The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order solitary-wave collisions are elastic, to the order of approximation of the extended KdV equation. In addition, the higher-order corrections to the phase shifts are found. To examine the exact nature of higher-order, solitary-wave collisions, numerical results for various special cases (including surface waves on shallow water) of the extended KdV equation are presented. The numerical results show evidence of inelastic behaviour well beyond the order of approximation of the extended KdV equation; after collision, a dispersive wavetrain of extremely small amplitude is found behind the smaller, higher-order solitary wave.

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The Gray-Scott model of cubic-autocatalysis with linear decay is coupled with diffusion and considered in a one-dimensional reactor (a reaction-diffusion cell). The boundaries of the reactor are permeable, so diffusion occurs from external reservoirs that contain fixed concentrations of the reactant and catalyst.The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations. The ordinary differential equations are then analysed to obtain semi-analytical results for the reaction-diffusion cell.Steady-state concentration profiles and bifurcation diagrams are obtained both explicitly, for the one-term method, and as the solution to a pair of transcendental equations, for the two-term method. Singularity theory is used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. Also, in the semi-analytical model, a fifth bifurcation diagram occurs in an extremely small parameter region; its size being O(10 −10 ).The region of parameter space, in which Hopf bifurcations can occur, is found by a local stability analysis of the semi-analytical model. An example of a stable limitcycle is also considered in detail. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.

Short-crested waves are defined as propagating surface gravity waves which are doublyperiodic in the horizontal plane. Linearly, the short-crested wave system we consider occurs when two progressive wavetrains of equal amplitude and frequency are propagating at an angle to each other.Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Harmonic resonance affects the solutions but Pade approximants can be used to estimate wave properties such as maximum wave steepness, frequency, kinetic energy and potential energy.The force exerted by waves being reflected by a seawall is also calculated. Our results for the maximum depth-integrated onshore wave force in the standing wave limit are compared with experiment. The maximum force exerted on a seawall occurs for a steep wave in shallow water incident at an oblique angle. Results are given for this maximum force.