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A Scheme for Generating Nonisospectral Integrable Hierarchies and Its Related Applications

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Abstract

Under the framework of the complex column-vector loop algebra C̃p, we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme. As applications of the scheme, we work out a nonisospectral integrable Schrödinger hierarchy and its expanding integrable model. The latter can be reduced to some nonisospectral generalized integrable Schrödinger systems, including the derivative nonlinear Schrödinger equation once obtained by Kaup and Newell. Specially, we obtain the famous Fokker-Plank equation and its generalized form, which has extensive applications in the stochastic dynamic systems. Finally, we investigate the Lie group symmetries, fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H3 and the matrix linear group SL(2,R) for the generalized Fokker-Plank equation (GFPE).

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, P. R. China

    Yu Feng Zhang & Xiang Zhi Zhang

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  1. Yu Feng Zhang
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  2. Xiang Zhi Zhang
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Correspondence to Xiang Zhi Zhang.

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Supported by the National Natural Science Foundation of China (Grant No. 11971475)

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Zhang, Y.F., Zhang, X.Z. A Scheme for Generating Nonisospectral Integrable Hierarchies and Its Related Applications. Acta. Math. Sin.-English Ser. 37, 707–730 (2021). https://doi.org/10.1007/s10114-021-0392-8

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  • DOI: https://doi.org/10.1007/s10114-021-0392-8

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