We study travelling-wave solutions of the generalized Heimburg–Jackson equation for electromechanical pulses in lipid membranes. The model includes both fourth-order spatial dispersion (coefficient m) and mixed space–time dispersion (coefficient h), which combine into an effective dispersion parameter \(m_{\textrm{eff}} = m + h c^{2}\) that depends on wave speed c. The travelling-wave reduction yields a Duffing-type equation with linear coefficient \(\alpha \) , quadratic nonlinearity \(\beta \) , and cubic nonlinearity \(\gamma \) . We derive exact solutions including kink fronts, Jacobi elliptic waves of sn, cn, and dn type, solitary \({{\,\textrm{sech}\,}}\) pulses, and rational blow-up profiles. Within this ansatz class, algebraic compatibility forces \(\beta = 0\) for the Jacobi elliptic and \({{\,\textrm{sech}\,}}\) families, corresponding to purely cubic elastic response near the membrane phase transition; periodic orbits of the reduced Duffing equation for \(\beta \ne 0\) lie outside this ansatz class and are constructed explicitly in Weierstrass elliptic form. Within the unshifted tanh-ansatz, kink waves require the equal-well condition \(9\alpha \gamma = 2\beta ^{2}\) ; the sign of \(m_{\textrm{eff}}\gamma \) controls admissibility for the derived solution families. Using phase-plane analysis, we classify trajectories by parameter regions. We relate kink solutions to structural phase transitions in membranes, periodic waves to trains of electromechanical pulses, and solitary waves to nerve-like solitons. The results make explicit the ansatz scope under which these exact solutions are admissible, and unify the classical and improved Heimburg–Jackson formulations within a single dispersive framework.
