01219nas 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-equations