@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 {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 {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}
}
@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 {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}
}
