Supersymmetric pairing of Lambert W-kink nerve impulses
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.