Asymptotic properties of solutions of the linear dispersion equation ut = uxxx in R × R+, and its (2k + 1)th-order generalisations are studied. General Hermitian spectral theory and asymptotic behaviour of its kernel, for the rescaled operator B = D3 + 1 3 yDy + 1 3 I, is developed, where a complete set of bi-orthonormal pair of eigenfunctions, {ψβ}, {ψ∗β }, are found. The results apply to the construction of VSS (very singular solutions) of the semilinear equation with absorption ut = uxxx − |u|p−1u in R × R+, where p > 1, which serves as a basic model for various applications, including the classic KdV area. Finally, the nonlinear dispersion equations such as ut = (|u|nu)xxx in R × R+, and ut = (|u|nu)xxx − |u|p−1u in R × R+, where n > 0, are studied and their “nonlinear eigenfunctions” are constructed. The basic tools include numerical methods and “homotopy-deformation” approaches, where the limits n → 0 and n → +∞ turn out to be fruitful. Local existence and uniqueness is proved and some bounds on the highly oscillatory tail are found. These odd-order models were not treated in existing mathematical literature, from the proposed point of view.