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In the Atmosphere and Oceanic Fluids: Scaling Transformations, Bilinear Forms, Bäcklund Transformations and Solitons for A Generalized Variable-Coefficient Korteweg-de Vries-Modified Korteweg-de Vries Equation

Abstract

The atmosphere is an evolutionary agent essential to the shaping of a planet, while in oceanic science and daily life, liquids are commonly seen. In this paper, we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere, oceanic fluids and plasmas. With symbolic computation, beginning with a presumption, we work out certain scaling transformations, bilinear forms through the binary Bell polynomials and our scaling transformations, N solitons (with N being a positive integer) via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons. In addition, Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out. Respective dependences and constraints on the variable/constant coefficients are discussed, while those coefficients correspond to the quadratic-nonlinear, cubic-nonlinear, dispersive, dissipative and line-damping effects in the atmosphere, oceanic fluids and plasmas.

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Correspondence to Yong-jiang Guo or Wen-rui Shan.

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This work was financially supported by the National Natural Science Foundation of China (Grant No. 11871116), the Fundamental Research Funds for the Central Universities of China (Grant No. 2019XD-A11), the BUPT Innovation and Entrepreneurship Support Program, Beijing University of Posts and Telecommunications, and the National Scholarship for Doctoral Students of China.

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Gao, Xy., Guo, Yj., Shan, Wr. et al. In the Atmosphere and Oceanic Fluids: Scaling Transformations, Bilinear Forms, Bäcklund Transformations and Solitons for A Generalized Variable-Coefficient Korteweg-de Vries-Modified Korteweg-de Vries Equation. China Ocean Eng 35, 518–530 (2021). https://doi.org/10.1007/s13344-021-0047-7

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Key words

  • atmosphere
  • oceanic fluids
  • plasmas
  • generalized variable-coefficient Korteweg-de Vries-modified Korteweg-de Vries equation
  • scaling transformations
  • bilinear forms
  • N solitons
  • auto-Bäcklund transformations