The Cauchy problem of the Ward equation (II)


Prof. Derchyi Wu  ( Academia Sinica )

2008 - 04 - 15 (Tue.)
14:00 - 15:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)

The self-dual Yang-Mills equations are a fundamental tool in four-manifold topology. Taking a one dimension reduction, one obtains the three-dimensional monopole equation [Hitchin,1988], whilst a two dimension reduction yields harmonic maps into Lie groups [Uhlenbeck, 1989]. These equations have generated much interest and been extensively studied. If we replace the background spaces by $R^{2,2}$ , and take a two dimension reduction of the self-dual Yang-Mills equations on $R^{1,1}$, then the harmonic maps can be reduced to integrable systems - the sine-Gordon equation [Uhlenbeck, 1992] and the wave map equation [Terng- Uhlenbeck, 2004]. In particular, these hyperbolic equations can be solved by the inverse scattering method. The Ward equation is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills equations on $R^{2,2}$.It is an integrable system, and differs only slightly from the wave map equations. In this report, using basic tools in Fourier analysis, we will solve the inverse scattering problem and the Cauchy problem of the Ward equation with non-small data. This is a generalization of results of Villarroel, Fokas, Ioannidou, Dai, Terng and Uhlenbeck.