seminar: On universality of critical behaviour in Hamiltonian PDEs

speaker: Boris Dubrovin (SISSA)

abstract: We study the behaviour of solutions to systems of Hamiltonian PDEs of the form $$ u_t =A(u) u_x + B₂(u; u_x, u_{xx}) + B₃(u; u_x, u_{xx}, u_{xxx})+\dots $$ in a neighborhood of the point of gradient catastrophe of the "unperturbed" system of quasilinear PDEs of the form $$ u_t =A(u)u_x. $$ Here for every $k=2, 3, \dots$ $B_k(u; u_x, \dots, u^{(k)})$ is a graded homogeneous polynomial of degree k in the derivatives $u_x$, \dots, $u^{(k)}$, assuming that $$ \deg u^{(m)}=m, \quad m=1, 2, \dots $$ We argue that for slow varying initial data this behaviour asymptotically does not depend on the solution, up to shifts, rescalings and other simple transformations, and is described by certain special solutions to Painleve'-type equations. In the talk we will describe the universality types for some simple examples of Hamiltonian PDEs and give numerical evidences supporting our unversality conjecture.

timetable:
Sat 1 Sep, 10:00 - 11:00, Lizzanello
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