Long-term stability for KdV solitons in weighted $H^{s}$ spaces
Published in Communications on Pure and Applied Analysis, 2017
Recommended citation: B. Pigott and S. Raynor, "Long-term stability for KdV solitons in weighted $H^{s}$ spaces." Commun. Pure Appl. Anal.. 16 (2) (2017). https://doi.org/10.3934/cpaa.2017020
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the I-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.
B. Pigott and S. Raynor, "Long-term stability for KdV solitons in weighted $H^{s}$ spaces." Commun. Pure Appl. Anal.. 16 (2) (2017).