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

Abstract

A type of prolongation structure for several general systems is discussed. They are based on a set of one-forms in which the underlying structure group of the integrability condition corresponds to the Lie algebra of SL(2, R), O(3) or SU (3). Each will be considered in turn and the latter two systems represent larger 3 × 3 cases. This geometric approach is applied to all three of these systems to obtain prolongation structures explicitly. In both 3 × 3 cases, the prolongation structure is reduced to the situation of three smaller 2 × 2 problems. Many types of conservation laws can be obtained at different stages of the development, and at the end, a single result is developed to show how this can be done. law MSCs: 35A30, 32A25, 35C05 I. 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 2 × 2 SL(2, R) 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 2 ×2 problems.Of greater complexity are a pair of 3 × 3 problems which will be considered next. In particular, it is shown how to construct an O(3) system based on three constituent one-forms as well as an SU(3) system composed of eight fundamental one-forms. The former has not appeared and in both cases, all of the results are presented explicitly. The approach is quite unified and so once the formalism is established for the SL(2, R) system, the overall procedure can be carried over to the other Lie algebra cases as well.Of course, the 3 × 3 problems will yield more types of conservation laws. The prolongation structures of the 3 × 3 problems will be observed to be reduceable to the same type as the 2 × 2 system considered at the beginning. However, the Riccati representations become much more complicated [8]. The generalization of this formalism to an n × n problem then becomes straightforward ...