Spectral properties and time decay for an Airy operator with potential (Q1377544)

The author considers the following linearization of the Korteweg-de Vries equation about a solitary wave: \[ \partial_tv= Av,\quad v(x,0)= f(x),\tag{1} \] where \(Av\equiv\partial_x(-\partial^2_x+ c- u)v\), and \(u(x)= 3c\text{ sech}^2(x\sqrt c)\). The linear initial-value problem (1) is studied in Sobolev spaces \(W_1\), \(W_2\) which involve spatial weights that grow algebraically as \(x\to\infty\) and decay algebraically or exponentially as \(x\to-\infty\). For suitably restricted initial data \(f\), the solution \(e^{At}f\) of (1) is shown to satisfy the decay estimate \[ \| e^{At}f\|_{W_2}\leq C(W_1, W_2)t^{-r}\| f\|_{W_1}\quad\forall t>0, \] where the decay rate \(r\) depends upon the amount of spatial localization that is given up in going from \(W_1\) to \(W_2\).