01-45 Ian M Davies, Aubrey Truman, Huaizhong Zhao

Stochastic Heat and Burgers Equations and Their Singularities - Geometrical and Analytical Properties (The Fish and the Butterfly, and Why.) (2290K, Postscript) Jan 31, 01

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Abstract. Arnol'd and Thom's beautiful classification of caustics (shockwaves for Burgers equation) suggests a similar one for the corresponding wavefronts for the heat equation. For instance, the wavefront corresponding to the simplest semi-cubical parabolic Cusp is the Tricorn and that corresponding to the Butterfly is the Fish. The Tricorn meets the semi-cubical parabola in three cusps, the Fish meets the Butterfly in three cusped curves and touches it along a straight line. We give here a general theorem for Hamiltonian systems characterising how the level surfaces of Hamiliton's principal function meet the caustic explaining the way that the Butterfly and Fish meet and a myriad of similar results. We show how these results can be applied to the stochastic Burgers equation by using earlier results of Truman and Zhao. We also explain how the characterisation of caustics and wavefronts carries over to the stochastic case and how our results generalise to this situation.

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