Conformal weldings and Dispersionless Toda hierarchy

Details Conformal weldings and Dispersionless Toda hierarchy Conformal weldings and Dispersionless Toda hierarchy

2008-08-05

arxiv.org/abs/0808.0727v1

Physics

Mathematical Physics

31 pages

Scientific paper

Given a $C^1$ homeomorphism of the unit circle $\gamma$, let $f$ and $g$ be respectively the normalized conformal maps from the unit disc and its exterior so that $\gamma= g^{-1}\circ f$ on the unit circle. In this article, we show that by suitably defined time variables, the evolutions of the pairs $(g, f)$ and $(g^{-1}, f^{-1})$ can be described by an infinite set of nonlinear partial differential equations known as dispersionless Toda hierarchy. Relations to the integrable structure of conformal maps first studied by Wiegmann and Zabrodin \cite{WZ} are discussed. An extension of the hierarchy which contains both our solution and the solution of \cite{WZ} is defined.

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