TY - JOUR T1 - Isomonodromy deformations at an irregular singularity with coalescing eigenvalues JF - Duke Math. J. Y1 - 2019 A1 - Giordano Cotti A1 - Boris Dubrovin A1 - Davide Guzzetti AB -

We consider an n×n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=∞, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. When the system also has a Fuchsian singularity at z=0, we show under minimal vanishing conditions on the residue matrix at z=0 that isomonodromic deformations can be extended to the whole polydisc, including Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

PB - Duke University Press VL - 168 UR - https://doi.org/10.1215/00127094-2018-0059 ER - TY - JOUR T1 - On an isomonodromy deformation equation without the Painlevé property Y1 - 2014 A1 - Boris Dubrovin A1 - Andrey Kapaev AB - We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlev\'e property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data. PB - Maik Nauka-Interperiodica Publishing UR - http://hdl.handle.net/1963/6466 N1 - 34 pages, 8 figures, references added U1 - 6410 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems JF - Matematische Annalen 349 (2011) 75-115 Y1 - 2011 A1 - Guido Carlet A1 - Boris Dubrovin A1 - Luca Philippe Mertens AB - We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold. PB - Springer UR - http://hdl.handle.net/1963/3584 U1 - 716 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Integrable functional equations and algebraic geometry JF - Duke Mathematical Journal. Volume: 76, Issue: 2, Pages: 645-668 Y1 - 1994 A1 - Boris Dubrovin A1 - A.S. Fokas A1 - P.M. Santini PB - SISSA UR - http://hdl.handle.net/1963/6482 U1 - 6428 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - CHAP T1 - Integrable systems and classification of 2D topological field theories T2 - Integrable systems : the Verdier memorial conference : actes du colloque international de Luminy / Olivier Babelon, Pierre Cartier, Yvette Kosmann-Schwarzbach editors. - Boston [etc.] : Birkhauser, c1993. - p. 313-359 Y1 - 1993 A1 - Boris Dubrovin AB - In this paper we consider from the point of view of differential geometry and of the\\r\\ntheory of integrable systems the so-called WDVV equations as defining relations of 2-\\r\\ndimensional topological field theory. A complete classification of massive topological conformal\\r\\nfield theories (TCFT) is obtained in terms of monodromy data of an auxillary\\r\\nlinear operator with rational coefficients. Procedure of coupling of a TCFT to topological\\r\\ngravity is described (at tree level) via certain integrable bihamiltonian hierarchies of\\r\\nhydrodynamic type and their τ -functions. A possible role of bihamiltonian formalism in\\r\\ncalculation of high genus corrections is discussed. As a biproduct of this discussion new\\r\\nexamples of infinite dimensional Virasoro-type Lie algebras and their nonlinear analogues\\r\\nare constructed. As an algebro-geometrical applications it is shown that WDVV is just the\\r\\nuniversal system of integrable differential equations (high order analogue of the Painlev´e-\\r\\nVI) specifying periods of Abelian differentials on Riemann surfaces as functions on moduli\\r\\nof these surfaces. JF - Integrable systems : the Verdier memorial conference : actes du colloque international de Luminy / Olivier Babelon, Pierre Cartier, Yvette Kosmann-Schwarzbach editors. - Boston [etc.] : Birkhauser, c1993. - p. 313-359 PB - SISSA SN - 0817636536 UR - http://hdl.handle.net/1963/6478 U1 - 6432 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Integrable systems in topological field theory JF - Nuclear Physics B. Volume 379, Issue 3, 1992, pages : 627-689 Y1 - 1992 A1 - Boris Dubrovin AB - Integrability of the system of PDE for dependence on coupling parameters of the (tree-level) primary partition function in massive topological field theories, being imposed by the associativity of the perturbed primary chiral algebra, is proved. In the conformal case it is shown that all the topological field theories are classified as solutions of a universal high-order Painlevé-type equation. Another integrable hierarchy (of systems of hydrodynamic type) is shown to describe coupling to gravity of the matter sector of any topological field theory. Different multicritical models with the given structure of primary correlators are identified with particular self-similar solutions of the hierarchy. The partition function of any of the models is calculated as the corresponding tau-function of the hierarchy. PB - SISSA UR - http://hdl.handle.net/1963/6477 U1 - 6433 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER -