130,792 research outputs found
Notes on non-generic isomonodromy deformations
No full textSome of the main results of [Cotti G., Dubrovin B., Guzzetti D., Duke Math. J., to appear, arXiv:1706.04808], concerning non-generic isomonodromy deformations of a certain linear differential system with irregular singularity and coalescing eigenvalues, are reviewed from the point of view of Pfaffian systems, making a distinction between weak and strong isomonodromic deformations. Such distinction has a counterpart in the case of Fuchsian systems, which is well known as Schlesinger and non-Schlesinger deformations, reviewed in Appendix A
Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities
No full textWe consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters u= (u1, ... , un) , which are eigenvalues of the leading matrix at the irregular singularity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger-type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters u1, ... , un. The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues
Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity
No full textWe 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 Com- plex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an n × n linear system of ODEs with an irregular singularity of Poincar ́e rank 1 at z = ∞ and Fuchsian singularity at z = 0, holomorphically depending on parameter t within a polydisk in Cn centered at t = 0. The eigenvalues of the leading matrix at ∞, which is diagonal, coalesce along a coalescence locus ∆ contained in the polydisk. Under minimal vanishing conditions on the residue matrix at z = 0, we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singular- ity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.] that isomonodromic deformations can be extended to the whole polydisk, including ∆, 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 ∆, where it simplifies. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting ∆, and if suitable entries of the Stokes matrices vanish, then ∆ is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlev ́e equations
Pole Distribution of PVI Transcendents close to a Critical Point
No full textThe distribution of the poles of branches of the Painleve' VI transcendents associated to semi-simple Frobenius manifolds is determined close to a critical point. It is shown that the poles accumulate at the critical point, asymptotically along two rays. The example of the Frobenius manifold given by the quantum cohomology of the two-dimensional complex projective space is also considered
Stokes matrices and monodromy for the quantum cohomology of projective spaces
No full textIn this paper we compute Stokes matrices and monodromy of the quantum
cohomology of projective spaces. This problem can be formulated in a “classical” framework,
as the problem of computation of Stokes matrices and monodromy of differential
equations with regular and irregular singularities.We prove that the Stokes’ matrix of the
quantum cohomology coincides with the Gram matrix in the theory of derived categories
of coherent sheaves.We also study the monodromy group of the quantum cohomology
and we show that it is related to hyperbolic triangular groups
The asymptotic behaviour of the Fourier transform of orthogonal polynomials II: Iterated Function Systems and Quantum Mechanics
No full textWe study measures generated by systems of linear iterated functions, their Fourier transforms, and those of their orthogonal polynomials. We characterize the asymptotic behaviours of their discrete and continuous averages.
Further related quantities are analyzed, and relevance of this analysis to quantum mechanics is briefly discussed
A Review of the Sixth Painlevé Equation
No full textFor the Painleve' 6 transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of poles close to a critical poin
On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painleve’ 6 Equation
No full textIn this paper we find a class of solutions of the sixth Painlevé equation appearing in
the theory of WDVV equations. This class covers almost all the monodromy data associated to
the equation, except one point in the space of the data. We describe the critical behavior close to
the critical points in terms of two parameters and we find the relation among the parameters at
the different critical points (connection problem). We also study the critical behavior of Painlevé
transcendents in the elliptic representation
Solving the sixth painlevé equation: Towards the classification of all the ritical behaviors and the connection formulae
No full textThe critical behavior of a three real parameter class of solutions of the sixth Painlev´e
equation is computed, and parametrized in terms of monodromy data of the associated
2 × 2 matrix linear Fuchsian system of ODE. The class may contain solutions with poles
accumulating at the critical point. The study of this class closes a gap in the description
of the transcendents in an one to one correspondence with the monodromy data. These
transcendents are reviewed in the paper. Some formulas that relate the monodromy data
to the critical behaviors of the four real (two complex) parameter class of solutions
are missing in the literature, so they are computed here. A computational procedure
to write the full expansion of the four and three real parameter class of solutions is
proposed
The Logarithmic Asymptotics of the Sixth Painleve’ Equation
No full textWe compute the monodromy group associated with the solutions of the sixth
Painlev´e equation with a logarithmic asymptotic behavior at a critical point and
we characterize the asymptotic behavior in terms of the monodromy itsel
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