00506nas a2200145   4500008004100000022001400041245011400055210006900169300001200238490000800250100001900258700002000277700001300297856005000310       2016                        eng d  a0167-278900aCorrelation functions of the KdV hierarchy and applications to intersection numbers over $\overline\CalM_g,n$0 aCorrelation functions of the KdV hierarchy and applications to i  a30–570 v3271 aBertola, Marco1 aDubrovin, Boris1 aYang, Di  uhttp://dx.doi.org/10.1016/j.physd.2016.04.00801219nas a2200145   4500008004100000245008300041210006900124260001000193520068100203100002000884700001800904700002100922700001800943856011200961       2013                        en  d00aOn critical behaviour in systems of Hamiltonian partial differential equations0 acritical behaviour in systems of Hamiltonian partial differentia  bSISSA3 a<p>We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_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.</p>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/1963/646500899nas 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/1963/1997