8 research outputs found
On critical behaviour in systems of Hamiltonian partial differential equations
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\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture
Thermodynamic Limit and Dispersive Regularisation in Matrix Models
We show that Hermitian matrix models support the occurrence of a new type of
phase transition characterised by dispersive regularisation of the order
parameter near the critical point. Using the identification of the partition
function with a solution of a reduction of the Toda hierarchy, known as
Volterra system, we argue that the singularity is resolved by the onset of a
multi-dimensional dispersive shock of the order parameter in the space of
coupling constants. This analysis explains the origin and mechanism leading to
the emergence of chaotic behaviours observed in M^6 matrix models and extends
its validity to even nonlinearity of arbitrary order.Comment: 8 pages, 7 figures. Sections have been reorganised and expanded.
Order of figures and captions have been modified. Accepted for publication in
Physical Review