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 - RPRT T1 - Local moduli of semisimple Frobenius coalescent structures Y1 - 2018 A1 - Giordano Cotti A1 - Boris Dubrovin A1 - Davide Guzzetti AB -There is a conjectural relation, formulated by the second author, between the enumerative geometry of a wide class of smooth projective varieties and their derived category of coherent sheaves. In particular, there is an increasing interest for an explicit description of certain local invariants, called monodromy data, of semisimple quantum cohomologies in terms of characteristic classes of exceptional collections in the derived categories. Being intentioned to address this problem, which, to our opinion, is still not well understood, we have realized that some issues in the theory of Frobenius manifolds need to be preliminarily clarified, and that an extension of the theory itself is necessary, in view of the fact that quantum cohomologies of certain classes of homogeneous spaces may show a coalescence phenomenon.

PB - SISSA UR - http://preprints.sissa.it/handle/1963/35304 U1 - 35610 U2 - Mathematics U4 - 1 U5 - MAT/03 ER - TY - JOUR T1 - Correlation functions of the KdV hierarchy and applications to intersection numbers over $\overline\CalM_g,n$ JF - Phys. D Y1 - 2016 A1 - Marco Bertola A1 - Boris Dubrovin A1 - Di Yang VL - 327 UR - http://dx.doi.org/10.1016/j.physd.2016.04.008 ER - TY - JOUR T1 - Simple Lie Algebras and Topological ODEs JF - Int. Math. Res. Not. Y1 - 2016 A1 - Marco Bertola A1 - Boris Dubrovin A1 - Di Yang VL - 2016 ER - TY - RPRT T1 - Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials Y1 - 2015 A1 - Boris Dubrovin A1 - Ian A.B. Strachan A1 - Youjin Zhang A1 - Dafeng Zuo AB - For the root systems of type Bl, Cl and Dl, we generalize the result of [7] by showing the existence of Frobenius manifold structures on the orbit spaces of the extended affine Weyl groups that correspond to any vertex of the Dynkin diagram instead of a particular choice made in [7]. It also depends on certain additional data. We also construct LG superpotentials for these Frobenius manifold structures. PB - SISSA UR - http://preprints.sissa.it/handle/1963/35316 U1 - 35625 U2 - Mathematics U4 - 1 U5 - MAT/07 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 - Minimal Liouville gravity correlation numbers from Douglas string equation Y1 - 2014 A1 - Alexander Belavin A1 - Boris Dubrovin A1 - Baur Mukhametzhanov AB - We continue the study of $(q,p)$ Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of \cite{Moore:1991ir}, \cite{Belavin:2008kv}, where Lee-Yang series $(2,2s+1)$ was studied, to $(3,3s+p_0)$ Minimal Liouville Gravity, where $p_0=1,2$. We demonstrate that there exist such coordinates $\tau_{m,n}$ on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates $\tau_{m,n}$ are related in a non-linear fashion to the natural coupling constants $\lambda_{m,n}$ of the perturbations of Minimal Lioville Gravity by the physical operators $O_{m,n}$. We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature \cite{Goulian:1990qr}, \cite{Zamolodchikov:2005sj}, \cite{Belavin:2006ex}. PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34588 U1 - 34795 U2 - Physics U4 - 2 ER - TY - RPRT T1 - On critical behaviour in systems of Hamiltonian partial differential equations Y1 - 2013 A1 - Boris Dubrovin A1 - Tamara Grava A1 - Christian Klein A1 - Antonio Moro AB -We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

PB - SISSA U1 - 7280 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Classical double, R-operators, and negative flows of integrable hierarchies JF - Theoretical and Mathematical Physics. Volume 172, Issue 1, July 2012, Pages 911-931 Y1 - 2012 A1 - Boris Dubrovin A1 - Taras V. Skrypnyk AB - Using the classical double G of a Lie algebra g equipped with the classical R-operator, we define two sets of functions commuting with respect to the initial Lie–Poisson bracket on g and its extensions. We consider examples of Lie algebras g with the “Adler–Kostant–Symes” R-operators and the two corresponding sets of mutually commuting functions in detail. Using the constructed commutative Hamiltonian flows on different extensions of g, we obtain zero-curvature equations with g-valued U–V pairs. The so-called negative flows of soliton hierarchies are among such equations. We illustrate the proposed approach with examples of two-dimensional Abelian and non-Abelian Toda field equations. PB - SISSA UR - http://hdl.handle.net/1963/6468 U1 - 6413 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - On the critical behavior in nonlinear evolutionary PDEs with small viscocity JF - Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460 Y1 - 2012 A1 - Boris Dubrovin A1 - Maria Elaeva AB - We address the problem of general dissipative regularization of the quasilinear transport equation. We argue that the local behavior of solutions to the regularized equation near the point of gradient catastrophe for the transport equation is described by the logarithmic derivative of the Pearcey function, a statement generalizing the result of A.M.Il\\\'in \\\\cite{ilin}. We provide some analytic arguments supporting such conjecture and test it numerically. PB - SISSA UR - http://hdl.handle.net/1963/6465 U1 - 6409 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - On the genus two free energies for semisimple Frobenius manifolds JF - Russian Journal of Mathematical Physics. Volume 19, Issue 3, September 2012, Pages 273-298 Y1 - 2012 A1 - Boris Dubrovin A1 - Si-Qi Liu A1 - Youjin Zhang AB - We represent the genus two free energy of an arbitrary semisimple Frobenius\\r\\nmanifold as a sum of contributions associated with dual graphs of certain\\r\\nstable algebraic curves of genus two plus the so-called \\\"genus two G-function\\\".\\r\\nConjecturally the genus two G-function vanishes for a series of important\\r\\nexamples of Frobenius manifolds associated with simple singularities as well as\\r\\nfor ${\\\\bf P}^1$-orbifolds with positive Euler characteristics. We explain the\\r\\nreasons for such Conjecture and prove it in certain particular cases. PB - SISSA UR - http://hdl.handle.net/1963/6464 N1 - 36 pages, 3 figures U1 - 6411 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 - Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations JF - Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290 Y1 - 2011 A1 - Boris Dubrovin A1 - M.V. Pavlov A1 - Sergei A. Zykov KW - Frobenius manifold AB - We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions. PB - Springer UR - http://hdl.handle.net/1963/6430 U1 - 6367 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations JF - SIAM J. Appl. Math. 71 (2011) 983-1008 Y1 - 2011 A1 - Boris Dubrovin A1 - Tamara Grava A1 - Christian Klein AB - This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117–139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\\\\epsilon^2$, where $\\\\epsilon^2$ is the small dispersion parameter. Corrections up to order $\\\\epsilon^4$ are obtained and tested numerically. PB - SIAM UR - http://hdl.handle.net/1963/4951 U1 - 4732 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - JOUR T1 - Hamiltonian PDEs: deformations, integrability, solutions JF - Journal of Physics A: Mathematical and Theoretical. Volume 43, Issue 43, 29 October 2010, Article number 434002 Y1 - 2010 A1 - Boris Dubrovin AB - We review recent classification results on the theory of systems of nonlinear\\r\\nHamiltonian partial differential equations with one spatial dimension, including\\r\\na perturbative approach to the integrability theory of such systems, and discuss\\r\\nuniversality conjectures describing critical behaviour of solutions to such\\r\\nsystems. PB - SISSA UR - http://hdl.handle.net/1963/6469 U1 - 6414 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - CHAP T1 - Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions T2 - New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276. Y1 - 2009 A1 - Boris Dubrovin AB - We begin with presentation of classi cation results in the theory of Hamiltonian\\r\\nPDEs with one spatial dimension depending on a small parameter. Special\\r\\nattention is paid to the deformation theory of integrable hierarchies, including an\\r\\nimportant subclass of the so-called integrable hierarchies of the topological type\\r\\nassociated with semisimple Frobenius manifolds. Many well known equations of\\r\\nmathematical physics, such as KdV, NLS, Toda, Boussinesq etc., belong to this\\r\\nsubclass, but there are many new integrable PDEs, some of them being of interest\\r\\nfor applications. Connections with the theory of Gromov{Witten invariants\\r\\nand random matrices are outlined. We then address the problem of comparative\\r\\nstudy of singularities of solutions to the systems of first order quasilinear\\r\\nPDEs and their Hamiltonian perturbations containing higher derivatives. We\\r\\nformulate Universality Conjectures describing different types of critical behavior\\r\\nof perturbed solutions near the point of gradient catastrophe of the unperturbed\\r\\none. JF - New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276. PB - SISSA SN - 978-90-481-2810-5 UR - http://hdl.handle.net/1963/6470 U1 - 6415 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the \\\\it tritronquée solution to the Painlevé-I equation JF - J. Nonlinear Sci. 19 (2009) 57-94 Y1 - 2009 A1 - Boris Dubrovin A1 - Tamara Grava A1 - Christian Klein AB - We argue that the critical behaviour near the point of ``gradient catastrophe\\\" of the solution to the Cauchy problem for the focusing nonlinear Schr\\\\\\\"odinger equation $ i\\\\epsilon \\\\psi_t +\\\\frac{\\\\epsilon^2}2\\\\psi_{xx}+ |\\\\psi|^2 \\\\psi =0$ with analytic initial data of the form $\\\\psi(x,0;\\\\epsilon) =A(x) e^{\\\\frac{i}{\\\\epsilon} S(x)}$ is approximately described by a particular solution to the Painlev\\\\\\\'e-I equation. UR - http://hdl.handle.net/1963/2525 U1 - 1593 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures JF - Adv. Math. 219 (2008) 780-837 Y1 - 2008 A1 - Boris Dubrovin A1 - Liu Si-Qi A1 - Zhang Youjin AB - The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations. UR - http://hdl.handle.net/1963/2523 U1 - 1595 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Hamiltonian partial differential equations and Frobenius manifolds JF - Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010 Y1 - 2008 A1 - Boris Dubrovin AB - In the first part of this paper the theory of Frobenius manifolds\\r\\nis applied to the problem of classification of Hamiltonian systems of partial\\r\\ndifferential equations depending on a small parameter. Also developed is\\r\\na deformation theory of integrable hierarchies including the subclass of\\r\\nintegrable hierarchies of topological type. Many well-known examples\\r\\nof integrable hierarchies, such as the Korteweg–de Vries, non-linear\\r\\nSchr¨odinger, Toda, Boussinesq equations, and so on, belong to this\\r\\nsubclass that also contains new integrable hierarchies. Some of these new\\r\\nintegrable hierarchies may be important for applications. Properties of the\\r\\nsolutions to these equations are studied in the second part. Consideration\\r\\nis given to the comparative study of the local properties of perturbed and\\r\\nunperturbed solutions near a point of gradient catastrophe. A Universality\\r\\nConjecture is formulated describing the various types of critical behaviour\\r\\nof solutions to perturbed Hamiltonian systems near the point of gradient\\r\\ncatastrophe of the unperturbed solution. PB - SISSA UR - http://hdl.handle.net/1963/6471 U1 - 6416 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Canonical structure and symmetries of the Schlesinger equations JF - Comm. Math. Phys. 271 (2007) 289-373 Y1 - 2007 A1 - Boris Dubrovin A1 - Marta Mazzocco AB - The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m×m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation ofthe general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates. UR - http://hdl.handle.net/1963/1997 U1 - 2199 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On the reductions and classical solutions of the Schlesinger equations JF - Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9 (2007) 157-187 Y1 - 2007 A1 - Boris Dubrovin A1 - Marta Mazzocco AB - The Schlesinger equations S(n,m) describe monodromy preserving\\r\\ndeformations of order m Fuchsian systems with n+1 poles. They\\r\\ncan be considered as a family of commuting time-dependent Hamiltonian\\r\\nsystems on the direct product of n copies of m×m matrix algebras\\r\\nequipped with the standard linear Poisson bracket. In this paper we address\\r\\nthe problem of reduction of particular solutions of “more complicated”\\r\\nSchlesinger equations S(n,m) to “simpler” S(n′,m′) having n′ < n\\r\\nor m′ < m. PB - SISSA UR - http://hdl.handle.net/1963/6472 U1 - 6418 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - RPRT T1 - Extended affine Weyl groups and Frobenius manifolds -- II Y1 - 2006 A1 - Boris Dubrovin A1 - Zhang Youjin A1 - Zuo Dafeng AB - For the root system of type $B_l$ and $C_l$, we generalize the result of \\\\cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of \\\\cite{DZ1998}. UR - http://hdl.handle.net/1963/1787 U1 - 2757 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations JF - Comm. Pure Appl. Math. 59 (2006) 559-615 Y1 - 2006 A1 - Boris Dubrovin A1 - Liu Si-Qi A1 - Zhang Youjin AB - We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \\\\cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives. UR - http://hdl.handle.net/1963/2535 U1 - 1583 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour Y1 - 2006 A1 - Boris Dubrovin AB - Hamiltonian perturbations of the simplest hyperbolic equation $u_t + a(u) u_x=0$ are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE. JF - Comm. Math. Phys. 267 (2006) 117-139 UR - http://hdl.handle.net/1963/1786 U1 - 2758 U2 - Mathematics U3 - Mathematical Physics ER - TY - CHAP T1 - On universality of critical behaviour in Hamiltonian PDEs T2 - Geometry, topology, and mathematical physics : S.P. Novikov\\\'s seminar : 2006-2007 / V.M. Buchstaber, I.M. Krichever, editors. - Providence, R.I. : American Mathematical Society, 2008. - pages : 59-109 Y1 - 2006 A1 - Boris Dubrovin AB - Our main goal is the comparative study of singularities of solutions to\\r\\nthe systems of rst order quasilinear PDEs and their perturbations containing higher\\r\\nderivatives. The study is focused on the subclass of Hamiltonian PDEs with one\\r\\nspatial dimension. For the systems of order one or two we describe the local structure\\r\\nof singularities of a generic solution to the unperturbed system near the point of\\r\\n\\\\gradient catastrophe\\\" in terms of standard objects of the classical singularity theory;\\r\\nwe argue that their perturbed companions must be given by certain special solutions\\r\\nof Painlev e equations and their generalizations. JF - Geometry, topology, and mathematical physics : S.P. Novikov\\\'s seminar : 2006-2007 / V.M. Buchstaber, I.M. Krichever, editors. - Providence, R.I. : American Mathematical Society, 2008. - pages : 59-109 PB - American Mathematical Society SN - 978-0-8218-4674-2 UR - http://hdl.handle.net/1963/6491 U1 - 6417 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - CHAP T1 - WDVV equations and Frobenius manifolds T2 - Encyclopedia of Mathematical Physics. Vol 1 A : A-C. Oxford: Elsevier, 2006, p. 438-447 Y1 - 2006 A1 - Boris Dubrovin JF - Encyclopedia of Mathematical Physics. Vol 1 A : A-C. Oxford: Elsevier, 2006, p. 438-447 PB - SISSA SN - 0125126611 UR - http://hdl.handle.net/1963/6473 U1 - 6419 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - On almost duality for Frobenius manifolds JF - Amer. Math. Soc. Transl. 212 (2004)\\n75-132. Y1 - 2004 A1 - Boris Dubrovin AB - We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg - Witten duality. UR - http://hdl.handle.net/1963/2543 U1 - 1576 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On analytic families of invariant tori for PDEs JF - Astérisque. Issue 297, 2004, Pages 35-65 Y1 - 2004 A1 - Boris Dubrovin AB - We propose to apply a version of the classical Stokes\\r\\nexpansion method to the perturbative construction of invariant tori for\\r\\nPDEs corresponding to solutions quasiperiodic in space and time variables.\\r\\nWe argue that, for integrable PDEs all but finite number of the\\r\\nsmall divisors arising in the perturbative analysis cancel. As an illustrative\\r\\nexample we establish such cancellations for the case of KP equation.\\r\\nIt is proved that, under mild assumptions about decay of the magnitude\\r\\nof the Fourier modes all analytic families of finite-dimensional invariant\\r\\ntori for KP are given by the Krichever construction in terms of thetafunctions\\r\\nof Riemann surfaces. We also present an explicit construction\\r\\nof infinite dimensional real theta-functions and corresponding quasiperiodic\\r\\nsolutions to KP as sums of infinite number of interacting plane\\r\\nwaves. PB - SISSA UR - http://hdl.handle.net/1963/6474 U1 - 6420 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - The Extended Toda Hierarchy JF - Moscow Math. J. 4 (2004)\\n313-332. Y1 - 2004 A1 - Guido Carlet A1 - Boris Dubrovin A1 - Zhang Youjin UR - http://hdl.handle.net/1963/2542 U1 - 1577 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Virasoro Symmetries of the Extended Toda Hierarchy JF - Comm. Math.\\nPhys. 250 (2004) 161-193. Y1 - 2004 A1 - Boris Dubrovin A1 - Zhang Youjin AB - We prove that the extended Toda hierarchy of \\\\cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\\\\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy. UR - http://hdl.handle.net/1963/2544 U1 - 1575 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Monodromy of certain Painlevé-VI transcendents and reflection groups JF - Invent. Math. 141 (2000) 55-147 Y1 - 2000 A1 - Boris Dubrovin A1 - Marta Mazzocco AB - We study the global analytic properties of the solutions of a particular family of Painleve\\\' VI equations with the parameters $\\\\beta=\\\\gamma=0$, $\\\\delta={1\\\\over2}$ and $\\\\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve\\\' VI equation. PB - Springer UR - http://hdl.handle.net/1963/2882 U1 - 1818 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Frobenius manifolds and Virasoro constraints JF - Selecta Math. (N.S.) 5 (1999) 423-466 Y1 - 1999 A1 - Boris Dubrovin A1 - Zhang Youjin AB - For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus $\\\\leq 1$ Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology. PB - Springer UR - http://hdl.handle.net/1963/2883 U1 - 1817 U2 - Mathematics U3 - Mathematical Physics ER - TY - CHAP T1 - Painlevé transcendents in two-dimensional topological field theory T2 - The Painlevé property : one century later / Robert Conte ed. - New York : Springer-Verlag, 1999. - (CRM series in mathematical physics). - p. 287-412 Y1 - 1999 A1 - Boris Dubrovin JF - The Painlevé property : one century later / Robert Conte ed. - New York : Springer-Verlag, 1999. - (CRM series in mathematical physics). - p. 287-412 PB - Springer SN - 0-387-98888-2 UR - http://hdl.handle.net/1963/3238 U1 - 1463 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Recurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory Y1 - 1999 A1 - Boris Dubrovin A1 - Andrei Ya A Maltsev AB -We present here the iteration procedure for the determination of free energy ǫ2-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times t1, t2, . . . to find the ǫ2-expansion of u(x, t1, t2, . . .) after the infinite number of shifts of u(x, 0, 0, . . .) ≡ x along t1, t2, . . . in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to un(x).

JF - arXiv:solv-int/990400 PB - SISSA UR - http://hdl.handle.net/1963/6489 U1 - 6421 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation JF - Comm. Math. Phys. 198 (1998) 311-361 Y1 - 1998 A1 - Boris Dubrovin A1 - Zhang Youjin AB - We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov - Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity. PB - Springer UR - http://hdl.handle.net/1963/3696 U1 - 609 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Differential geometry of the space of orbits of a Coxeter group JF - J. Differential Geometry Suppl.4 (1998) 181-211 Y1 - 1998 A1 - Boris Dubrovin AB - Differential-geometric structures on the space of orbits of a finite Coxeter group, determined by Groth\\\\\\\'endieck residues, are calculated. This gives a construction of a 2D topological field theory for an arbitrary Coxeter group. PB - International Press UR - http://hdl.handle.net/1963/3562 U1 - 1140 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Extended affine Weyl groups and Frobenius manifolds JF - Compositio Mathematica. Volume 111, Issue 2, 1998, Pages 167-219 Y1 - 1998 A1 - Boris Dubrovin A1 - Youjin Zhang PB - Kluwer UR - http://hdl.handle.net/1963/6486 U1 - 6424 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - CHAP T1 - Geometry and analytic theory of Frobenius manifolds T2 - Proceedings of the International Congress of Mathematicians : Berlin 1998, August 18 - 27. II, Invited lectures. - Bielefeld : Universität Bielefeld, Fakultät für Mathematik cop. 1998. - pages : 315-326 Y1 - 1998 A1 - Boris Dubrovin AB - Main mathematical applications of Frobenius manifolds are\\r\\nin the theory of Gromov - Witten invariants, in singularity theory, in\\r\\ndifferential geometry of the orbit spaces of reflection groups and of their\\r\\nextensions, in the hamiltonian theory of integrable hierarchies. The theory\\r\\nof Frobenius manifolds establishes remarkable relationships between\\r\\nthese, sometimes rather distant, mathematical theories. JF - Proceedings of the International Congress of Mathematicians : Berlin 1998, August 18 - 27. II, Invited lectures. - Bielefeld : Universität Bielefeld, Fakultät für Mathematik cop. 1998. - pages : 315-326 UR - http://hdl.handle.net/1963/6488 U1 - 6422 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - CHAP T1 - Flat pencils of metrics and Frobenius manifolds T2 - Integrable systems and algebraic geometry : proceedings of the Taniguchi symposium 1997, Kobe, June 30 - July 4, 1997 and Research Institute for Mathematical Sciences, Kyoto University, July 7 - 11, 1997 / eds. M.-H. Saito, Y. Shimizu and K. Ueno. - Sing Y1 - 1997 A1 - Boris Dubrovin AB - This paper is based on the author\\\'s talk at 1997 Taniguchi Symposium \\\"Integrable Systems and Algebraic Geometry\\\". We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold $M$ appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space $L(M)$. This elucidates the relations between Frobenius manifolds and integrable hierarchies. JF - Integrable systems and algebraic geometry : proceedings of the Taniguchi symposium 1997, Kobe, June 30 - July 4, 1997 and Research Institute for Mathematical Sciences, Kyoto University, July 7 - 11, 1997 / eds. M.-H. Saito, Y. Shimizu and K. Ueno. - Sing PB - World Scientific SN - 981-02-3266-7 UR - http://hdl.handle.net/1963/3237 U1 - 1065 U2 - Mathematics U3 - Mathematical Physics ER - TY - CHAP T1 - Functionals of the Peierls - Fröhlich Type and the Variational Principle for the Whitham Equations T2 - Solitons, geometry, and topology : on the crossroad / V. M. Buchstaber, S. P. Novikov editors.- Providence : American Mathematical Society, 1997. - ( American mathematical society translations. Series 2. - vol. 179). - pages : 35-44 Y1 - 1997 A1 - Boris Dubrovin JF - Solitons, geometry, and topology : on the crossroad / V. M. Buchstaber, S. P. Novikov editors.- Providence : American Mathematical Society, 1997. - ( American mathematical society translations. Series 2. - vol. 179). - pages : 35-44 PB - American Mathematical Society SN - 0821806661 UR - http://hdl.handle.net/1963/6485 U1 - 6425 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Three-Phase Solutions of the Kadomtsev - Petviashvili Equation JF - Studies in Applied Mathematics. Year : 1997 ; Volume: 99 ; Issue: 2 ; Pages: 137-203 Y1 - 1997 A1 - Boris Dubrovin A1 - Ron Flickinger A1 - Harvey Segur AB - The Kadomtsev]Petviashvili KP. equation is known to admit explicit periodic\\r\\nand quasiperiodic solutions with N independent phases, for any integer\\r\\nN, based on a Riemann theta-function of N variables. For Ns1 and 2,\\r\\nthese solutions have been used successfully in physical applications. This\\r\\narticle addresses mathematical problems that arise in the computation of\\r\\ntheta-functions of three variables and with the corresponding solutions of\\r\\nthe KP equation. We identify a set of parameters and their corresponding\\r\\nranges, such that e¨ery real-valued, smooth KP solution associated with a\\r\\nRiemann theta-function of three variables corresponds to exactly one choice\\r\\nof these parameters in the proper range. Our results are embodied in a\\r\\nprogram that computes these solutions efficiently and that is available to the\\r\\nreader. We also discuss some properties of three-phase solutions. PB - SISSA UR - http://hdl.handle.net/1963/6484 U1 - 6426 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - CHAP T1 - Geometry of 2D topological field theories T2 - Integrable systems and quantum groups : lectures given at the 1st session of the Centro internazionale matematico estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 14-22, 1995 / R. Donagi, B. Dubrovin, E. Frenkel... [et al.] ; editors, M. Francavig Y1 - 1995 A1 - Boris Dubrovin AB - These notes are devoted to the theory of “equations of associativity”\\r\\ndescribing geometry of moduli spaces of 2D topological field theories. JF - Integrable systems and quantum groups : lectures given at the 1st session of the Centro internazionale matematico estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 14-22, 1995 / R. Donagi, B. Dubrovin, E. Frenkel... [et al.] ; editors, M. Francavig PB - SISSA SN - 3-540-60542-8 UR - http://hdl.handle.net/1963/6483 U1 - 6427 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - RPRT T1 - Algebraic-geometrical Darboux coordinates in R-matrix formalism Y1 - 1994 A1 - P. Diener A1 - Boris Dubrovin PB - SISSA UR - http://hdl.handle.net/1963/3655 U1 - 650 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 - Dispersion relations for non-linear waves and the Schottky problem T2 - Important developments in soliton theory / A. S. Fokas, V. E. Zakharov (eds.) - Berlin : Springer-Verlag, 1993. - pages : 86-98 Y1 - 1993 A1 - Boris Dubrovin AB - An approach to the Schottky problem of specification of periods of holomorphic differentials\\r\\non Riemann surfaces (or, equivalently, specification of Jacobians among all principaly\\r\\npolarized Abelian varieties) based on the theory of Kadomtsev - Petviashvili equation,\\r\\nis discussed. JF - Important developments in soliton theory / A. S. Fokas, V. E. Zakharov (eds.) - Berlin : Springer-Verlag, 1993. - pages : 86-98 PB - SISSA SN - 3540559132 UR - http://hdl.handle.net/1963/6480 U1 - 6430 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - Geometry and integrability of topological-antitopological fusion JF - Communications in Mathematical Physics. Volume 152, Issue 3, March 1993, Pages 539-564 Y1 - 1993 A1 - Boris Dubrovin AB - Integrability of equations of topological-antitopological fusion (being proposed\\r\\nby Cecotti and Vafa) describing the ground state metric on a given 2D topological\\r\\nfield theory (TFT) model, is proved. For massive TFT models these equations\\r\\nare reduced to a universal form (being independent on the given TFT model) by\\r\\ngauge transformations. For massive perturbations of topological conformal field theory\\r\\nmodels the separatrix solutions of the equations bounded at infinity are found\\r\\nby the isomonodromy deformations method. Also it is shown that the ground state\\r\\nmetric together with some part of the underlined TFT structure can be parametrized\\r\\nby pluriharmonic maps of the coupling space to the symmetric space of real positive\\r\\ndefinite quadratic forms. PB - SISSA UR - http://hdl.handle.net/1963/6481 U1 - 6429 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA 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 - CHAP T1 - Topological conformal field theory from the point of view of integrable systems T2 - Integrable quantum field theories / edited by L. Bonora ... \et al.! - New York : Plenum Press, 1993. - page : 283 - 302. Y1 - 1993 A1 - Boris Dubrovin AB - Recent results on classification of massive topological conformal field theories (TCFT) in terms of monodromy data of auxiliary linear operators with rational coefficients are presented. Procedure of coupling of a TCFT to topological gravity is described (at tree-level approximation) via certain integrable hierarchies of hydrodynamic type and their tau-functions. It is explained how the calculation of the ground state metric on TCFT can be interpreted in terms of harmonic maps. Also a construction of some models via Coxeter groups is described. JF - Integrable quantum field theories / edited by L. Bonora ... \et al.! - New York : Plenum Press, 1993. - page : 283 - 302. PB - SISSA SN - 0306445344 UR - http://hdl.handle.net/1963/6479 N1 - NATO ASI series / B ;v. 310 U1 - 6431 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Hamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models JF - Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207 Y1 - 1992 A1 - Boris Dubrovin AB - We show that the bi-hamiltonian structure of the averaged Gelfand-Dikii\\r\\nhierarchy is involved in the Landau-Ginsburg topological models (for An-Series):\\r\\nthe Casimirs for the first P.B. give the correct coupling parameters for the perturbed\\r\\ntopological minimal model; the correspondence {coupling parameters} ~ {primary\\r\\nfields} is determined by the second P.B. The partition function (at the tree level) and\\r\\nthe chiral algebra for LG models are calculated for any genus g. PB - SISSA UR - http://hdl.handle.net/1963/6476 U1 - 6434 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 - TY - RPRT T1 - Differential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory Y1 - 1991 A1 - Boris Dubrovin AB - We construct flat Riemannian metrics on moduli spaces of algebraic curves with marked meromorphic function. This gives a new class of exact algebraic-geometry solutions to certain non-linear equations in terms of functions on the moduli spaces. We show that the Riemannian metrics on the moduli spaces coincide with two-point correlators in topological conformal field theory and calculate the partition function for A_n model for arbitrary genus. A universal method for constructing complete families of conservation laws for Whitham-type hierarchies of PDEs is also proposed. JF - Preprint n.117, Scuola Normale Superiore, Pisa, November 1991, 31 pp. Published in: Surveys in Differential Geometry , Vol. IV (1999), p. 213 - 238. PB - Scuola Normale Superiore di Pisa UR - http://hdl.handle.net/1963/6475 U1 - 6435 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER -