UCLA

This work explores the existence and behavior of solutions to the Korteweg–de Vries equation on the line for large perturbations of certain classical solutions. First, we show that given a suitable solution $V(t,x)$, KdV is globally well-posed for initial data $u(0,x) \in V (0,x) + H^{-1}(\mathbb{R})$. Our conditions on $V$ do include regularity but do not impose any assumptions on spatial asymptotics. In particular, we show that smooth periodic and step-like profiles $V(0,x)$ satisfy our hypotheses.

Our second main objective is to prove a variational characterization of KdV multisolitons. Maddocks and Sachs used that $n$-solitons are local constrained minimizers of the polynomial conserved quantities in order to prove that $n$-solitons are orbitally stable in $H^n(\mathbb{R})$. We show that multisolitons are the unique global constrained minimizers for this problem. We then use this characterization to provide a new proof of the Maddocks–Sachs orbital stability result via concentration compactness.