The nonlinear dynamics of empty multi-bubbles with the same distance and initial conditions are studied analytically through a modified Rayleigh–Plesset equation. The collapse time and analytical solution are derived under various initial conditions. In particular, when considering a positive initial vibration velocity, the exact analytical expression for the maximal radius is obtained by solving a cubic algebraic equation. To the best of our knowledge, this is the first time that a parabolic function has been used to construct the parametric analytical solution for this case. This type of function is able to simulate the collapse motion whereby the bubble radius first grows to the maximal radius and then decays to zero. The limiting behavior of the resulting analytical results for multi-bubbles (including the collapse time, analytical solution, and maximal radius) is also investigated, enabling the corresponding analytical results for single bubbles to be deduced in the limit as the distance between the multi-bubbles approaches infinity. In addition, the dynamical characteristics and qualitative analysis of these bubbles and the effects of the relevant physical parameters are studied.

中文翻译：

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空球多泡在水动力空化中的非线性动力学分析研究

通过修正的Rayleigh-Plesset方程对具有相同距离和初始条件的空多气泡的非线性动力学进行了分析研究。崩溃时间和解析解在各种初始条件下得出。特别地，当考虑正的初始振动速度时，通过求解三次代数方程获得最大半径的精确解析表达式。据我们所知，这是首次使用抛物线函数构造这种情况的参数解析解。这种类型的功能能够模拟塌陷运动，从而气泡半径首先增大到最大半径，然后衰减到零。多气泡产生的分析结果的极限行为（包括崩溃时间，分析解，并且还研究了最大半径），从而可以在多个气泡之间的距离接近无穷远时，在极限中推导单个气泡的相应分析结果。此外，还研究了这些气泡的动力学特性和定性分析以及相关物理参数的影响。