Critical asymptotic behavior for the Korteweg–de Vries equation and in random matrix theory

Summary

We discuss universality in random matrix theory and in the study of Hamiltonian partial differential equations. We focus on universality of critical behavior and we compare results in unitary random matrix ensembles with their counterparts for the Korteweg–de Vries equation, emphasizing the similarities between both subjects.

1. Introduction

It has been observed and conjectured that the critical behavior of solutions to Hamiltonian perturbations of hyperbolic and elliptic systems of partial differential equations near points of gradient catastrophe is asymptotically independent of the chosen initial data and independent of the chosen equation [Dubrovin 2006; Dubrovin et al. 2009]. A classical example of a Hamiltonian perturbation of a hyperbolic equation which exhibits such universal behavior, is the Korteweg–de Vries (KdV) equation

If one is interested in the behavior of KdV solutions in the small dispersion limit ∈ → 0, it is natural to study first the inviscid Burgers’ or Hopf equation u_t + 6uu_x = 0. Given smooth initial data u(x, 0) = u₀(x) decaying at ±∞, the solution of this equation is, for t sufficiently small, given by the method of characteristics: we have u(x,t) = u(ξ(x,t)), where ξ(x,t) is given as the solution to the equation It is easily derived from this implicit form of the solution that the x-derivative of u(x,t) blows up at time

which is called the time of gradient catastrophe. After this time, the Hopf solution u(x,t) ceases to exist in the classical sense. For t slightly smaller than the critical time tc the KdV solution starts to oscillate as shown numerically in [Grava and Klein 2007]. For t > t_c the KdV solution develops a train of rapid oscillations of wavelength of order ∈. In general, the asymptotics for the KdV solution as ∈→0 can be described in terms of an equilibrium problem, discovered by Lax and Levermore [1983a; 1983b; 1983c; Lax et al. 1993].

The support of the solution of the equilibrium problem, which depends on x and t, consists of a finite or infinite union of intervals [Grava 2004; Deift et al. 1998b], and the endpoints evolve according to the Whitham equations [Whitham 1974; Flaschka et al. 1980]. For t < t_c, the support of the equilibrium problem consists of one interval and the KdV solution as ∈ → 0 is approximated by the Hopf solution.