We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

}, author = {Boris Dubrovin and Tamara Grava and Christian Klein and Antonio Moro} } @article {2012, title = {Classical double, R-operators, and negative flows of integrable hierarchies}, journal = {Theoretical and Mathematical Physics. Volume 172, Issue 1, July 2012, Pages 911-931}, year = {2012}, publisher = {SISSA}, abstract = {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{\textendash}Poisson bracket on g and its extensions. We consider examples of Lie algebras g with the {\textquotedblleft}Adler{\textendash}Kostant{\textendash}Symes{\textquotedblright} 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{\textendash}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.}, doi = {10.1007/s11232-012-0086-6}, url = {http://hdl.handle.net/1963/6468}, author = {Boris Dubrovin and Taras V. Skrypnyk} } @article {2012, title = {On the critical behavior in nonlinear evolutionary PDEs with small viscocity}, journal = {Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460}, year = {2012}, publisher = {SISSA}, abstract = {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.}, doi = {10.1134/S106192081204005X}, url = {http://hdl.handle.net/1963/6465}, author = {Boris Dubrovin and Maria Elaeva} } @article {2012, title = {On the genus two free energies for semisimple Frobenius manifolds}, journal = {Russian Journal of Mathematical Physics. Volume 19, Issue 3, September 2012, Pages 273-298}, number = {arXiv:1205.5990;}, year = {2012}, note = {36 pages, 3 figures}, publisher = {SISSA}, abstract = {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.}, doi = {10.1134/S1061920812030028}, url = {http://hdl.handle.net/1963/6464}, author = {Boris Dubrovin and Si-Qi Liu and Youjin Zhang} } @article {2011, title = {Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems}, journal = {Matematische Annalen 349 (2011) 75-115}, number = {SISSA;12/2009/FM}, year = {2011}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s00208-010-0509-3}, url = {http://hdl.handle.net/1963/3584}, author = {Guido Carlet and Boris Dubrovin and Luca Philippe Mertens} } @article {2011, title = {Linearly degenerate Hamiltonian PDEs and a new class of solutions to the WDVV associativity equations}, journal = {Functional Analysis and Its Applications. Volume 45, Issue 4, December 2011, Pages 278-290}, year = {2011}, publisher = {Springer}, abstract = {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.}, keywords = {Frobenius manifold}, doi = {10.1007/s10688-011-0030-9}, url = {http://hdl.handle.net/1963/6430}, author = {Boris Dubrovin and M.V. Pavlov and Sergei A. Zykov} } @article {2011, title = {Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations}, journal = {SIAM J. Appl. Math. 71 (2011) 983-1008}, number = {arXiv:1101.0268;}, year = {2011}, publisher = {SIAM}, abstract = {This article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117{\textendash}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 Painlev{\'e}-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.}, doi = {10.1137/100819783}, url = {http://hdl.handle.net/1963/4951}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2010, title = {Hamiltonian PDEs: deformations, integrability, solutions}, journal = {Journal of Physics A: Mathematical and Theoretical. Volume 43, Issue 43, 29 October 2010, Article number 434002}, year = {2010}, publisher = {SISSA}, abstract = {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.}, doi = {10.1088/1751-8113/43/43/434002}, url = {http://hdl.handle.net/1963/6469}, author = {Boris Dubrovin} } @inbook {2009, title = {Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions}, booktitle = {New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276.}, year = {2009}, publisher = {SISSA}, organization = {SISSA}, abstract = {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.}, isbn = {978-90-481-2810-5}, url = {http://hdl.handle.net/1963/6470}, author = {Boris Dubrovin} } @article {2009, title = {On universality of critical behaviour in the focusing nonlinear Schr{\"o}dinger equation, elliptic umbilic catastrophe and the {\\\\it tritronqu{\'e}e} solution to the Painlev{\'e}-I equation}, journal = {J. Nonlinear Sci. 19 (2009) 57-94}, number = {arXiv.org;0704.0501}, year = {2009}, abstract = {We argue that the critical behaviour near the point of {\textquoteleft}{\textquoteleft}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.}, doi = {10.1007/s00332-008-9025-y}, url = {http://hdl.handle.net/1963/2525}, author = {Boris Dubrovin and Tamara Grava and Christian Klein} } @article {2008, title = {Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures}, journal = {Adv. Math. 219 (2008) 780-837}, number = {arXiv.org;0710.3115}, year = {2008}, abstract = {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.}, doi = {10.1016/j.aim.2008.06.009}, url = {http://hdl.handle.net/1963/2523}, author = {Boris Dubrovin and Liu Si-Qi and Zhang Youjin} } @article {2008, title = {Hamiltonian partial differential equations and Frobenius manifolds}, journal = {Russian Mathematical Surveys. Volume 63, Issue 6, 2008, Pages 999-1010}, year = {2008}, publisher = {SISSA}, abstract = {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{\textendash}de Vries, non-linear\\r\\nSchr{\textasciidieresis}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.}, doi = {10.1070/RM2008v063n06ABEH004575}, url = {http://hdl.handle.net/1963/6471}, author = {Boris Dubrovin} } @article {2007, title = {Canonical structure and symmetries of the Schlesinger equations}, journal = {Comm. Math. Phys. 271 (2007) 289-373}, number = {arXiv.org;math/0311261v4}, year = {2007}, abstract = {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{\texttimes}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.}, doi = {10.1007/s00220-006-0165-3}, url = {http://hdl.handle.net/1963/1997}, author = {Boris Dubrovin and Marta Mazzocco} } @article {2007, title = {On the reductions and classical solutions of the Schlesinger equations}, journal = {Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9 (2007) 157-187}, year = {2007}, publisher = {SISSA}, abstract = {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{\texttimes}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 {\textquotedblleft}more complicated{\textquotedblright}\\r\\nSchlesinger equations S(n,m) to {\textquotedblleft}simpler{\textquotedblright} S(n',m') having n' < n\\r\\nor m' < m.}, url = {http://hdl.handle.net/1963/6472}, author = {Boris Dubrovin and Marta Mazzocco} } @article {2006, title = {Extended affine Weyl groups and Frobenius manifolds -- II}, number = {SISSA;90/2005/FM}, year = {2006}, abstract = {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}.}, url = {http://hdl.handle.net/1963/1787}, author = {Boris Dubrovin and Zhang Youjin and Zuo Dafeng} } @article {2006, title = {On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour}, number = {SISSA;89/2005/FM}, year = {2006}, abstract = {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.}, doi = {10.1007/s00220-006-0021-5}, url = {http://hdl.handle.net/1963/1786}, author = {Boris Dubrovin} } @article {2006, title = {On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations}, journal = {Comm. Pure Appl. Math. 59 (2006) 559-615}, number = {arXiv.org;math/0410027}, year = {2006}, abstract = {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.}, doi = {10.1002/cpa.20111}, url = {http://hdl.handle.net/1963/2535}, author = {Boris Dubrovin and Liu Si-Qi and Zhang Youjin} } @inbook {2006, title = {On universality of critical behaviour in Hamiltonian PDEs}, booktitle = {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}, number = {American Mathematical Society translations, ISSN 0065-9290;ser. 2;v. 224}, year = {2006}, publisher = {American Mathematical Society}, organization = {American Mathematical Society}, abstract = {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.}, isbn = {978-0-8218-4674-2}, url = {http://hdl.handle.net/1963/6491}, author = {Boris Dubrovin} } @inbook {2006, title = {WDVV equations and Frobenius manifolds}, booktitle = {Encyclopedia of Mathematical Physics. Vol 1 A : A-C. Oxford: Elsevier, 2006, p. 438-447}, year = {2006}, publisher = {SISSA}, organization = {SISSA}, isbn = {0125126611}, url = {http://hdl.handle.net/1963/6473}, author = {Boris Dubrovin} } @article {2004, title = {On almost duality for Frobenius manifolds}, journal = {Amer. Math. Soc. Transl. 212 (2004)\\n75-132.}, number = {arXiv.org;math/0307374}, year = {2004}, abstract = {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.}, url = {http://hdl.handle.net/1963/2543}, author = {Boris Dubrovin} } @article {2004, title = {On analytic families of invariant tori for PDEs}, journal = {Ast{\'e}risque. Issue 297, 2004, Pages 35-65}, year = {2004}, publisher = {SISSA}, abstract = {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.}, url = {http://hdl.handle.net/1963/6474}, author = {Boris Dubrovin} } @article {2004, title = {The Extended Toda Hierarchy}, journal = {Moscow Math. J. 4 (2004)\\n313-332.}, number = {arXiv.org;nlin/0306060}, year = {2004}, url = {http://hdl.handle.net/1963/2542}, author = {Guido Carlet and Boris Dubrovin and Zhang Youjin} } @article {2004, title = {Virasoro Symmetries of the Extended Toda Hierarchy}, journal = {Comm. Math.\\nPhys. 250 (2004) 161-193.}, number = {arXiv.org;math/0308152}, year = {2004}, abstract = {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.}, doi = {10.1007/s00220-004-1084-9}, url = {http://hdl.handle.net/1963/2544}, author = {Boris Dubrovin and Zhang Youjin} } @article {2000, title = {Monodromy of certain Painlev{\'e}-VI transcendents and reflection groups}, journal = {Invent. Math. 141 (2000) 55-147}, number = {arXiv.org;math/9806056v1}, year = {2000}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/PL00005790}, url = {http://hdl.handle.net/1963/2882}, author = {Boris Dubrovin and Marta Mazzocco} } @article {1999, title = {Frobenius manifolds and Virasoro constraints}, journal = {Selecta Math. (N.S.) 5 (1999) 423-466}, number = {SISSA;84/98/FM}, year = {1999}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s000290050053}, url = {http://hdl.handle.net/1963/2883}, author = {Boris Dubrovin and Zhang Youjin} } @inbook {1999, title = {Painlev{\'e} transcendents in two-dimensional topological field theory}, booktitle = {The Painlev{\'e} property : one century later / Robert Conte ed. - New York : Springer-Verlag, 1999. - (CRM series in mathematical physics). - p. 287-412}, number = {SISSA;24/98/FM}, year = {1999}, publisher = {Springer}, organization = {Springer}, isbn = {0-387-98888-2}, url = {http://hdl.handle.net/1963/3238}, author = {Boris Dubrovin} } @article {10445, title = {Recurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory}, number = {arXiv:solv-int/990400;}, year = {1999}, institution = {SISSA}, abstract = {We present here the iteration procedure for the determination of free energy ǫ2-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times t1, t2, . . . to find the ǫ2-expansion of u(x, t1, t2, . . .) after the infinite number of shifts of u(x, 0, 0, . . .) = x along t1, t2, . . . in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to un(x).

}, url = {http://hdl.handle.net/1963/6489}, author = {Boris Dubrovin and Andrei Ya A Maltsev} } @article {1998, title = {Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation}, journal = {Comm. Math. Phys. 198 (1998) 311-361}, number = {SISSA;152/97/FM}, year = {1998}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s002200050480}, url = {http://hdl.handle.net/1963/3696}, author = {Boris Dubrovin and Zhang Youjin} } @article {1998, title = {Differential geometry of the space of orbits of a Coxeter group}, journal = {J. Differential Geometry Suppl.4 (1998) 181-211}, number = {SISSA;29/93/FM}, year = {1998}, publisher = {International Press}, abstract = {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.}, url = {http://hdl.handle.net/1963/3562}, author = {Boris Dubrovin} } @article {1998, title = {Extended affine Weyl groups and Frobenius manifolds}, journal = {Compositio Mathematica. Volume 111, Issue 2, 1998, Pages 167-219}, year = {1998}, publisher = {Kluwer}, doi = {10.1023/A:1000258122329}, url = {http://hdl.handle.net/1963/6486}, author = {Boris Dubrovin and Youjin Zhang} } @inbook {1998, title = {Geometry and analytic theory of Frobenius manifolds}, booktitle = {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}, number = {Documenta mathematica;Extra vol.}, year = {1998}, abstract = {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.}, url = {http://hdl.handle.net/1963/6488}, author = {Boris Dubrovin} } @inbook {1997, title = {Flat pencils of metrics and Frobenius manifolds}, booktitle = {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}, number = {arXiv.org;math/9803106v1}, year = {1997}, publisher = {World Scientific}, organization = {World Scientific}, abstract = {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.}, isbn = {981-02-3266-7}, url = {http://hdl.handle.net/1963/3237}, author = {Boris Dubrovin} } @inbook {1997, title = {Functionals of the Peierls - Fr{\"o}hlich Type and the Variational Principle for the Whitham Equations}, booktitle = {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}, number = {American mathematical society translations. Series 2.;v.179}, year = {1997}, publisher = {American Mathematical Society}, organization = {American Mathematical Society}, isbn = {0821806661}, url = {http://hdl.handle.net/1963/6485}, author = {Boris Dubrovin} } @article {1997, title = {Three-Phase Solutions of the Kadomtsev - Petviashvili Equation}, journal = {Studies in Applied Mathematics. Year : 1997 ; Volume: 99 ; Issue: 2 ; Pages: 137-203}, year = {1997}, publisher = {SISSA}, abstract = {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{\textasciidieresis}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.}, doi = {10.1111/1467-9590.00059}, url = {http://hdl.handle.net/1963/6484}, author = {Boris Dubrovin and Ron Flickinger and Harvey Segur} } @inbook {1995, title = {Geometry of 2D topological field theories}, booktitle = {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}, year = {1995}, publisher = {SISSA}, organization = {SISSA}, abstract = {These notes are devoted to the theory of {\textquotedblleft}equations of associativity{\textquotedblright}\\r\\ndescribing geometry of moduli spaces of 2D topological field theories.}, isbn = {3-540-60542-8}, url = {http://hdl.handle.net/1963/6483}, author = {Boris Dubrovin} } @article {1994, title = {Algebraic-geometrical Darboux coordinates in R-matrix formalism}, number = {SISSA;88/1994/FM}, year = {1994}, institution = {SISSA}, url = {http://hdl.handle.net/1963/3655}, author = {P. Diener and Boris Dubrovin} } @article {10521, title = {Integrable functional equations and algebraic geometry}, journal = {Duke Mathematical Journal. Volume: 76, Issue: 2, Pages: 645-668}, year = {1994}, publisher = {SISSA}, doi = {10.1215/S0012-7094-94-07623-0}, url = {http://hdl.handle.net/1963/6482}, author = {Boris Dubrovin and A.S. Fokas and P.M. Santini} } @inbook {1993, title = {Dispersion relations for non-linear waves and the Schottky problem}, booktitle = {Important developments in soliton theory / A. S. Fokas, V. E. Zakharov (eds.) - Berlin : Springer-Verlag, 1993. - pages : 86-98}, year = {1993}, publisher = {SISSA}, organization = {SISSA}, abstract = {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.}, isbn = {3540559132}, url = {http://hdl.handle.net/1963/6480}, author = {Boris Dubrovin} } @article {1993, title = {Geometry and integrability of topological-antitopological fusion}, journal = {Communications in Mathematical Physics. Volume 152, Issue 3, March 1993, Pages 539-564}, year = {1993}, publisher = {SISSA}, abstract = {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.}, doi = {10.1007/BF02096618}, url = {http://hdl.handle.net/1963/6481}, author = {Boris Dubrovin} } @inbook {1993, title = {Integrable systems and classification of 2D topological field theories}, booktitle = {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}, year = {1993}, publisher = {SISSA}, organization = {SISSA}, abstract = {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{\textasciiacute}e-\\r\\nVI) specifying periods of Abelian differentials on Riemann surfaces as functions on moduli\\r\\nof these surfaces.}, isbn = {0817636536}, url = {http://hdl.handle.net/1963/6478}, author = {Boris Dubrovin} } @inbook {1993, title = {Topological conformal field theory from the point of view of integrable systems}, booktitle = {Integrable quantum field theories / edited by L. Bonora ... \et al.! - New York : Plenum Press, 1993. - page : 283 - 302.}, year = {1993}, note = {NATO ASI series / B ;v. 310}, publisher = {SISSA}, organization = {SISSA}, abstract = {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.}, isbn = {0306445344}, url = {http://hdl.handle.net/1963/6479}, author = {Boris Dubrovin} } @article {1992, title = {Hamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models}, journal = {Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207}, year = {1992}, publisher = {SISSA}, abstract = {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.}, doi = {10.1007/BF02099286}, url = {http://hdl.handle.net/1963/6476}, author = {Boris Dubrovin} } @article {1992, title = {Integrable systems in topological field theory}, journal = {Nuclear Physics B. Volume 379, Issue 3, 1992, pages : 627-689}, year = {1992}, publisher = {SISSA}, abstract = {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 Painlev{\'e}-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.}, url = {http://hdl.handle.net/1963/6477}, author = {Boris Dubrovin} } @article {1991, title = {Differential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory}, year = {1991}, institution = {Scuola Normale Superiore di Pisa}, abstract = {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.}, url = {http://hdl.handle.net/1963/6475}, author = {Boris Dubrovin} }