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On initial data of the monopole equation

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Abstract

The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in ℝ2,2 to ℝ2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used by Dai and Terng to construct monopoles with continuous scattering data, and then the equation can be linearized by the scattering data, allowing one to use the inverse scattering method to solve the Cauchy problem with rapidly decaying small initial data. In this paper, we use the terminology of holomorphic bundle and transversality of certain maps, parametrized by initial data, to give more initial data, with which we can use scattering method to solve the Cauchy problem of the monopole equation up to gauge transformation.

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Authors and Affiliations

  1. School of Mathematical Sciences, Yangzhou University, Yangzhou, 225002, P. R. China

    Hong Yu Wang & Xiu Juan Zhu

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  1. Hong Yu Wang
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  2. Xiu Juan Zhu
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Correspondence to Hong Yu Wang.

Additional information

Supported by NSFC Grant No. 10671171 and NSF (Jiangsu Province, China) Grant No. BK2007073

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Wang, H.Y., Zhu, X.J. On initial data of the monopole equation. Acta. Math. Sin.-English Ser. 25, 2127–2132 (2009). https://doi.org/10.1007/s10114-009-7427-x

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  • DOI: https://doi.org/10.1007/s10114-009-7427-x

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