A one-dimensional analysis of real and complex turbulence and the Maxwell set for the stochastic Burgers equation

Abstract

The inviscid limit of the Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimizing Hamilton–Jacobi function and the classical mechanical caustic and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one-dimensional) action function using a circle of ideas due to Arnol'd, Cayley and Klein. We characterize those parts of the caustic which are singular, and give an explicit expression for the cusp density on caustics and level surfaces. By considering the double points of level surfaces we find an explicit formula for the Maxwell set in the two-dimensional polynomial case, and we extend this to higher dimensions using a double discriminant of the reduced action, solving a long-standing problem for Hamiltonian dynamical systems. When the pre-level surface touches the pre-caustic, the geometry (number of cusps) on the level surface changes infinitely rapidly causing 'real turbulence'. Using an idea of Klein, it is shown that the geometry (number of swallowtails) on the caustic also changes infinitely rapidly when the real part of the pre-caustic touches its complex counterpart, causing 'complex turbulence'. These are both inherently stochastic in nature, and we determine their intermittence in terms of the recurrent behaviour of two processes.