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.

Date of Award | 1 Nov 2008 |
---|

Original language | English |
---|

Awarding Institution | |
---|

Supervisor | Victor Galaktionov (Supervisor) |
---|

- nonlinear dispersion
- PDEs
- odd-order
- VSS

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

Fernandes, R. (Author). 1 Nov 2008

Student thesis: Doctoral Thesis › PhD