Extension of the pole decomposition for the multidimensional Burgers equation.

It is shown that the generalizations to more than one space dimension of the pole decomposition for the Burgers equation with finite viscosity nu and no force are of the form u=-2nu inverted Delta ln P, where the P's are explicitly known algebraic (or trigonometric) polynomials in the space variables with polynomial (or exponential) dependence on time. Such solutions have pole singularities on the complex algebraic varieties.

Integrable structure of interface dynamics

We establish the equivalence of 2D contour dynamics to the dispersionless limit of the integrable Toda hierarchy constrained by a string equation. Remarkably, the same hierarchy underlies 2D quantum gravity.