The integrable dynamics of discrete and continuous curves

We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch during the motion, iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. Well known examples of integrable evolution equations, like the nonlinear Schrödinger and the sine-Gordon equations, as well as their discrete analogues, are derived in this general framework. 1 A historical introduction One of the classical problems of the XIX-century geometers was the study of the connection between differential geometry of submanifolds and nonlinear (integrable) PDE’s. For instance, Liouville found the general solution of the equation (known now as the Liouville equation) which describes minimal surfaces in E3 [1]. Bianchi solved the general Goursat problem for the sine-Gordon (SG) equation [2], which encodes the whole geometry of the pseudospherical surfaces. Moreover the method of construction of a new pseudospherical surface from a given one, proposed by Bianchi [3], gives rise to the Bäcklund transformation for the SG equation [4]. The connection between geometry and integrable PDE’s became even deeper when Hasimoto [5] found the transformation between the equations governing the curvature and torsion of a nonstretching thin vortex filament moving in an incompressible fluid and the NLS equation. Several authors, including Lamb [6], Lakshmanan [7], Sasaki [8], Chern and Tenenblat [9] related the Zakharov-Shabat(ZS) [10] spectral problem and the associated Ablowitz-Kaup-Newell-Segur(AKNS) hierarchy [11] to the motion of curves in E3 or to the pseudospherical surfaces and certain foliations on them. Almost at that time Sym introduced the soliton surfaces approach, in which the powerful tools of the IST method are used to construct explicit formulas for the immersions ∗This work was supported by KBN grant 2-0168-91-01, by the INFN and by the 1994 agreement between Warsaw and Rome Universities