Low Regularity Stability for Subcritical Generalized Korteweg-de Vries Equations

In this thesis we prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H^{s}_{x}(\mathbb{R})$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.

Link to Thesis

B. Pigott, "Low Regularity Stability for Subcritical Generalized Korteweg-de Vries Equations." Ph.D. Thesis, University of Toronto. (2011).