00978nas a2200121 4500008004300000245018700043210006900230520046200299100002000761700001800781700002100799856003600820 2009 en_Ud 00aOn universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the {\\\\it tritronquée} solution to the Painlevé-I equation0 auniversality of critical behaviour in the focusing nonlinear Sch3 aWe argue that the critical behaviour near the point of ``gradient catastrophe\\\" of the solution to the Cauchy problem for the focusing nonlinear Schr\\\\\\\"odinger equation $ i\\\\epsilon \\\\psi_t +\\\\frac{\\\\epsilon^2}2\\\\psi_{xx}+ |\\\\psi|^2 \\\\psi =0$ with analytic initial data of the form $\\\\psi(x,0;\\\\epsilon) =A(x) e^{\\\\frac{i}{\\\\epsilon} S(x)}$ is approximately described by a particular solution to the Painlev\\\\\\\'e-I equation.1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian uhttp://hdl.handle.net/1963/252501070nas a2200121 4500008004100000020002200041245006200063210005900125260003400184520067400218100002000892856003600912 2006 en d a978-0-8218-4674-200aOn universality of critical behaviour in Hamiltonian PDEs0 auniversality of critical behaviour in Hamiltonian PDEs bAmerican Mathematical Society3 aOur main goal is the comparative study of singularities of solutions to\\r\\nthe systems of rst order quasilinear PDEs and their perturbations containing higher\\r\\nderivatives. The study is focused on the subclass of Hamiltonian PDEs with one\\r\\nspatial dimension. For the systems of order one or two we describe the local structure\\r\\nof singularities of a generic solution to the unperturbed system near the point of\\r\\n\\\\gradient catastrophe\\\" in terms of standard objects of the classical singularity theory;\\r\\nwe argue that their perturbed companions must be given by certain special solutions\\r\\nof Painlev e equations and their generalizations.1 aDubrovin, Boris uhttp://hdl.handle.net/1963/6491