| Abstract: |
| It is well-known that the self-dual Yang-Mills (SDYM) equation is a fundamental equation in conformal field theory as well as a general 4D equation in integrable systems. In Yang`s formulation, one can first solve the unreduced SDYM equation in a general complex 4D space and then implement reductions so that solutions meet the reality conditions and gauge conditions in the real 4D spaces. In this talk, I will show that the unreduced SDYM equation can be formulated from the matrix KP hierarchy and the matrix AKNS hierarchy. Such formulations are based on the Cauchy matrix scheme and solutions for the unreduced SDYM equation can be constructed by solving the Sylvester equations. These new structures enable us to obtain new solutions for the SU(N) SDYM equation in the different real 4D spaces (with different signatures) as well as to get some integrable equations arising from reductions of the SDYM equation. I will also introduce the reductions of solutions and connections (with other equations, e.g. the Fokas-Lenells equation). This talk is mainly based on joint works with Shangshuai Li and Changzheng Qu. |
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