Analytical Study of Weakly Damped Shock Structures in the Adlam–Allen Model

We investigate the formation of damped oscillatory shock structures in a cold, weakly collisional plasma using the hydromagnetic Adlam–Allen (AA) model. By incorporating a small, constant dissipation term $$\nu $$ motivated by effective electron–ion scattering in the transverse direction, we derive a nonlinear second-order differential equation governing the magnetic field evolution. Using the Sagdeev pseudo-potential method and Jacobian linearization, we systematically classify the resulting structures in the phase space, revealing the conditions under which stable spiral and saddle-type solutions arise. Furthermore, by linearizing the governing equation near equilibrium points, we obtain explicit expressions for the field amplitudes, which exhibit damped harmonic behavior. Our analytical results offer a complementary perspective to prior numerical studies and provide deeper insight into the nature of weakly damped hydromagnetic shock waves relevant to space and laboratory plasmas.