Resonant solutions of the periodically forced KdV equation

This thesis examines the steady state solutions for the periodically forced Korteweg de Vries (fKdV) equation with general forcing. The solutions of the fKdV equation are not well known, and their study has been primarily on problems in which the forcing is a single sinusoidal mode. This research focuses on adapting pre-existing results for this topic to a general forcing in an attempt to broaden the understanding of the solutions for the generally forced KdV equation.;Numerical solutions are used to verify results. First, undamped and damped leading order approximations are shown to correspond well to the numerical results. Next, the variation of the detuning parameter is investigated and leads to parametric continuations through a rich array of steady solutions. These qualitatively distinct non-dispersive solutions give rise to issues of peak overcrowding, solution domain saturation, and simultaneous multiple peak formation.;Examination of regions of stability shows that it is classified by peak location in the solution, in relation to the extrema of the non-dispersive solution, and extends the known stability characteristics for the single mode case. Regions of solution stability are dependent on stability within each layer, yielding ranges of the detuning parameter which correspond to either stable or unstable solutions.;Using singular perturbation methods and multiple scale techniques, general matched, uniformly valid, leading order approximations are determined for the damped and undamped forced KdV equation. Following this, variation of the detuning parameter is examined, more specifically, analytical determination of peak formation and movement, as the solution is continuously varied.