AI 中文总结
研究基于改进模型将神经冲动建模为机电密度波，通过因式分解方法得到兰伯特W扭结孤子解，揭示超对称结构并构造伙伴孤子，建立生物膜非线性机电波传播与超对称量子力学联系，为分析神经膜扰动提供理论基础。
AI 中文摘要
在改进的海姆堡-杰克逊模型中，神经冲动可被建模为机电密度波。包含高阶多项式非线性会导致一个具有三阶和四阶非线性的广义布辛涅斯克方程，在行波约化下可简化为李纳型方程。应用因式分解方法可得到精确的兰伯特W扭结孤子解，其代表膜熔化转变附近的局部非线性密度波。该因式分解不仅提供精确解，还揭示了潜在的超对称结构。相关算子满足类似于超对称量子力学的代数关系，从而能够构造一个伙伴孤子。这种超对称配对在生物膜中的非线性机电波传播与超对称量子力学方法之间建立了一种新颖且前所未有的联系。所得框架为分析神经膜中机械诱导的扰动及其非线性传播提供了理论基础，对理解创伤性脑损伤背后的生物力学机制具有潜在意义。
英文摘要
Nerve impulses can be modelled as electromechanical density waves within the improved Heimburg-Jackson model. The inclusion of higher-order polynomial nonlinearities leads to a generalized Boussinesq equation with third and fourth order nonlinearities that, under a traveling-wave reduction, reduces to a Liénard-type equation. Applying a factorization method yields exact Lambert W-kink soliton solutions that represent localized nonlinear density waves near the membrane melting transition. Beyond providing exact solutions, the factorization uncovers an underlying supersymmetric structure. The associated operators satisfy algebraic relations analogous to those of supersymmetric quantum mechanics, thereby enabling the construction of a partner soliton. This supersymmetric pairing establishes a novel and previously unexplored connection between nonlinear electromechanical wave propagation in biological membranes and supersymmetric quantum-mechanical methods. The resulting framework offers a theoretical foundation for analysing mechanically induced perturbations and their nonlinear propagation in nerve membranes, with potential implications for understanding the biomechanical mechanisms underlying traumatic brain injury.
