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Riemann-Hilbert approach to TD equation with nonzero boundary condition

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Abstract

We extend the Riemann-Hilbert approach to the TD equation, which is a highly nonlinear differential integrable equation. Zero boundary condition at infinity for the TD equation is not suitable. Inverse scattering transform for this equation involves the singular Riemann-Hilbert problem, which means that the sectionally analytic functions have singularities on the boundary curve. Regularization procedures of the singular Riemann-Hilbert problem for two cases, the general case and the case for reflectionless potentials, are considered. Solitonic solutions to the TD equation are given.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11471295, 11331008).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Junyi Zhu, Linlin Wang & Xianguo Geng

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  1. Junyi Zhu
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  2. Linlin Wang
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  3. Xianguo Geng
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Correspondence to Junyi Zhu.

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Zhu, J., Wang, L. & Geng, X. Riemann-Hilbert approach to TD equation with nonzero boundary condition. Front. Math. China 13, 1245–1265 (2018). https://doi.org/10.1007/s11464-018-0729-5

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  • DOI: https://doi.org/10.1007/s11464-018-0729-5

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