Integrable nonlinear evolution equations with time-dependent coefficients

Abstract

A simple and straightforward method for generating completely integrable nonlinear evolution equations with time-dependent coefficients is presented herein. For the equations under consideration, the solutions to those given by vector fields which are independent of time are given, thus explicit links between equations are obtained. As an application of the proposed method it is shown that the linear superposition, with arbitrary time-dependent coefficients, of different members of an integrable hierarchy is again integrable. Furthermore, it turns out that for some integrable equations [like the Korteweg–de Vries (KdV), the Benjamin–Ono (BO), or the Kadomtsev–Petviashvili (KP) the resolvent operator of lower order flows can be explicitly obtained from that of any higher order flow. Those flows (demonstrated for the KdV) which can be generated by Lie homomorphisms coming from first order problems are completely classified. Many well-known equations which can be found in the literature are of that type. As an application of such a first order link a direct link is given from KdV to the cylindrical KdV, and from there to the KP with nontrivial dependence on the second spatial variable.