TY - CHAP
T1 - Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions
T2 - New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276.
Y1 - 2009
A1 - Boris Dubrovin
AB - 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.
JF - New Trends in Mathematical Physics : Selected contributions of the XVth International Congress on Mathematical Physics, Springer Netherlands, 2009, pp. 231-276.
PB - SISSA
SN - 978-90-481-2810-5
UR - http://hdl.handle.net/1963/6470
U1 - 6415
U2 - Mathematics
U4 - 1
U5 - MAT/03 GEOMETRIA
ER -