On universality of critical behaviour in Hamiltonian PDEs
by
Boris Dubrovin
SISSA, Trieste, Italy

We study the behaviour of solutions to systems of Hamiltonian PDEs of the form

ut = A(u) ux + B2(u; ux, uxx) + B3(u; ux, uxx, uxxx)+...

in a neighborhood of the point of gradient catastrophe of the "unperturbed" system of quasilinear PDEs of the form

ut = A(u)ux.

Here for every k=2, 3, ... B_k(u; u_x, ..., u^(k)) is a graded homogeneous polynomial of degree k in the derivatives u_x, ..., u^(k), assuming that

degu(m)=m,     m=1, 2, ...

We argue that for slowly 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 Painlevé-type equations. In the talk we will describe the universality types for some simple examples of Hamiltonian PDEs and give numerical evidences supporting our universality conjecture.

Date received: July 12, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-38.