Perturbed KdV equations and their integrability properties

Abstract

In this thesis we investigate the integrability properties of the regularized long-wave (RLW) equation and modified regularized long-wave (mRLW) equation as perturbations of the integrable Korteweg-de Vries (KdV) equation. We study various properties of numerical mRLW three-soliton scattering and compare these with the corresponding RLW soliton solutions. We find that the numerical mRLW solitons behave much like integrable solitons in the sense that the only result of the three-soliton interaction is the phase shift each soliton experiences, which is approximately equal to the sum of pairwise phase shifts. Furthermore, we investigate the so-called quasi-integrability properties of these RLW and mRLW simulations. Using both analytical and numerical methods, we argue that these models possess an infinite amount of asymptotically conserved charges, i.e., quasi-conserved charges, which are observed in multi-soliton interactions. Finally, we also simulate numerical RLW and mRLW solutions in the presence of additional perturbing terms. This allows us to study soliton-radiation interactions and we find that for certain perturbations, these interactions preserve the quasi-conservation laws to a certain extend.