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Very singular solutions of odd-order PDEs, with linear and nonlinear dispersion


Reference:

Fernandes, R. S., 2009. Very singular solutions of odd-order PDEs, with linear and nonlinear dispersion. Thesis (Doctor of Philosophy (PhD)). University of Bath.

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    Abstract

    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.

    Details

    Item Type Thesis (Doctor of Philosophy (PhD))
    CreatorsFernandes, R. S.
    Uncontrolled Keywordsnonlinear dispersion, pdes, odd-order, vss
    DepartmentsFaculty of Science > Mathematical Sciences
    StatusPublished
    ID Code17913

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