Motion of strings, embedding problem and soliton equations

Abstract

The motion of a flexible string of constant length in E ³ in interaction, corresponding to a variety of physical situations, is considered. It is pointed out that such a system could be studied in terms of the embedding problem in differential geometry, either as a moving helical space curve in E ³ or by the embedding equations of two dimensional surfaces in E ³. The resulting integrability equations are identifiable with standard soliton equations such as the non-linear Schrödinger, modified K-dV, sine-Gordon, Lund-Regge equations, etc. On appropriate reductions the embedding equations in conjunction with suitable local space-time and/or gauge symmetries reproduce the AKNS-type eigenvalue equations and Riccati equations associated with soliton equations. The group theoretical properties follow naturally from these studies. Thus the above procedure gives a simple geometric interpretation to a large class of the soliton possessing nonlinear evolution equations and at the same time solves the underlying string equations.