Supersymmetric pairing of Lambert W-kink nerve impulses

Nerve impulses can be modelled as electromechanical density waves within the improved Heimburg–Jackson model. Including higher-order polynomial nonlinearities leads to a generalized Boussinesq equation which, by using the travelling-wave reduction, becomes a constant-coefficient Liénard equation with a fifth-order polynomial nonlinearity. Although the resulting equation fails the Painlevé test for integrability, its resonances allow local solutions to be constructed. Noncommutative factorization of the Liénard equation yields analytical travelling-wave solutions near the membrane melting transition, including asymmetric Lambert W -kink solitons. Reversing the factorization ordering generates a supersymmetric partner with generally different amplitude and width. The explicit supersymmetric pairing of these nonlinear density-wave solutions, together with the associated non-shape-invariant supersymmetric potential, constitutes the main novelty of this work. In the context of nerve-membrane dynamics, this factorization structure can be interpreted as an effective modification of the generalized Boussinesq description of nonlinear density-wave propagation. The analytical solutions are verified numerically, and their linear stability is analysed. These results provide a mathematical framework linking higher-order nonlinearities in nerve membranes with asymmetric kink propagation, supersymmetric pairing, and stability.