TY - JOUR
T1 - Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures
JF - Adv. Math. 219 (2008) 780-837
Y1 - 2008
A1 - Boris Dubrovin
A1 - Liu Si-Qi
A1 - Zhang Youjin
AB - The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations.
UR - http://hdl.handle.net/1963/2523
U1 - 1595
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - RPRT
T1 - Extended affine Weyl groups and Frobenius manifolds -- II
Y1 - 2006
A1 - Boris Dubrovin
A1 - Zhang Youjin
A1 - Zuo Dafeng
AB - For the root system of type $B_l$ and $C_l$, we generalize the result of \\\\cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of \\\\cite{DZ1998}.
UR - http://hdl.handle.net/1963/1787
U1 - 2757
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations
JF - Comm. Pure Appl. Math. 59 (2006) 559-615
Y1 - 2006
A1 - Boris Dubrovin
A1 - Liu Si-Qi
A1 - Zhang Youjin
AB - We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \\\\cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.
UR - http://hdl.handle.net/1963/2535
U1 - 1583
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - The Extended Toda Hierarchy
JF - Moscow Math. J. 4 (2004)\\n313-332.
Y1 - 2004
A1 - Guido Carlet
A1 - Boris Dubrovin
A1 - Zhang Youjin
UR - http://hdl.handle.net/1963/2542
U1 - 1577
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Virasoro Symmetries of the Extended Toda Hierarchy
JF - Comm. Math.\\nPhys. 250 (2004) 161-193.
Y1 - 2004
A1 - Boris Dubrovin
A1 - Zhang Youjin
AB - We prove that the extended Toda hierarchy of \\\\cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\\\\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.
UR - http://hdl.handle.net/1963/2544
U1 - 1575
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Frobenius manifolds and Virasoro constraints
JF - Selecta Math. (N.S.) 5 (1999) 423-466
Y1 - 1999
A1 - Boris Dubrovin
A1 - Zhang Youjin
AB - For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus $\\\\leq 1$ Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology.
PB - Springer
UR - http://hdl.handle.net/1963/2883
U1 - 1817
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation
JF - Comm. Math. Phys. 198 (1998) 311-361
Y1 - 1998
A1 - Boris Dubrovin
A1 - Zhang Youjin
AB - We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov - Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.
PB - Springer
UR - http://hdl.handle.net/1963/3696
U1 - 609
U2 - Mathematics
U3 - Mathematical Physics
ER -