Geometric Methods to Investigate Prolongation Structures for Differential Systems with Applications to Integrable Systems

A type of prolongation structure for several general systems is discussed. They are based on a set of one forms for which the underlying structure group of the integrability condition corresponds to the Lie algebra of , , and . Each will be considered in turn and the latter two systems represent larger cases. This geometric approach is applied to all of the three of these systems to obtain prolongation structures explicitly. In both cases, the prolongation structure is reduced to the situation of three smaller problems. 1. Introduction Geometric approaches have been found useful in producing a great variety of results for nonlinear partial differential equations [1]. A specific geometric approach discussed here has been found to produce a very elegant, coherent, and unified understanding of many ideas in nonlinear physics by means of fundamental differential geometric concepts. In fact, relationships between a geometric interpretation of soliton equations, prolongation structure, Lax pairs, and conservation laws can be clearly realized and made use of. The interest in the approach, its generality, and the results it produces do not depend on a specific equation at the outset. The formalism in terms of differential forms [2] can encompass large classes of nonlinear partial differential equation, certainly the AKNS systems [3, 4], and it allows the production of generic expressions for infinite numbers of conservation laws. Moreover, it leads to the consequence that many seemingly different equations turn out to be related by a gauge transformation. Here the discussion begins by studying prolongation structures for a system discussed first by Sasaki [5, 6] and Crampin [7] to present and illustrate the method. This will also demonstrate the procedure and the kind of prolongation results that emerge. It also provides a basis from which to work out larger systems since they can generally be reduced to problems. Of greater complexity are a pair of problems which will be considered next. It is shown how to construct an system based on three constituent one forms as well as an system composed of eight fundamental one forms. The former has not appeared. The problem is less well known than the problem; however, some work has appeared in [8, 9]. In the first of these, the solitons degenerate to the AKNS solitons in three ways, while in the latter nondegenerate case, they do not. All of the results are presented explicitly; that is, the coefficients of all the forms and their higher exterior derivatives are calculated and given explicitly. Maple is used to do this