TY - JOUR
T1 - On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the \\\\it tritronquée solution to the Painlevé-I equation
JF - J. Nonlinear Sci. 19 (2009) 57-94
Y1 - 2009
A1 - Boris Dubrovin
A1 - Tamara Grava
A1 - Christian Klein
AB - We 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.
UR - http://hdl.handle.net/1963/2525
U1 - 1593
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - CHAP
T1 - On universality of critical behaviour in Hamiltonian PDEs
T2 - Geometry, topology, and mathematical physics : S.P. Novikov\\\'s seminar : 2006-2007 / V.M. Buchstaber, I.M. Krichever, editors. - Providence, R.I. : American Mathematical Society, 2008. - pages : 59-109
Y1 - 2006
A1 - Boris Dubrovin
AB - Our 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.
JF - Geometry, topology, and mathematical physics : S.P. Novikov\\\'s seminar : 2006-2007 / V.M. Buchstaber, I.M. Krichever, editors. - Providence, R.I. : American Mathematical Society, 2008. - pages : 59-109
PB - American Mathematical Society
SN - 978-0-8218-4674-2
UR - http://hdl.handle.net/1963/6491
U1 - 6417
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -