Fluctuating solitons of the KdV equations

We introduce a new AKNS three-component system, which is convenient for finding periodic and/or almost periodic solutions to the hierarchy of the KdV equations. It conserves the spectral functions which determine the spectrum of the auxiliary Schrödinger equation containing the solutions of the Korteweg-de Vries equations as potentials. By means of the Darboux and Abraham-Moses transformations we derive new solutions of the KdV hierarchy, which can be grasped as solitons on the fluctuating background.Some parts of the paper were delivered in the talk at the III Potsdam-V Kiev international workshop on nonlinear processes in physics, Potsdam (USA), 1–11 August, 1991.