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Bogoyavlensky–modified KdV hierarchy and toroidal Lie algebra \(\textrm{sl}^\textrm{tor}_{2}\)

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Abstract

By principal representation of toroidal Lie algebra \(\mathrm{sl^{tor}_2}\), we construct an integrable system: Bogoyavlensky–modified KdV (B–mKdV) hierarchy, which is \((2+1)\)-dimensional generalization of modified KdV hierarchy. Firstly, bilinear equations of B–mKdV hierarchy are obtained by fermionic representation of \(\mathrm{sl^{tor}_2}\) and boson–fermion correspondence, which are rewritten into Hirota bilinear forms. Also Fay-like identities of B–mKdV hierarchy are derived. Then from B–mKdV bilinear equations, we investigate Lax structure, which is another equivalent formulation of B–mKdV hierarchy. Conversely, we also derive B–mKdV bilinear equations from Lax structure. Other equivalent formulations of wave functions and dressing operator are needed when discussing bilinear equations and Lax structure. After that, Miura links between Bogoyavlensky–KdV hierarchy and B–mKdV hierarchy are discussed. Finally, we construct soliton solutions of B–mKdV hierarchy.

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Acknowledgements

This work is supported by the National Natural Science Foundations of China (Nos. 12171472 and 12261072) and “ Qinglan Project" of Jiangsu Universities. We thank Professor Chao–Zhong Wu (SYSU, China) for long-term encouragements and supports.

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  1. School of Mathematics, Sun Yat-sen University, Guangzhou, 510000, Guangdong, China

    Yi Yang

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, China

    Jipeng Cheng

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  1. Yi Yang
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Correspondence to Jipeng Cheng.

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Yang, Y., Cheng, J. Bogoyavlensky–modified KdV hierarchy and toroidal Lie algebra \(\textrm{sl}^\textrm{tor}_{2}\). Lett Math Phys 114, 50 (2024). https://doi.org/10.1007/s11005-024-01798-9

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