  We consider a modified Ricci flow equation whose stationary solutions include
Einstein and Ricci soliton metrics, and we study the linear stability of those
solutions relative to the flow. After deriving various criteria that imply
linear stability, we turn our attention to left-invariant soliton metrics on
(non-compact) simply connected solvable Lie groups and prove linear stability
of many such metrics. These include an open set of two-step solvsolitons, all
two-step nilsolitons, two infinite families of three-step solvable Einstein
metrics, all nilsolitons of dimensions six or less, and all solvable Einstein
metrics of dimension seven or less with codimension-one nilradical. For each
linearly stable metric, dynamical stability follows a generalization of the
techniques of Guenther, Isenberg, and Knopf.
