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.

1 aCotti, Giordano1 aDubrovin, Boris1 aGuzzetti, Davide uhttps://doi.org/10.1215/00127094-2018-005901258nas a2200133 4500008004100000245006300041210006300104260001000167520083800177100002001015700002001035700002101055856004801076 2018 en d00aLocal moduli of semisimple Frobenius coalescent structures0 aLocal moduli of semisimple Frobenius coalescent structures bSISSA3 aThere 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.

1 aCotti, Giordano1 aDubrovin, Boris1 aGuzzetti, Davide uhttp://preprints.sissa.it/handle/1963/3530400506nas a2200145 4500008004100000022001400041245011400055210006900169300001200238490000800250100001900258700002000277700001300297856005000310 2016 eng d a0167-278900aCorrelation functions of the KdV hierarchy and applications to intersection numbers over $\overline\CalM_g,n$0 aCorrelation functions of the KdV hierarchy and applications to i a30–570 v3271 aBertola, Marco1 aDubrovin, Boris1 aYang, Di uhttp://dx.doi.org/10.1016/j.physd.2016.04.00800397nas a2200121 4500008004100000245004500041210004500086490000900131100001900140700002000159700001300179856008300192 2016 eng d00aSimple Lie Algebras and Topological ODEs0 aSimple Lie Algebras and Topological ODEs0 v20161 aBertola, Marco1 aDubrovin, Boris1 aYang, Di uhttps://www.math.sissa.it/publication/simple-lie-algebras-and-topological-odes00912nas a2200145 4500008004100000245010700041210006900148260001000217520041300227100002000640700002400660700001800684700001600702856004800718 2015 en d00aExtended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials0 aExtended affine Weyl groups of BCD type Frobenius manifolds and bSISSA3 aFor 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.1 aDubrovin, Boris1 aStrachan, Ian, A.B.1 aZhang, Youjin1 aZuo, Dafeng uhttp://preprints.sissa.it/handle/1963/3531600957nas a2200121 4500008004100000245007500041210006900116260004100185520053400226100002000760700001900780856003600799 2014 en d00aOn an isomonodromy deformation equation without the Painlevé property0 aisomonodromy deformation equation without the Painlevé property bMaik Nauka-Interperiodica Publishing3 aWe 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.1 aDubrovin, Boris1 aKapaev, Andrey uhttp://hdl.handle.net/1963/646601625nas a2200133 4500008004100000245007900041210006900120260001300189520117000202100002301372700002001395700002501415856005101440 2014 en d00aMinimal Liouville gravity correlation numbers from Douglas string equation0 aMinimal Liouville gravity correlation numbers from Douglas strin bSpringer3 aWe 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}.1 aBelavin, Alexander1 aDubrovin, Boris1 aMukhametzhanov, Baur uhttp://urania.sissa.it/xmlui/handle/1963/3458801219nas a2200145 4500008004100000245008300041210006900124260001000193520068100203100002000884700001800904700002100922700001800943856011200961 2013 en d00aOn critical behaviour in systems of Hamiltonian partial differential equations0 acritical behaviour in systems of Hamiltonian partial differentia bSISSA3 aWe 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.

1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian1 aMoro, Antonio uhttps://www.math.sissa.it/publication/critical-behaviour-systems-hamiltonian-partial-differential-equations01113nas a2200121 4500008004100000245008000041210006900121260001000190520071100200100002000911700002400931856003600955 2012 en d00aClassical double, R-operators, and negative flows of integrable hierarchies0 aClassical double Roperators and negative flows of integrable hie bSISSA3 aUsing 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.1 aDubrovin, Boris1 aSkrypnyk, Taras, V. uhttp://hdl.handle.net/1963/646800864nas a2200121 4500008004100000245008100041210006900122260001000191520046700201100002000668700001800688856003600706 2012 en d00aOn the critical behavior in nonlinear evolutionary PDEs with small viscocity0 acritical behavior in nonlinear evolutionary PDEs with small visc bSISSA3 aWe 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.1 aDubrovin, Boris1 aElaeva, Maria uhttp://hdl.handle.net/1963/646500982nas a2200133 4500008004100000245007000041210006300111260001000174520057500184100002000759700001500779700001800794856003600812 2012 en d00aOn the genus two free energies for semisimple Frobenius manifolds0 agenus two free energies for semisimple Frobenius manifolds bSISSA3 aWe 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.1 aDubrovin, Boris1 aLiu, Si-Qi1 aZhang, Youjin uhttp://hdl.handle.net/1963/646400770nas a2200133 4500008004300000245007400043210006900117260001300186520033600199100001800535700002000553700002700573856003600600 2011 en_Ud 00aInfinite-dimensional Frobenius manifolds for 2 + 1 integrable systems0 aInfinitedimensional Frobenius manifolds for 2 1 integrable syste bSpringer3 aWe 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.1 aCarlet, Guido1 aDubrovin, Boris1 aMertens, Luca Philippe uhttp://hdl.handle.net/1963/358400923nas a2200145 4500008004100000245010600041210006900147260001300216520043000229653002300659100002000682700001700702700002200719856003600741 2011 en d00aLinearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations0 aLinearly degenerate Hamiltonian PDEs and a new class of solution bSpringer3 aWe 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.10aFrobenius manifold1 aDubrovin, Boris1 aPavlov, M.V.1 aZykov, Sergei, A. uhttp://hdl.handle.net/1963/643001547nas a2200133 4500008004100000245008700041210006900128260000900197520111200206100002001318700001801338700002101356856003601377 2011 en d00aNumerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations0 aNumerical Study of breakup in generalized Kortewegde Vries and K bSIAM3 aThis 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.1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/495100685nas a2200109 4500008004100000245006100041210005800102260001000160520034900170100002000519856003600539 2010 en d00aHamiltonian PDEs: deformations, integrability, solutions0 aHamiltonian PDEs deformations integrability solutions bSISSA3 aWe 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/646901528nas a2200121 4500008004100000020002200041245010900063210006900172260001000241520109900251100002001350856003601370 2009 en d a978-90-481-2810-500aHamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions0 aHamiltonian perturbations of hyperbolic PDEs from classification bSISSA3 aWe 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647000978nas a2200121 4500008004300000245018700043210006900230520046200299100002000761700001800781700002100799856003600820 2009 en_Ud 00aOn universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the {\\\\it tritronquée} solution to the Painlevé-I equation0 auniversality of critical behaviour in the focusing nonlinear Sch3 aWe 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.1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/252500701nas a2200121 4500008004300000245009900043210006900142520027900211100002000490700001500510700001800525856003600543 2008 en_Ud 00aFrobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures0 aFrobenius Manifolds and Central Invariants for the Drinfeld Soko3 aThe 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.1 aDubrovin, Boris1 aSi-Qi, Liu1 aYoujin, Zhang uhttp://hdl.handle.net/1963/252301495nas a2200109 4500008004100000245007100041210006900112260001000181520113800191100002001329856003601349 2008 en d00aHamiltonian partial differential equations and Frobenius manifolds0 aHamiltonian partial differential equations and Frobenius manifol bSISSA3 aIn 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647100899nas a2200109 4500008004300000245006800043210006800111520053400179100002000713700002000733856003600753 2007 en_Ud 00aCanonical structure and symmetries of the Schlesinger equations0 aCanonical structure and symmetries of the Schlesinger equations3 aThe 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.1 aDubrovin, Boris1 aMazzocco, Marta uhttp://hdl.handle.net/1963/199700931nas a2200121 4500008004100000245007500041210006800116260001000184520053900194100002000733700002000753856003600773 2007 en d00aOn the reductions and classical solutions of the Schlesinger equations0 areductions and classical solutions of the Schlesinger equations bSISSA3 aThe 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.1 aDubrovin, Boris1 aMazzocco, Marta uhttp://hdl.handle.net/1963/647200687nas a2200121 4500008004300000245006200043210005900105520031100164100002000475700001800495700001600513856003600529 2006 en_Ud 00aExtended affine Weyl groups and Frobenius manifolds -- II0 aExtended affine Weyl groups and Frobenius manifolds II3 aFor 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}.1 aDubrovin, Boris1 aYoujin, Zhang1 aDafeng, Zuo uhttp://hdl.handle.net/1963/178700819nas a2200097 4500008004300000245011600043210006900159520043700228100002000665856003600685 2006 en_Ud 00aOn Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour0 aHamiltonian perturbations of hyperbolic systems of conservation 3 aHamiltonian 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/178601265nas a2200121 4500008004300000245012600043210006900169520081600238100002001054700001501074700001801089856003601107 2006 en_Ud 00aOn Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations0 aHamiltonian perturbations of hyperbolic systems of conservation 3 aWe 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.1 aDubrovin, Boris1 aSi-Qi, Liu1 aYoujin, Zhang uhttp://hdl.handle.net/1963/253501070nas a2200121 4500008004100000020002200041245006200063210005900125260003400184520067400218100002000892856003600912 2006 en d a978-0-8218-4674-200aOn universality of critical behaviour in Hamiltonian PDEs0 auniversality of critical behaviour in Hamiltonian PDEs bAmerican Mathematical Society3 aOur 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/649100318nas a2200109 4500008004100000020001500041245004300056210004300099260001000142100002000152856003600172 2006 en d a012512661100aWDVV equations and Frobenius manifolds0 aWDVV equations and Frobenius manifolds bSISSA1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647300665nas a2200097 4500008004300000245004600043210004300089520037900132100002000511856003600531 2004 en_Ud 00aOn almost duality for Frobenius manifolds0 aalmost duality for Frobenius manifolds3 aWe 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/254301230nas a2200109 4500008004100000245005200041210004900093260001000142520091200152100002001064856003601084 2004 en d00aOn analytic families of invariant tori for PDEs0 aanalytic families of invariant tori for PDEs bSISSA3 aWe 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647400305nas a2200109 4500008004300000245003200043210002800075100001800103700002000121700001800141856003600159 2004 en_Ud 00aThe Extended Toda Hierarchy0 aExtended Toda Hierarchy1 aCarlet, Guido1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/254201068nas a2200109 4500008004300000245005500043210005500098520073100153100002000884700001800904856003600922 2004 en_Ud 00aVirasoro Symmetries of the Extended Toda Hierarchy0 aVirasoro Symmetries of the Extended Toda Hierarchy3 aWe 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.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/254401068nas a2200121 4500008004300000245007400043210007000117260001300187520067000200100002000870700002000890856003600910 2000 en_Ud 00aMonodromy of certain Painlevé-VI transcendents and reflection groups0 aMonodromy of certain PainlevéVI transcendents and reflection gro bSpringer3 aWe 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.1 aDubrovin, Boris1 aMazzocco, Marta uhttp://hdl.handle.net/1963/288200734nas a2200121 4500008004300000245004900043210004900092260001300141520038400154100002000538700001800558856003600576 1999 en_Ud 00aFrobenius manifolds and Virasoro constraints0 aFrobenius manifolds and Virasoro constraints bSpringer3 aFor 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.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/288300382nas a2200109 4500008004300000020001800043245007200061210007000133260001300203100002000216856003600236 1999 en_Ud a0-387-98888-200aPainlevé transcendents in two-dimensional topological field theory0 aPainlevé transcendents in twodimensional topological field theor bSpringer1 aDubrovin, Boris uhttp://hdl.handle.net/1963/323800981nas a2200121 4500008004100000245011300041210007000154260001000224520054300234100002000777700002600797856003600823 1999 en d00aRecurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory0 aRecurrent procedure for the determination of the free energy ε2 bSISSA3 aWe 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).

1 aDubrovin, Boris1 aMaltsev, Andrei, Ya A uhttp://hdl.handle.net/1963/648900862nas a2200121 4500008004300000245008700043210006900130260001300199520045400212100002000666700001800686856003600704 1998 en_Ud 00aBihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation0 aBihamiltonian Hierarchies in 2D Topological Field Theory At OneL bSpringer3 aWe 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.1 aDubrovin, Boris1 aYoujin, Zhang uhttp://hdl.handle.net/1963/369600609nas a2200109 4500008004300000245006800043210006800111260002400179520024000203100002000443856003600463 1998 en_Ud 00aDifferential geometry of the space of orbits of a Coxeter group0 aDifferential geometry of the space of orbits of a Coxeter group bInternational Press3 aDifferential-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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/356200348nas a2200109 4500008004100000245005600041210005600097260001100153100002000164700001800184856003600202 1998 en d00aExtended affine Weyl groups and Frobenius manifolds0 aExtended affine Weyl groups and Frobenius manifolds bKluwer1 aDubrovin, Boris1 aZhang, Youjin uhttp://hdl.handle.net/1963/648600741nas a2200097 4500008004100000245005600041210005600097520043400153100002000587856003600607 1998 en d00aGeometry and analytic theory of Frobenius manifolds0 aGeometry and analytic theory of Frobenius manifolds3 aMain 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/648800967nas a2200121 4500008004300000020001800043245005200061210005200113260002100165520060300186100002000789856003600809 1997 en_Ud a981-02-3266-700aFlat pencils of metrics and Frobenius manifolds0 aFlat pencils of metrics and Frobenius manifolds bWorld Scientific3 aThis 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/323700430nas a2200109 4500008004100000020001500041245010400056210007000160260003400230100002000264856003600284 1997 en d a082180666100aFunctionals of the Peierls - Fröhlich Type and the Variational Principle for the Whitham Equations0 aFunctionals of the Peierls Fröhlich Type and the Variational Pri bAmerican Mathematical Society1 aDubrovin, Boris uhttp://hdl.handle.net/1963/648501348nas a2200133 4500008004100000245006700041210006400108260001000172520093800182100002001120700002001140700001801160856003601178 1997 en d00aThree-Phase Solutions of the Kadomtsev - Petviashvili Equation0 aThreePhase Solutions of the Kadomtsev Petviashvili Equation bSISSA3 aThe 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.1 aDubrovin, Boris1 aFlickinger, Ron1 aSegur, Harvey uhttp://hdl.handle.net/1963/648400493nas a2200121 4500008004100000020001800041245004600059210004600105260001000151520015400161100002000315856003600335 1995 en d a3-540-60542-800aGeometry of 2D topological field theories0 aGeometry of 2D topological field theories bSISSA3 aThese notes are devoted to the theory of “equations of associativity”\\r\\ndescribing geometry of moduli spaces of 2D topological field theories.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/648300368nas a2200109 4500008004300000245006800043210006600111260001000177100001500187700002000202856003600222 1994 en_Ud 00aAlgebraic-geometrical Darboux coordinates in R-matrix formalism0 aAlgebraicgeometrical Darboux coordinates in Rmatrix formalism bSISSA1 aDiener, P.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/365500381nas a2200121 4500008004100000245005900041210005900100260001000159100002000169700001600189700001800205856003600223 1994 en d00aIntegrable functional equations and algebraic geometry0 aIntegrable functional equations and algebraic geometry bSISSA1 aDubrovin, Boris1 aFokas, A.S.1 aSantini, P.M. uhttp://hdl.handle.net/1963/648200684nas a2200121 4500008004100000020001500041245007100056210006900127260001000196520030000206100002000506856003600526 1993 en d a354055913200aDispersion relations for non-linear waves and the Schottky problem0 aDispersion relations for nonlinear waves and the Schottky proble bSISSA3 aAn 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/648001159nas a2200109 4500008004100000245006900041210006800110260001000178520080500188100002000993856003601013 1993 en d00aGeometry and integrability of topological-antitopological fusion0 aGeometry and integrability of topologicalantitopological fusion bSISSA3 aIntegrability 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/648101540nas a2200121 4500008004100000020001500041245007500056210006900131260001000200520115200210100002001362856003601382 1993 en d a081763653600aIntegrable systems and classification of 2D topological field theories0 aIntegrable systems and classification of 2D topological field th bSISSA3 aIn 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647800952nas a2200121 4500008004100000020001500041245008400056210006900140260001000209520055500219100002000774856003600794 1993 en d a030644534400aTopological conformal field theory from the point of view of integrable systems0 aTopological conformal field theory from the point of view of int bSISSA3 aRecent 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647900882nas a2200109 4500008004100000245009500041210006900136260001000205520050100215100002000716856003600736 1992 en d00aHamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models0 aHamiltonian formalism of Whithamtype hierarchies and topological bSISSA3 aWe 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647601141nas a2200109 4500008004100000245005100041210005100092260001000143520082200153100002000975856003600995 1992 en d00aIntegrable systems in topological field theory0 aIntegrable systems in topological field theory bSISSA3 aIntegrability 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/647701022nas a2200109 4500008004100000245012700041210006900168260003700237520058200274100002000856856003600876 1991 en d00aDifferential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory0 aDifferential geometry of moduli spaces and its applications to s bScuola Normale Superiore di Pisa3 aWe 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.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/6475