Plasma Shock Layer Equations

The paper proves that travelling-wave structures of ut + f(u)x = (A(u,λ)ux)x and viscous profiles for ut + f(u)x = ε(D(u,μ)ux)x reduce, after one integration and Rankine–Hugoniot, to first-order heteroclinic ODEs; this is stated explicitly with equations (4.6)–(4.7) and Theorem 4.2 in the PDF. The candidate solution reproduces the same reduction correctly. For the 1D steady two-fluid Hall–MHD shock layer, the paper derives the eight-equation ODE system (with ζ2 = u dB2/dx, ζ3 = u dB3/dx) and defines the closures σ, χ, β; the candidate’s system and parameter definitions agree with the paper, including the optional setting P2* = P3* = 0. Only a minor notational slip (ν instead of u in two lines) appears in the candidate text; otherwise the arguments coincide. Key correspondences: heteroclinic reduction and constant C determined by Rankine–Hugoniot (paper: , ), and the final 1D steady shock-layer system with σ, χ, β and energy/Poynting terms (paper: , ).