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Guzzetti Davide - One of the best experts on this subject based on the ideXlab platform.

  • Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities
    2021
    Co-Authors: Guzzetti Davide
    Abstract:

    We consider a Pfaffian System expressing isomonodromy of an irregular System of Okubo type, depending on complex deformation parameters u=(u_1,...,u_n), which are eigenvalues of the leading matrix at the irregular singuilarity. 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 u_1,...,u_n. 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 references [4] and [20] 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 [4] and [20] to the isomonodromic case with coalescences/confluences, allow to prove by means of Laplace transform the main result of reference [11], which is the analytic theory of non-generic isomonodromic deformations of the irregular System with coalescing eigenvalues.Comment: 57 pages, 4 figure

  • Isomonodromic deformations along a stratum of the coalescence locus
    2021
    Co-Authors: Guzzetti Davide
    Abstract:

    We consider deformations of a differential System with Poincare' rank 1 at infinity and Fuchsian singularity at zero along a stratum of a coalescence locus. We give necessary and sufficient conditions for the deformation to be strongly isomonodromic, both as an explicit Pfaffian System (integrable deformation) and as a non linear System of PDEs on the residue matrix A at the Fuchsian singularity. This construction is complementary to that of [10]. For the specific System here considered, the results generalize those of [20], by giving up the generic conditions, and those of [3], by giving up the Lidskii generic assumption. The importance of the case here considered originates form its possible implications in the study of strata of Dubrovin-Frobenius manifolds and F-manifolds.Comment: 38 pages, 4 figure

  • The sixth Painleve' equation as isomonodromy deformation of an irregular System: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds
    2021
    Co-Authors: Degano Gabriele, Guzzetti Davide
    Abstract:

    We consider a 3-dimensional Pfaffian System, whose z-component is a differential System with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlev\'e equation PVI. The coefficients of the System will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [44, 61]. We then express in terms of the Stokes matrices of the 3x3 irregular System the monodromy invariants p_{jk}=Tr(M_jM_k) of the 2-dimensional isomonodromic Fuchsian System with four singularities, traditionally associated to PVI [23, 55] and used to solve the non-linear connection problem. Several years after [44, 61], the authors of [14] showed that the computation of the monodromy data of a class of irregular Systems may be facilitated in case of coalescing eigenvalues. This coalescence corresponds to the critical points (fixed singularities) of PVI. In the second part of the paper, we classify the branches of PVI transcendents holomorphic at a critical point such that the analyticity and semisimplicity properties described in [14] are satisfied, and we compute the associated Stokes matrices and the invariants p_{jk}. Finally, we compute the monodromy data parametrizing the chamber of a 3-dim Dubrovin-Frobenius manifold associated with a transcendent holomorphic at x=0.Comment: 69 pages, 1 table, 1 figur

Florian Litzinger - One of the best experts on this subject based on the ideXlab platform.

  • Optimal regularity for two-dimensional Pfaffian Systems and the fundamental theorem of surface theory.
    The Journal of Geometric Analysis, 2020
    Co-Authors: Florian Litzinger
    Abstract:

    We prove that a Pfaffian System with coefficients in the critical space $L^2_\mathrm{loc}$ on a simply connected open subset of $\mathbb{R}^2$ has a non-trivial solution in $W^{1,2}_\mathrm{loc}$ if the coefficients are antisymmetric and satisfy a compatibility condition. As an application of this result, we show that the fundamental theorem of surface theory holds for prescribed first and second fundamental forms of optimal regularity in the classes $W^{1,2}_\mathrm{loc}$ and $L^2_\mathrm{loc}$, respectively, that satisfy a compatibility condition equivalent to the Gauss-Codazzi-Mainardi equations. Finally, we give a weak compactness theorem for surface immersions in the class $W^{2,2}_\mathrm{loc}$.

Litzinger Florian - One of the best experts on this subject based on the ideXlab platform.

  • Optimal regularity for two-dimensional Pfaffian Systems and the fundamental theorem of surface theory
    'Springer Science and Business Media LLC', 2020
    Co-Authors: Litzinger Florian
    Abstract:

    We prove that a Pfaffian System with coefficients in the critical space $L^2_\mathrm{loc}$ on a simply connected open subset of $\mathbb{R}^2$ has a non-trivial solution in $W^{1,2}_\mathrm{loc}$ if the coefficients are antisymmetric and satisfy a compatibility condition. As an application of this result, we show that the fundamental theorem of surface theory holds for prescribed first and second fundamental forms of optimal regularity in the classes $W^{1,2}_\mathrm{loc}$ and $L^2_\mathrm{loc}$, respectively, that satisfy a compatibility condition equivalent to the Gauss-Codazzi-Mainardi equations. Finally, we give a weak compactness theorem for surface immersions in the class $W^{2,2}_\mathrm{loc}$.Comment: 14 pages, v2: Section 3 revised. To appear in The Journal of Geometric Analysi

Teruhisa Tsuda - One of the best experts on this subject based on the ideXlab platform.

  • hypergeometric solution of a certain polynomial hamiltonian System of isomonodromy type
    arXiv: Classical Analysis and ODEs, 2010
    Co-Authors: Teruhisa Tsuda
    Abstract:

    In our previous work, a unified description as polynomial Hamiltonian Systems was established for a broad class of the Schlesinger Systems including the sixth Painleve equation and Garnier Systems. The main purpose of this paper is to present particular solutions of this Hamiltonian System in terms of a certain generalization of Gauss' hypergeometric function. Key ingredients of the argument are the linear Pfaffian System derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian System.

Tsuda Teruhisa - One of the best experts on this subject based on the ideXlab platform.

  • Hypergeometric solution of a certain polynomial Hamiltonian System of isomonodromy type
    2010
    Co-Authors: Tsuda Teruhisa
    Abstract:

    In our previous work, a unified description as polynomial Hamiltonian Systems was established for a broad class of the Schlesinger Systems including the sixth Painleve equation and Garnier Systems. The main purpose of this paper is to present particular solutions of this Hamiltonian System in terms of a certain generalization of Gauss' hypergeometric function. Key ingredients of the argument are the linear Pfaffian System derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian System.Comment: 16 pages, no figur