are shown to coincide with the higher Birkhoff invariants determining the local asymptotics near to irregular singular points, together with the pole loci. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, extended by the exponents of formal monodromy, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves, locally, may be identified as the orbits of a free abelian group. The projection of the infinitesimal isomonodromic deformations vector fields to the quotient manifold under this action gives the commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.

nlin.SI↗

arXiv2022

Singular limits of certain Hilbert-Schmidt integral operators

In this paper we study the small-$λ$ spectral asymptotics of an integral operator $\mathscr{K}$ defined on two multi-intervals $J$ and $E$, when the multi-intervals touch each other (but their interiors are disjoint). The operator $\mathscr{K}$ is closely related to the multi-interval Finite Hilbert Transform (FHT). This case can be viewed as a singular limit of self-adjoint Hilbert-Schmidt integral operators with so-called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when $\text{dist}(J,E)>0$, and $\mathscr{K}$ is of the Hilbert-Schmidt class, was studied in an earlier paper by the authors. The main assumption in this paper is that $U=J\cup E$ is a single interval. We show that the eigenvalues of $\mathscr{K}$, if they exist, do not accumulate at $λ=0$. Combined with the results in an earlier paper by the authors, this implies that $H_p$, the subspace of discontinuity (the span of all eigenfunctions) of $\mathscr{K}$, is finite dimensional and consists of functions that are smooth in the interiors of $J$ and $E$. We also obtain an approximation to the kernel of the unitary transformation that diagonalizes $\mathscr{K}$, and obtain a precise estimate of the exponential instability of inverting $\mathscr{K}$. Our work is based on the method of Riemann-Hilbert problem and the nonlinear steepest-descent method of Deift and Zhou.

math.FA↗

arXiv2021

Fredholm Pfaffian $τ$-functions for orthogonal isospectral and isomonodromic systems

We extend the approach to $τ$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems. The combinatorial expansion of the $τ$-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and the dependence on the flow (or deformation) parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal $τ$-functions, both in the Drinfeld-Sokolov and isomonodromic case, become squares of $τ$-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial $τ$-functions of orthogonal Drinfeld-Sokolov type and of isomonodromic ones with four regular singular points.

math-ph↗

arXiv2020

Symplectic extensions of the Kirillov-Kostant and Goldman Poisson structures and Fuchsian systems

We revisit symplectic properties of the monodromy map for Fuchsian systems on the Riemann sphere. We extend previous results of Hitchin, Alekseev-Malkin and Korotkin-Samtleben where it was shown that the monodromy map is a Poisson morphism between the Kirillov-Kostant Poisson structure on the space of coefficients, on one side, and the Goldman bracket on the monodromy character variety on the other. The extension is provided by defining larger spaces on both sides which are equipped with symplectic structures naturally projecting to the canonical ones. On the coefficient side our symplectic structure corresponds to a non-degenerate quadratic Poisson structure expressed via the rational dynamical $r$-matrix; it reduces to the Kirillov-Kostant bracket when projected to the standard space. On the monodromy side we get a symplectic structure which induces the symplectic structure of Alekseev-Malkin on the leaves of the Goldman Poisson bracket. We prove that the monodromy map provides a symplectomorphism using the formalism of Malgrange and one of the authors. As a corollary we prove the recent conjecture by A.Its, O.Lisovyy and A.Prokhorov in its "strong" version while the original "weak" version is derived from previously known results. We show also that the isomonodromic Jimbo-Miwa tau-function is intimately related to a generating function of such transformation.

math-ph↗

arXiv2020

Generating weighted Hurwitz numbers

Multicurrent correlators associated to KP $τ$-functions of hypergeometric type are used as generating functions for weighted Hurwitz numbers. These are expressed as formal Taylor series and used to compute generic, simple, rational and quantum weighted single Hurwitz numbers.

math-ph↗

arXiv2020

Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals

The quantum spectral curve equation associated to KP $τ$-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding Sato Grassmannian element are shown to be Meijer $G$-functions, or their asymptotic series. Using their Mellin integral representation the $τ$-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral.

math-ph↗

arXiv2018

Hodge and Prym tau functions, Jenkins-Strebel differentials and combinatorial model of $\mathcal M_{g,n}$

The principal goal of the paper is to apply the approach inspired by the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Jenkins-Strebel differentials. The line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles $\mathcal L_j$ and the sections are constructed in terms of tau functions. The combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau-function provides a section of a circle bundle. The phase of the ratio of the Prym and Hodge tau functions gives a section of the $κ_1$-circle bundle. By evaluating the increment of the phase around co-dimension $2$ sub-complexes, we identify the Poincaré\ dual cycles to the Chern classes of the corresponding line bundles: they are expressed explicitly as combination of Witten's cycle $W_5$ and Kontsevich's boundary. This provides combinatorial analogues of Mumford's relations on $\mathcal M_{g,n}$ and Penner's relations in the hyperbolic combinatorial model. The free homotopy classes of loops around $W_5$ are interpreted as pentagon moves while those of loops around Kontsevich's boundary as combinatorial Dehn twists. Throughout the paper we exploit the classical description of the combinatorial model in terms of Jenkins--Strebel differentials, parametrized in terms of {\it homological coordinates}; we also show that they provide Darboux coordinates for the symplectic structure introduced by Kontsevich. We also express the latter in clear geometric terms as the intersection pairing in the odd homology of the canonical double cover.

math-ph↗

arXiv2017

Universal Peregrine soliton structure in nonlinear pulse compression in optical fiber

We present experimental evidence of the universal emergence of the Peregrine soliton predicted in the semi-classical (zero-dispersion) limit of the focusing nonlinear Schrödinger equation [Comm. Pure Appl. Math. {\bf 66}, 678 (2012)]. Experiments studying higher-order soliton propagation in optical fiber use an optical sampling oscilloscope and frequency-resolved optical gating to characterise intensity and phase around the first point of soliton compression and the results show that the properties of the compressed pulse and background pedestal can be interpreted in terms of the Peregrine soliton. Experimental and numerical results reveal that the universal mechanism under study is highly robust and can be observed over a broad range of parameters, and experiments are in very good agreement with numerical simulations.

nlin.PS↗

arXiv2013

The gap probabilities of the tacnode, Pearcey and Airy point processes, their mutual relationship and evaluation

We express the gap probabilities of the tacnode process as the ratio of two Fredholm determinants; the denominator is the standard Tracy-Widom distribution, while the numerator is the Fredholm determinant of a very explicit kernel constructed with Airy functions and exponentials. The formula allows us to apply the theory of numerical evaluation of Fredholm determinants and thus produce numerical results for the gap probabilities. In particular we investigate numerically how, in different regimes, the Pearcey process degenerates to the Airy one, and the tacnode degenerates to the Pearcey and Airy ones.

math-ph↗

arXiv2013

Cauchy-Laguerre two-matrix model and the Meijer-G random point field

We apply the general theory of Cauchy biorthogonal polynomials developed previously by the authors, to the case associated with Laguerre measures. In particular, we obtain explicit formulae in terms of Meijer-G functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the Cauchy two-matrix model associated with them. The central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists, and is given by a new determinantal two--level random point field, the "Meijer-G random field". We conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of Laguerre weights. We express the joint distributions of the smallest eigenvalues in terms of suitable Fredholm determinants and evaluate them numerically. We also show that in a suitable limit, the Meijer-G random field converges to the Bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the Laguerre ensemble.

math.PR↗

arXiv2011

Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case

We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels à la Its-Izergin-Korepin-Slavnov (IIKS) and hence related to suitable Riemann-Hilbert problems, thus extending the known results for the single-time case. We focus on the Airy and Pearcey processes. As an example of applications we re-deduce a third order PDE, found by Adler and van Moerbeke, for the two-time Airy process.

math-ph↗

arXiv2011

The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach

We consider the gap probability for the Pearcey and Airy processes; we set up a Riemann--Hilbert approach (different from the standard one) whereby the asymptotic analysis for large gap/large time of the Pearcey process is shown to factorize into two independent Airy processes using the Deift-Zhou steepest descent analysis. Additionally we relate the theory of Fredholm determinants of integrable kernels and the theory of isomonodromic tau function. Using the Riemann-Hilbert problem mentioned above we construct a suitable Lax pair formalism for the Pearcey gap probability and re-derive the two nonlinear PDEs recently found and additionally find a third one not reducible to those.

math-ph↗

arXiv2011

Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation

We extend the formalism of integrable operators a' la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi-infinite interval and to matrix integral operators with a kernel of the form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painleve' II (recently introduced by Retakh and Rubtsov), a related noncommutative equation of Painleve' type. We construct a particular family of solutions of the noncommutative Painleve' II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painleve' II. Such a solution plays the same role as its commutative counterpart relative to the Tracy-Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.

math-ph↗

arXiv2009

On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights

For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \leq p \leq \infty$. It is shown that eventually all but a uniformly bounded number of the roots of the $L^p$-optimal polynomials lie within a small neighborhood of the support of a certain equilibrium measure; asymptotics for the $n$th roots of the $L^p$ norms are also provided. The case $p=\infty$ is well known and corresponds to weighted Chebyshev polynomials; the case $p=2$ corresponding to orthogonal polynomials as well as any other $1\leq p <\infty$ is our contribution.

math.CA↗

arXiv2009

Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve

We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schroedinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the semiclassical parameter epsilon) with modulated amplitude and phase, the space-time plane subdivides into regions of qualitatively different behavior, with the boundary between them consisting typically of collection of piecewise smooth arcs (breaking curve(s)). In the first region the evolution of the potential is ruled by modulation equations (Whitham equations), but for every value of the space variable x there is a moment of transition (breaking), where the solution develops fast, quasi-periodic behavior, i.e., the amplitude becomes also fastly oscillating at scales of order epsilon. The very first point of such transition is called the point of gradient catastrophe. We study the detailed asymptotic behavior of the left and right edges of the interface between these two regions at any time after the gradient catastrophe. The main finding is that the first oscillations in the amplitude are of nonzero asymptotic size even as epsilon tends to zero, and they display two separate natural scales; of order epsilon in the parallel direction to the breaking curve in the (x,t)-plane, and of order epsilon ln(epsilon) in a transversal direction. The study is based upon the inverse-scattering method and the nonlinear steepest descent method.

nlin.SI↗

arXiv2009

Peakons and Cauchy Biorthogonal Polynomials

The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.

nlin.SI↗

arXiv2009

Cauchy Biorthogonal Polynomials

The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in term of a Riemann-Hilbert problem.

math-ph↗