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.

%B Duke Math. J. %I Duke University Press %V 168 %P 967–1108 %8 04 %G eng %U https://doi.org/10.1215/00127094-2018-0059 %R 10.1215/00127094-2018-0059 %0 Report %D 2018 %T Local moduli of semisimple Frobenius coalescent structures %A Giordano Cotti %A Boris Dubrovin %A Davide Guzzetti %XThere 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.

%I SISSA %G en %U http://preprints.sissa.it/handle/1963/35304 %1 35610 %2 Mathematics %4 1 %# MAT/03 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-01-16T11:11:03Z No. of bitstreams: 1 Preprint_Davide.pdf: 1242726 bytes, checksum: 527d898383b9f997856370b14965bbdc (MD5) %0 Journal Article %J Phys. D %D 2016 %T Correlation functions of the KdV hierarchy and applications to intersection numbers over $\overline\CalM_g,n$ %A Marco Bertola %A Boris Dubrovin %A Di Yang %B Phys. D %V 327 %P 30–57 %G eng %U http://dx.doi.org/10.1016/j.physd.2016.04.008 %R 10.1016/j.physd.2016.04.008 %0 Journal Article %J Int. Math. Res. Not. %D 2016 %T Simple Lie Algebras and Topological ODEs %A Marco Bertola %A Boris Dubrovin %A Di Yang %B Int. Math. Res. Not. %V 2016 %G eng %0 Report %D 2015 %T Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials %A Boris Dubrovin %A Ian A.B. Strachan %A Youjin Zhang %A Dafeng Zuo %X 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. %I SISSA %G en %U http://preprints.sissa.it/handle/1963/35316 %1 35625 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-05-23T10:53:54Z No. of bitstreams: 1 1510.08690.pdf: 433692 bytes, checksum: 0cd6111d36bb3b57262400e1f1cebe3a (MD5) %0 Journal Article %D 2014 %T On an isomonodromy deformation equation without the Painlevé property %A Boris Dubrovin %A Andrey Kapaev %X 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. %I Maik Nauka-Interperiodica Publishing %G en %U http://hdl.handle.net/1963/6466 %1 6410 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-08T11:29:56Z No. of bitstreams: 1 1301.7211v2.pdf: 415681 bytes, checksum: f606a3b5df4b201ae37a9d6fa1b79016 (MD5) %R 10.1134/S1061920814010026 %0 Journal Article %D 2014 %T Minimal Liouville gravity correlation numbers from Douglas string equation %A Alexander Belavin %A Boris Dubrovin %A Baur Mukhametzhanov %X 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}. %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34588 %1 34795 %2 Physics %4 2 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-09-28T14:45:32Z No. of bitstreams: 1 art2014.pdf: 914777 bytes, checksum: 6f6d70a301b22d993393d16ad5abef80 (MD5) %R 10.1007/JHEP01(2014)156 %0 Report %D 2013 %T On critical behaviour in systems of Hamiltonian partial differential equations %A Boris Dubrovin %A Tamara Grava %A Christian Klein %A Antonio Moro %XWe 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.

%I SISSA %G en %1 7280 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Tamara Grava (grava@sissa.it) on 2014-01-14T18:10:19Z No. of bitstreams: 1 EHfinal_3.pdf: 7760169 bytes, checksum: 1e98e693fbceb1268a5acd269dd9b03e (MD5) %0 Journal Article %J Theoretical and Mathematical Physics. Volume 172, Issue 1, July 2012, Pages 911-931 %D 2012 %T Classical double, R-operators, and negative flows of integrable hierarchies %A Boris Dubrovin %A Taras V. Skrypnyk %X 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. %B Theoretical and Mathematical Physics. Volume 172, Issue 1, July 2012, Pages 911-931 %I SISSA %G en %U http://hdl.handle.net/1963/6468 %1 6413 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Approved for entry into archive by Lucio Lubiana (lubiana@sissa.it) on 2013-02-11T14:56:54Z (GMT) No. of bitstreams: 0 %R 10.1007/s11232-012-0086-6 %0 Journal Article %J Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460 %D 2012 %T On the critical behavior in nonlinear evolutionary PDEs with small viscocity %A Boris Dubrovin %A Maria Elaeva %X 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. %B Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460 %I SISSA %G en %U http://hdl.handle.net/1963/6465 %1 6409 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-08T11:28:04Z\\nNo. of bitstreams: 1\\ndubrovin_elaeva.pdf: 8618184 bytes, checksum: 3c56d1922af7001581ca4ceb4c79cc3b (MD5) %R 10.1134/S106192081204005X %0 Journal Article %J Russian Journal of Mathematical Physics. Volume 19, Issue 3, September 2012, Pages 273-298 %D 2012 %T On the genus two free energies for semisimple Frobenius manifolds %A Boris Dubrovin %A Si-Qi Liu %A Youjin Zhang %X 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. %B Russian Journal of Mathematical Physics. Volume 19, Issue 3, September 2012, Pages 273-298 %I SISSA %G en %U http://hdl.handle.net/1963/6464 %1 6411 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-08T11:35:38Z\\nNo. of bitstreams: 1\\n1205.5990v1.pdf: 547320 bytes, checksum: 9c9d894fbe3241c632b6cab13379f9c9 (MD5) %R 10.1134/S1061920812030028 %0 Journal Article %J Matematische Annalen 349 (2011) 75-115 %D 2011 %T Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems %A Guido Carlet %A Boris Dubrovin %A Luca Philippe Mertens %X 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. %B Matematische Annalen 349 (2011) 75-115 %I Springer %G en_US %U http://hdl.handle.net/1963/3584 %1 716 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-03-09T09:29:31Z\\r\\nNo. of bitstreams: 1\\r\\n0902.1245.pdf: 343930 bytes, checksum: 421342a05bb8986c1bf542d6446f5792 (MD5) %R 10.1007/s00208-010-0509-3 %0 Journal Article %J Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290 %D 2011 %T Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations %A Boris Dubrovin %A M.V. Pavlov %A Sergei A. Zykov %K Frobenius manifold %X 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. %B Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290 %I Springer %G en %U http://hdl.handle.net/1963/6430 %1 6367 %2 Mathematics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-01-29T11:15:30Z No. of bitstreams: 1 dubrovin_linearly.pdf: 298813 bytes, checksum: 568feaa543b4082cc8e8fab4643dce71 (MD5) %R 10.1007/s10688-011-0030-9 %0 Journal Article %J SIAM J. Appl. Math. 71 (2011) 983-1008 %D 2011 %T Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations %A Boris Dubrovin %A Tamara Grava %A Christian Klein %X 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. %B SIAM J. Appl. Math. 71 (2011) 983-1008 %I SIAM %G en %U http://hdl.handle.net/1963/4951 %1 4732 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-27T12:18:45Z\\nNo. of bitstreams: 1\\n1101.0268v1.pdf: 522533 bytes, checksum: d9e2df220724f918ec3b888cef3593d4 (MD5) %R 10.1137/100819783 %0 Journal Article %J Journal of Physics A: Mathematical and Theoretical. Volume 43, Issue 43, 29 October 2010, Article number 434002 %D 2010 %T Hamiltonian PDEs: deformations, integrability, solutions %A Boris Dubrovin %X 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. %B Journal of Physics A: Mathematical and Theoretical. Volume 43, Issue 43, 29 October 2010, Article number 434002 %I SISSA %G en %U http://hdl.handle.net/1963/6469 %1 6414 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T14:36:26Z\\nNo. of bitstreams: 0 %R 10.1088/1751-8113/43/43/434002 %0 Book Section %B New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276. %D 2009 %T Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions %A Boris Dubrovin %X 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. %B New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276. %I SISSA %@ 978-90-481-2810-5 %G en %U http://hdl.handle.net/1963/6470 %1 6415 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T14:43:17Z\\nNo. of bitstreams: 1\\ndubrovin_icmp.pdf: 902220 bytes, checksum: a35a8999aa1c5c58113eda66180935d9 (MD5) %0 Journal Article %J J. Nonlinear Sci. 19 (2009) 57-94 %D 2009 %T On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the \\\\it tritronquée solution to the Painlevé-I equation %A Boris Dubrovin %A Tamara Grava %A Christian Klein %X 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. %B J. Nonlinear Sci. 19 (2009) 57-94 %G en_US %U http://hdl.handle.net/1963/2525 %1 1593 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-11T13:11:26Z\\nNo. of bitstreams: 1\\n0704.0501v3.pdf: 568190 bytes, checksum: cf8471fc01eea53ce252339eda81b3cd (MD5) %R 10.1007/s00332-008-9025-y %0 Journal Article %J Adv. Math. 219 (2008) 780-837 %D 2008 %T Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures %A Boris Dubrovin %A Liu Si-Qi %A Zhang Youjin %X 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. %B Adv. Math. 219 (2008) 780-837 %G en_US %U http://hdl.handle.net/1963/2523 %1 1595 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-11T12:52:04Z\\nNo. of bitstreams: 1\\n0710.3115v1.pdf: 569666 bytes, checksum: e3e72944ffd5f097ccaf975d2df90986 (MD5) %R 10.1016/j.aim.2008.06.009 %0 Journal Article %J Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010 %D 2008 %T Hamiltonian partial differential equations and Frobenius manifolds %A Boris Dubrovin %X 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. %B Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010 %I SISSA %G en %U http://hdl.handle.net/1963/6471 %1 6416 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T14:46:43Z\\nNo. of bitstreams: 1\\ndubrovin_2008_umn.pdf: 417000 bytes, checksum: 093ef887f6154d0c26771476cae72503 (MD5) %R 10.1070/RM2008v063n06ABEH004575 %0 Journal Article %J Comm. Math. Phys. 271 (2007) 289-373 %D 2007 %T Canonical structure and symmetries of the Schlesinger equations %A Boris Dubrovin %A Marta Mazzocco %X 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. %B Comm. Math. Phys. 271 (2007) 289-373 %G en_US %U http://hdl.handle.net/1963/1997 %1 2199 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-08-30T08:59:18Z\\nNo. of bitstreams: 1\\nmathDG0311261v4.pdf: 748258 bytes, checksum: c550bb118062fa82741da16a2735b68f (MD5) %R 10.1007/s00220-006-0165-3 %0 Journal Article %J Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9 (2007) 157-187 %D 2007 %T On the reductions and classical solutions of the Schlesinger equations %A Boris Dubrovin %A Marta Mazzocco %X 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. %B Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9 (2007) 157-187 %I SISSA %G en %U http://hdl.handle.net/1963/6472 %1 6418 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T14:52:15Z\\nNo. of bitstreams: 1\\ndubrovin_mazzocco_2007_irma.pdf: 300042 bytes, checksum: 85252eedf7d0fbbdf06061c20b471d44 (MD5) %0 Report %D 2006 %T Extended affine Weyl groups and Frobenius manifolds -- II %A Boris Dubrovin %A Zhang Youjin %A Zuo Dafeng %X 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}. %G en_US %U http://hdl.handle.net/1963/1787 %1 2757 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-03-30T13:34:27Z\\nNo. of bitstreams: 1\\n90FM-2005.pdf: 254524 bytes, checksum: abdf833c0cf0b01fb79663489dd6acc2 (MD5) %0 Report %D 2006 %T On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour %A Boris Dubrovin %X 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. %B Comm. Math. Phys. 267 (2006) 117-139 %G en_US %U http://hdl.handle.net/1963/1786 %1 2758 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-03-30T13:22:52Z\\nNo. of bitstreams: 1\\n89FM-2005.pdf: 250067 bytes, checksum: 1e057f524c879ec57fa25833b141b6b6 (MD5) %R 10.1007/s00220-006-0021-5 %0 Journal Article %J Comm. Pure Appl. Math. 59 (2006) 559-615 %D 2006 %T On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations %A Boris Dubrovin %A Liu Si-Qi %A Zhang Youjin %X 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. %B Comm. Pure Appl. Math. 59 (2006) 559-615 %G en_US %U http://hdl.handle.net/1963/2535 %1 1583 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-18T13:01:41Z\\nNo. of bitstreams: 1\\n0410027v2.pdf: 508002 bytes, checksum: 4e8fc8db5fc7512dd54eb832cc52192d (MD5) %R 10.1002/cpa.20111 %0 Book Section %B 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 %D 2006 %T On universality of critical behaviour in Hamiltonian PDEs %A Boris Dubrovin %X 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. %B 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 %I American Mathematical Society %@ 978-0-8218-4674-2 %G en %U http://hdl.handle.net/1963/6491 %1 6417 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T14:49:10Z\\nNo. of bitstreams: 1\\ndubrovin_2008_ams.pdf: 671674 bytes, checksum: 08be98e78f30199030c7a4062cfcb8cf (MD5) %0 Book Section %B Encyclopedia of Mathematical Physics. Vol 1 A : A-C. Oxford: Elsevier, 2006, p. 438-447 %D 2006 %T WDVV equations and Frobenius manifolds %A Boris Dubrovin %B Encyclopedia of Mathematical Physics. Vol 1 A : A-C. Oxford: Elsevier, 2006, p. 438-447 %I SISSA %@ 0125126611 %G en %U http://hdl.handle.net/1963/6473 %1 6419 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T14:55:56Z\\nNo. of bitstreams: 1\\ndubrovin_2006_enc.pdf: 16392102 bytes, checksum: 90fed38af94f1692c7c4375ca8bf30c3 (MD5) %0 Journal Article %J Amer. Math. Soc. Transl. 212 (2004)\\n75-132. %D 2004 %T On almost duality for Frobenius manifolds %A Boris Dubrovin %X 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. %B Amer. Math. Soc. Transl. 212 (2004)\\n75-132. %G en_US %U http://hdl.handle.net/1963/2543 %1 1576 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-21T12:40:01Z\\nNo. of bitstreams: 1\\n0307374v2.pdf: 617411 bytes, checksum: 7c8c52a06ea53850475a3e7055ce9af1 (MD5) %0 Journal Article %J Astérisque. Issue 297, 2004, Pages 35-65 %D 2004 %T On analytic families of invariant tori for PDEs %A Boris Dubrovin %X 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. %B Astérisque. Issue 297, 2004, Pages 35-65 %I SISSA %G en %U http://hdl.handle.net/1963/6474 %1 6420 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T14:58:35Z\\nNo. of bitstreams: 1\\ndubrovin_2004_asterisque.pdf: 318797 bytes, checksum: 3fe6968166d8ac662d44abd508bde574 (MD5) %0 Journal Article %J Moscow Math. J. 4 (2004)\\n313-332. %D 2004 %T The Extended Toda Hierarchy %A Guido Carlet %A Boris Dubrovin %A Zhang Youjin %B Moscow Math. J. 4 (2004)\\n313-332. %G en_US %U http://hdl.handle.net/1963/2542 %1 1577 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-21T12:22:27Z\\nNo. of bitstreams: 1\\n0306060v2.pdf: 242352 bytes, checksum: 7a4fc53dbf549eb7046e75be6c087368 (MD5) %0 Journal Article %J Comm. Math.\\nPhys. 250 (2004) 161-193. %D 2004 %T Virasoro Symmetries of the Extended Toda Hierarchy %A Boris Dubrovin %A Zhang Youjin %X 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. %B Comm. Math.\\nPhys. 250 (2004) 161-193. %G en_US %U http://hdl.handle.net/1963/2544 %1 1575 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-21T12:53:52Z\\nNo. of bitstreams: 1\\n0308152v2.pdf: 341202 bytes, checksum: 306b1f696e6ec01c0eabbeeeca895290 (MD5) %R 10.1007/s00220-004-1084-9 %0 Journal Article %J Invent. Math. 141 (2000) 55-147 %D 2000 %T Monodromy of certain Painlevé-VI transcendents and reflection groups %A Boris Dubrovin %A Marta Mazzocco %X 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. %B Invent. Math. 141 (2000) 55-147 %I Springer %G en_US %U http://hdl.handle.net/1963/2882 %1 1818 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-10T09:24:50Z\\nNo. of bitstreams: 1\\n9806056v1.pdf: 604033 bytes, checksum: f49a9c54ad1f165a94c653239d2c08dd (MD5) %R 10.1007/PL00005790 %0 Journal Article %J Selecta Math. (N.S.) 5 (1999) 423-466 %D 1999 %T Frobenius manifolds and Virasoro constraints %A Boris Dubrovin %A Zhang Youjin %X 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. %B Selecta Math. (N.S.) 5 (1999) 423-466 %I Springer %G en_US %U http://hdl.handle.net/1963/2883 %1 1817 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-10T09:34:57Z\\nNo. of bitstreams: 1\\n9808048v2.pdf: 352376 bytes, checksum: 12496406fad546b37d3f66131d2d51ee (MD5) %R 10.1007/s000290050053 %0 Book Section %B The Painlevé property : one century later / Robert Conte ed. - New York : Springer-Verlag, 1999. - (CRM series in mathematical physics). - p. 287-412 %D 1999 %T Painlevé transcendents in two-dimensional topological field theory %A Boris Dubrovin %B The Painlevé property : one century later / Robert Conte ed. - New York : Springer-Verlag, 1999. - (CRM series in mathematical physics). - p. 287-412 %I Springer %@ 0-387-98888-2 %G en_US %U http://hdl.handle.net/1963/3238 %1 1463 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-11-04T11:08:16Z\\nNo. of bitstreams: 1\\n9803107v2.pdf: 723839 bytes, checksum: 3066b99f8c826fa005eb229ddd03d8dc (MD5) %0 Report %D 1999 %T Recurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory %A Boris Dubrovin %A Andrei Ya A Maltsev %XWe 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).

%B arXiv:solv-int/990400 %I SISSA %G en %U http://hdl.handle.net/1963/6489 %1 6421 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:03:06Z No. of bitstreams: 1 dubrovin_maltsev_1999.pdf: 156918 bytes, checksum: b8e98c45d6ed83c8762446339a79418a (MD5) %0 Journal Article %J Comm. Math. Phys. 198 (1998) 311-361 %D 1998 %T Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation %A Boris Dubrovin %A Zhang Youjin %X 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. %B Comm. Math. Phys. 198 (1998) 311-361 %I Springer %G en_US %U http://hdl.handle.net/1963/3696 %1 609 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-27T12:07:27Z\\nNo. of bitstreams: 1\\n9712232v2.pdf: 436927 bytes, checksum: 47b60da0e628b29d60ba56f9071105ee (MD5) %R 10.1007/s002200050480 %0 Journal Article %J J. Differential Geometry Suppl.4 (1998) 181-211 %D 1998 %T Differential geometry of the space of orbits of a Coxeter group %A Boris Dubrovin %X 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. %B J. Differential Geometry Suppl.4 (1998) 181-211 %I International Press %G en_US %U http://hdl.handle.net/1963/3562 %1 1140 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-03-03T18:46:23Z\\nNo. of bitstreams: 1\\n29-93-FM.pdf: 243594 bytes, checksum: 4ed9f8d24e959397f629a8e6b6c8114d (MD5) %0 Journal Article %J Compositio Mathematica. Volume 111, Issue 2, 1998, Pages 167-219 %D 1998 %T Extended affine Weyl groups and Frobenius manifolds %A Boris Dubrovin %A Youjin Zhang %B Compositio Mathematica. Volume 111, Issue 2, 1998, Pages 167-219 %I Kluwer %G en %U http://hdl.handle.net/1963/6486 %1 6424 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:13:49Z\\nNo. of bitstreams: 1\\ndubrovin_zhang_1998_compositio.pdf: 321250 bytes, checksum: 49608b8d8e6245677c302db09170d85e (MD5) %R 10.1023/A:1000258122329 %0 Book Section %B 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 %D 1998 %T Geometry and analytic theory of Frobenius manifolds %A Boris Dubrovin %X 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. %B 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 %G en %U http://hdl.handle.net/1963/6488 %1 6422 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:05:37Z\\nNo. of bitstreams: 1\\nicm98.pdf: 184775 bytes, checksum: 33db28beb5919405d7396b9f6ba40c56 (MD5) %0 Book Section %B 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 %D 1997 %T Flat pencils of metrics and Frobenius manifolds %A Boris Dubrovin %X 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. %B 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 %I World Scientific %@ 981-02-3266-7 %G en_US %U http://hdl.handle.net/1963/3237 %1 1065 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-11-04T10:55:32Z\\nNo. of bitstreams: 1\\n9803106v1.pdf: 198515 bytes, checksum: bd38d2f5dc411787c3b0d0660cc0b783 (MD5) %0 Book Section %B 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 %D 1997 %T Functionals of the Peierls - Fröhlich Type and the Variational Principle for the Whitham Equations %A Boris Dubrovin %B 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 %I American Mathematical Society %@ 0821806661 %G en %U http://hdl.handle.net/1963/6485 %1 6425 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:16:27Z\\nNo. of bitstreams: 1\\npeierls.pdf: 108417 bytes, checksum: 9821dab3c135a3be76e7cb5cb4ec401e (MD5) %0 Journal Article %J Studies in Applied Mathematics. Year : 1997 ; Volume: 99 ; Issue: 2 ; Pages: 137-203 %D 1997 %T Three-Phase Solutions of the Kadomtsev - Petviashvili Equation %A Boris Dubrovin %A Ron Flickinger %A Harvey Segur %X 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. %B Studies in Applied Mathematics. Year : 1997 ; Volume: 99 ; Issue: 2 ; Pages: 137-203 %I SISSA %G en %U http://hdl.handle.net/1963/6484 %1 6426 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:19:00Z\\nNo. of bitstreams: 1\\ndubrovin_flickinger_segur.pdf: 1081636 bytes, checksum: a10c5af7339b1422cb469d18823c5c92 (MD5) %R 10.1111/1467-9590.00059 %0 Book Section %B 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 %D 1995 %T Geometry of 2D topological field theories %A Boris Dubrovin %X These notes are devoted to the theory of “equations of associativity”\\r\\ndescribing geometry of moduli spaces of 2D topological field theories. %B 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 %I SISSA %@ 3-540-60542-8 %G en %U http://hdl.handle.net/1963/6483 %1 6427 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:22:39Z\\nNo. of bitstreams: 1\\nmonte.pdf: 1437681 bytes, checksum: ecddb1d7310dbda2a35388d25609d3aa (MD5) %0 Report %D 1994 %T Algebraic-geometrical Darboux coordinates in R-matrix formalism %A P. Diener %A Boris Dubrovin %I SISSA %G en_US %U http://hdl.handle.net/1963/3655 %1 650 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-06-19T09:50:48Z\\nNo. of bitstreams: 1\\nconvert_SCAN-9409240.pdf: 910778 bytes, checksum: 4c51b62625fe5debfdaa55aa2db61ff8 (MD5) %0 Journal Article %J Duke Mathematical Journal. Volume: 76, Issue: 2, Pages: 645-668 %D 1994 %T Integrable functional equations and algebraic geometry %A Boris Dubrovin %A A.S. Fokas %A P.M. Santini %B Duke Mathematical Journal. Volume: 76, Issue: 2, Pages: 645-668 %I SISSA %G en %U http://hdl.handle.net/1963/6482 %1 6428 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:25:35Z\\nNo. of bitstreams: 1\\ndubrovin_fokas_santini_1994.pdf: 19581899 bytes, checksum: 25c6a7352fa60a95d891bf8ae238f20c (MD5) %R 10.1215/S0012-7094-94-07623-0 %0 Book Section %B Important developments in soliton theory / A. S. Fokas, V. E. Zakharov (eds.) - Berlin : Springer-Verlag, 1993. - pages : 86-98 %D 1993 %T Dispersion relations for non-linear waves and the Schottky problem %A Boris Dubrovin %X 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. %B Important developments in soliton theory / A. S. Fokas, V. E. Zakharov (eds.) - Berlin : Springer-Verlag, 1993. - pages : 86-98 %I SISSA %@ 3540559132 %G en %U http://hdl.handle.net/1963/6480 %1 6430 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:31:24Z\\nNo. of bitstreams: 1\\ndubrovin_1993_schottky.pdf: 162204 bytes, checksum: a23db6fee2ea5f3db9c3a3562ae196f5 (MD5) %0 Journal Article %J Communications in Mathematical Physics. Volume 152, Issue 3, March 1993, Pages 539-564 %D 1993 %T Geometry and integrability of topological-antitopological fusion %A Boris Dubrovin %X 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. %B Communications in Mathematical Physics. Volume 152, Issue 3, March 1993, Pages 539-564 %I SISSA %G en %U http://hdl.handle.net/1963/6481 %1 6429 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:28:15Z\\nNo. of bitstreams: 1\\ndubrovin_1993_cmp.pdf: 1218370 bytes, checksum: 6e082400ac3ed51e3b01ec532f9a598d (MD5) %R 10.1007/BF02096618 %0 Book Section %B 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 %D 1993 %T Integrable systems and classification of 2D topological field theories %A Boris Dubrovin %X 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. %B 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 %I SISSA %@ 0817636536 %G en %U http://hdl.handle.net/1963/6478 %1 6432 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:43:16Z\\nNo. of bitstreams: 1\\ndubrovin_1993_verdier.pdf: 341620 bytes, checksum: 7aeca49fd73426f29caffc6712ae6b94 (MD5) %0 Book Section %B Integrable quantum field theories / edited by L. Bonora ... \et al.! - New York : Plenum Press, 1993. - page : 283 - 302. %D 1993 %T Topological conformal field theory from the point of view of integrable systems %A Boris Dubrovin %X 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. %B Integrable quantum field theories / edited by L. Bonora ... \et al.! - New York : Plenum Press, 1993. - page : 283 - 302. %I SISSA %@ 0306445344 %G en %U http://hdl.handle.net/1963/6479 %1 6431 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:39:39Z No. of bitstreams: 1 dubrovin_1993_como.pdf: 5585529 bytes, checksum: ab3c2666b211166f3722fb8ef2b5716f (MD5) %0 Journal Article %J Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207 %D 1992 %T Hamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models %A Boris Dubrovin %X 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. %B Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207 %I SISSA %G en %U http://hdl.handle.net/1963/6476 %1 6434 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:47:41Z\\nNo. of bitstreams: 1\\ndubrovin_1992_cmp.pdf: 603608 bytes, checksum: 2f99a3ad40552bd6108f212c3ab48d36 (MD5) %R 10.1007/BF02099286 %0 Journal Article %J Nuclear Physics B. Volume 379, Issue 3, 1992, pages : 627-689 %D 1992 %T Integrable systems in topological field theory %A Boris Dubrovin %X 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. %B Nuclear Physics B. Volume 379, Issue 3, 1992, pages : 627-689 %I SISSA %G en %U http://hdl.handle.net/1963/6477 %1 6433 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:45:42Z\\nNo. of bitstreams: 1\\ndubrovin_1992_npb.pdf: 2696879 bytes, checksum: 163a16f37434f88134edaeaa900f7d1e (MD5) %0 Report %D 1991 %T Differential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory %A Boris Dubrovin %X 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. %B Preprint n.117, Scuola Normale Superiore, Pisa, November 1991, 31 pp. Published in: Surveys in Differential Geometry , Vol. IV (1999), p. 213 - 238. %I Scuola Normale Superiore di Pisa %G en %U http://hdl.handle.net/1963/6475 %1 6435 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-11T15:53:59Z\\nNo. of bitstreams: 1\\ndubrovin_1991_pisa.pdf: 53312802 bytes, checksum: d2f189e9a1439fd386fc50d3fa79b40b (MD5)