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Two types of analytical undular bore solutions, of the initial value problem for the modified Korteweg-de Vries (mKdV), are found. The first, an undular bore composed of cnoidal waves, is qualitatively similar to the bore found for the KdV equation, with solitons occurring at the leading edge and small amplitude linear waves occurring at the trailing edge. The second, a newly identified type of undular bore, consists of finite amplitude sinusiodal waves, which have a rational form. At the leading edge is the mKdV algebraic soliton, while, again, small amplitude linear waves occur at the trailing edge. The initial-boundary value (IBV) problem for the mKdV equation is also examined. The solutions of the initial value problem are used to construct approximate analytical solutions of the IBV problem. An alternative analytical solution for the IBV problem, based on the assumption of an uniform train of solitons, is also developed. The parameter regimes, in which the different types of solution occur, for both the initial value and IBV problem are identified and excellent comparisons are obtained between the numerical and approximate solutions, for both problems.

Solitary wave interaction and evolution for a higher-order Hirota equation is examined. The higher-order Hirota equation is asymptotically transformed to a higher-order member of the NLS hierarchy of integrable equations, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive higher-order one- and two-soliton solutions. It is shown that the interaction is asymptotically elastic and the higher-order corrections to the coordinate and phase shifts are derived. For the higher-order Hirota equation resonance occurs between the solitary waves and linear radiation, so soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and it is found that the tail vanishes when the algebraic relationship from the asymptotic theory is satisfied. Hence a two-parameter family of higher-order asymptotic embedded solitons exists. A comparison between the theoretical predictions and numerical solutions shows strong agreement for both solitary wave interaction, where the higher-order coordinate and phase shifts are compared, and solitary wave evolution, with comparisons made of the solitary wave tail.

Marangoni-Benard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Benard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.

The microwave heating of a one-dimensional, semi-infinite material with temperature-dependent material properties is studied. The governing equations, Maxwell's equations and the forced heat equation, are considered in the limit of high-frequency radiation and small thermal diffusivity. Particular power-law dependencies are chosen for the material parameters and an uniformly valid asymptotic solution is derived, using the techniques of strained coordinates and multiple scales. The asymptotic solution is compared with numerical solutions and an example involving the development of a hot-spot (a localised area of very high temperature) is examined.