| Abstract: |
| We discuss current results concerning the distance between solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations in appropriate Sobolev spaces. This family of equations includes, as special cases, the standard gKdV equation with power nonlinearities as well as weakly nonlinear perturbations of the KdV equation. The distance estimates are based on a crucial size estimate for local gKdV solutions. Consequently, these estimates predict that the dynamics of the gKdV and KdV equations remain close over long time intervals for initial amplitudes approaching unity, while providing an explicit rate of deviation for larger amplitudes. Furthermore, it is demonstrated that in the case of power nonlinearities and large solitonic initial data, the deviation between the integrable and non-integrable dynamics can be drastically reduced by incorporating suitable rotation effects via a rescaled KdV equation. As a result, the integrable dynamics stemming from the rescaled KdV equation may persist within the gKdV family of equations over remarkably long timescales. Finally, we comment on similar results regarding Nonlinear Schr\odinger (NLS) equations. |
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