Rigid body motions, space curves, prolongation structures, fiber bundles, and solitons

Abstract

The dynamics of a nonlinear string of constant length represented by a helical space curve may be studied through a consideration of the motion of an arbitrary rigid body along it. The resulting set of compatibility equations is shown to result in the class of nonlinear evolution equations solvable through the two component inverse scattering phenomenology. A class of pseudopotentials and prolongation structures follow naturally due to the intrinsic group structure of the phenomenon. This leads to an identification of the underlying fiber bundle structure and connection forms. Thus a unified picture emerges for a class of soliton possessing evolution equations.