Institution

Boston University

Abstract

The inviscid Burgers equation is part of an "integrable hierarchy" of PDEs, also known as dispersionless KP. Work by A.V. Zabrodin (2001) gave a geometric solution that uses moments of an analytic curve, but no explicit formulas; A. Boyarsky, A. Marshakov, O. Ruchayskiy, P. Wiegmann and A. Zabrodin (2001) identified one case in which the curve is algebraic and the solution can be made explicit. In joint work with Shigeki Matsutani (2008), explicit solutions were found, that use Klein's sigma function on any curve that is a cyclic cover of the Riemann sphere; these were extended (2019) to any smooth curve in Weierstrass canonical form. The independent variables are integrals on the Jacobian of the curve. The talk will present the definition of the sigma function and the explicit solutions, and review the Zabrodin construction to pose the question of the relationship between the two.

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Special-Functions Solutions of Burgers' Equation

The inviscid Burgers equation is part of an "integrable hierarchy" of PDEs, also known as dispersionless KP. Work by A.V. Zabrodin (2001) gave a geometric solution that uses moments of an analytic curve, but no explicit formulas; A. Boyarsky, A. Marshakov, O. Ruchayskiy, P. Wiegmann and A. Zabrodin (2001) identified one case in which the curve is algebraic and the solution can be made explicit. In joint work with Shigeki Matsutani (2008), explicit solutions were found, that use Klein's sigma function on any curve that is a cyclic cover of the Riemann sphere; these were extended (2019) to any smooth curve in Weierstrass canonical form. The independent variables are integrals on the Jacobian of the curve. The talk will present the definition of the sigma function and the explicit solutions, and review the Zabrodin construction to pose the question of the relationship between the two.