Geometry of Moving space curves associated with integrable equations: Connections and applications

Abstract

An unified approach to study possible connections between the geometry of moving space curves in three dimensional space and integrable nonlinear evolution equations in (1+1) dimensions, has been formulated. This unified formalism leads to an interesting result that each solution of such an evolution equation could be associated with two more distinct moving space curves in addition to the existing association with one moving curve. The formalism is applied to two nonlinear-evolution equations, viz., the nonlinear Schrodinger equation(NLS) and the lamb equation. An unified analysis is presented to show that in addition to Hasimoto function, two other complex functions that arises from the basic curve evolution equations. It is demonstrated that these can also satisfy various soliton equations. It leads to the result that each integrable equation is associated with three distinct classes of space curve motion, and has rich geometric structure. The three moving curves associated with the Lamb equation are obtained in terms of an exact solution m of the Belavin-Polyakov equation using an analog of the method developed in this study for obtaining the geometry of NLS. The explicit expressions for the three surfaces corresponding to the soliton solution and the three surfaces associated with the instanton solution are derived and displayed pictorially.