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-equations00948nas a2200121   4500008004100000245006000041210006000101260002200161520056500183100002400748700001800772856003600790       2012                        en  d00aThermodynamic phase transitions and shock singularities0 aThermodynamic phase transitions and shock singularities  bThe Royal Society3 aWe show that under rather general assumptions on the form of the entropy\\r\\nfunction, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of the characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of\\r\\nnon-hydrodynamic type such as the classical plasma and the ideal Bose gas are\\r\\nalso discussed.1 aDe Nittis, Giuseppe1 aMoro, Antonio  uhttp://hdl.handle.net/1963/6090