Stochastic heat and Burgers equations and their singularities. I. Geometrical properties

Abstract

Arnol’d and Thom’s beautiful classification of caustics (shockwaves) for the Burgers equation suggests a similar one for the wavefronts of the corresponding heat equation. We give here a general theorem for Hamiltonian systems characterizing how the level surfaces of Hamilton’s principal function (wavefronts) meet the caustic surface in both the deterministic and stochastic cases. We further show how these results can be applied to the stochastic Burgers equation by using earlier results of Truman and Zhao. The generic example of a caustic, appearing in the two-dimensional case, is the semicubical parabolic cusp with the corresponding zero level surface being a combination of a generalized hypocycloid and a line pair. We refer to these as the cusp and tricorn. The analogous butterfly caustic, in the three-dimensional case, has a cusped zero level surface, the fish, which meets the butterfly caustic in three cusped curves and touches it along a straight line. Our results explain in terms of classical mechanics the properties of the caustic and wavefront for these two archetypal examples and characterize the caustic-wavefront intersection for the general stochastic case. We discuss the application of these results to turbulence for the Burgers velocity field.