On the proximal dynamics between integrable and non-integrable members of a generalized Korteweg-de Vries family of equations

Abstract: The distance between the solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations is estimated in appropriate Sobolev spaces. Special cases of the latter family of equations include the standard gKdV equation with a power nonlinearity, as well as weakly nonlinear perturbations of the KdV equation. The distance estimates are established for initial data and nonlinearity parameters of arbitrary size. A crucial step for their derivation is the proof of a size estimate for local solutions to the gKdV family of equations, which is linear in the norm of the initial data. As a result, the distance estimates themselves predict that the dynamics between the gKdV and KdV equations remains close over long times for initial amplitudes even close to unity, while for larger amplitudes they predict an explicit rate of deviation of the dynamics between the compared equations. These theoretical results are illustrated via numerical simulations in the case of one-soliton and two-soliton initial data, which are in an excellent agreement with the theoretical predictions. Importantly, in the case of a power nonlinearity and large solitonic initial data, the deviation between the integrable and non-integrable dynamics can be drastically reduced upon incorporating suitable rotation effects by means of a rescaled KdV equation. As a result, the integrable dynamics stemming from the rescaled KdV equation may persist in the gKdV family of equations over remarkably long timescales.