A Conservation Law Treatment of Nonlinear KdV Hierarchies

Abstract

We study the hierarchy commonly defined as an infinite sequence of partial differential equations which begins with the Korteweg–de Vries equation and its modified version. An important feature of the hierarchy is its highly nonlinear property. In this regard, obtaining solutions for the members of the hierarchy poses a great problem. In this paper, we propose a method to allow for the construction of solutions to the full hierarchy. Our approach involves a recursion operator in the conservation law of the hierarchy. The efficiency of the method is demonstrated by selected examples. In certain cases, we obtain snoidal solutions.