Chaos in KdV-like equations

The correspondence between soliton solutions of nonlinear partial differential equations and homoclinic orbits in appropriately chosen phase spaces of these equations is well-known. Perturbations of homoclinic orbits can be studied by use of the Melnikov technique; this focusses on the splitting of such orbits into stable and unstable invariant manifolds and explains the emergence of chaotic phenomena via Smale horseshoes. In this thesis the Melnikov method is applied to the homoclinic orbits corresponding to solitons of the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (MKdV) equations. These equations are reduced to third order ordinary differential equations by a travelling wave ansatz, defining a three-dimensional phase space of the equivalent systems of three first order equations. The geometry of periodic and homoclinic orbits and their structural changes under perturbations is investigated. It turns out that the three dimensional phase spaces foliate into a continuous family of invariant two-dimensional subspaces. By integrating the equations to second order the analysis by Melnikov's method is restricted to these subspaces and is considerably simplified. The Melnikov integrals stemming from the periodic and dissipative part of the perturbations, determining the onset of chaos, are then evaluated for the reduced KdV and MKdV systems. They are used to calculate the critical ratios between perturbation amplitude and dissipation coefficient at which tangency between table and unstable manifolds occur. At these critical ratios the transition/bifurcation from regular to chaotic behaviour occurs. It is observed that the Melnikov function for the periodic perturbation of the MKdV case vanishes for certain perturbation frequencies and parameter values, as confirmed by numerical work. The apparent discrepancy between structurally unstable homoclinic orbits and stable solitons is discussed and it is shown that solitons can persist despite the splitting of their corresponding homoclinic orbits under perturbation. Finally, subharmonics and resonance for the periodic solutions under perturbations are investigated using fourth order averaging techniques applied over the solution periods, which reveals a period doubling bifurcation in the subharmonics.