AI 中文总结
本文借助Hirota双线性方法解析构造(2+1)维KD方程的呼吸子相互作用解，采用三维体素盒计数法证实其产生的结构具有分形性质，相关框架可为非线性波等领域研究提供参考。
AI 中文摘要
本文借助Hirota双线性方法研究了(2+1)维Konopelchenko-Dubrovsky（KD）方程中通过非线性呼吸子相互作用产生的分形结构。首先推导了该系统的双线性形式，随后通过合适的辅助函数解析构造了呼吸子相互作用解。研究表明，耦合非线性环境中呼吸子波的相互作用会产生高度复杂的多尺度图案，该图案在连续放大下呈现出自相似行为。为表征所得结构的几何复杂性，采用了基于三维体素的盒计数法，计算得到的维数为非整数，证实了所产生图案的分形性质。此外，还进行了相对误差分析、标准误差估计、自助法标准差计算以及收敛性分析，以检验所估计维数的稳健性和可重复性。本工作表明，耦合色散系统中的非线性呼吸子相互作用可能为分形几何和复杂多尺度结构的产生提供一种自然机制，本文开发的解析与定量相结合的框架或可为流体动力学、等离子体物理及非线性波传播中出现的非线性能量局域化和尺度依赖结构提供进一步的见解。
英文摘要
Fractal structures generated through nonlinear breather interactions are investigated for the $(2+1)$-dimensional Konopelchenko--Dubrovsky (KD) equation by means of the Hirota bilinear method. The bilinear form of the system is first derived, after which breather interaction solutions are constructed analytically through suitable auxiliary functions. It is shown that the interaction of breather waves in the coupled nonlinear environment gives rise to highly intricate multiscale patterns exhibiting self-similar behaviour under successive magnification. To characterize the geometric complexity of the obtained structures, a three-dimensional voxel-based box-counting method is employed. The computed dimensions are found to be non-integer, confirming the fractal nature of the generated patterns. In addition, relative error analysis, standard error estimation, bootstrap standard deviation and convergence analysis are performed to examine the robustness and reproducibility of the estimated dimensions. The present work suggests that nonlinear breather interactions in coupled dispersive systems may provide a natural mechanism for the emergence of fractal geometries and complex multiscale structures. The combined analytical and quantitative framework developed here may provide further insight into nonlinear energy localization and scale-dependent structures arising in fluid dynamics, plasma physics and nonlinear wave propagation.
