Kudryashov and Sinelshchikov’s method for solving the radial oscillation problem of multielectron bubbles in liquid helium


This work investigates the exact general analytical solution of the Rayleigh equation for multielectron bubbles in liquid helium using Kudryashov and Sinelshchikov’s method. We firstly obtain its first integral involving three negative exponential powers, then a specific Sundman transformation is proper established to transform it into an equation for the elliptic functions, and finally the analytical solution expressed by Weierstrass elliptic function is constructed appropriately. As applications, the derived analytical solution is used to test numerical algorithm, and also to construct the analytical expressions of the bubble oscillation period and the derivatives of the bubble radius. Further, the influence of the pressure on the helium is also discussed.

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This work is supported by the Key Scientific Research Projects of the Higher Education Institutions of Henan Province (20B410001, 19B460003), the Key Scientific and Technological Project of Henan Province (192102210222), and the Doctoral Fund of Henan Institute of Technology (KQ1860).

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Correspondence to Yupeng Qin.

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Qin, Y., Lou, Q., Wang, Z. et al. Kudryashov and Sinelshchikov’s method for solving the radial oscillation problem of multielectron bubbles in liquid helium. J Math Chem (2020). https://doi.org/10.1007/s10910-020-01145-y

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  • Rayleigh equation
  • Multielectron bubble in liquid helium
  • Analytical solution
  • Sundman transformation
  • Weierstrass elliptic function