Abstract
We study the tau function of the KP-hierarchy associated with an (n, 1) curve \(y^n=x-\alpha \). If \(\alpha =0\) the corresponding tau function is 1. On the other hand if \(\alpha \ne 0\) the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex operators to the latter we obtain soliton solutions on nonzero constant backgrounds.
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Acknowledgements
I would like to thank Saburo Kakei and Yasuhiro Ohta for valuable comments which they gave me at the conference “Varieties of Studies on Non-Linear Waves” held at Kyushu University in November 2019. I would also like to thank Yasuhiko Yamada for useful comments on the manuscript and Kanehisa Takasaki for pointing out the reference [7] and invaluable comments related with the dispersionless KP-hierarchy. Finally, I thank the anonymous referees for careful reading the manuscript and for important comments on the Galilean invariance of the KP-hierarchy which help me to improve the manuscript. This work was supported by JSPS KAKENHI Grant Number JP19K03528.
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Nakayashiki, A. Tau functions of (n, 1) curves and soliton solutions on nonzero constant backgrounds. Lett Math Phys 111, 85 (2021). https://doi.org/10.1007/s11005-021-01411-3
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DOI: https://doi.org/10.1007/s11005-021-01411-3