Nonlinear Schrödinger Family on Moving Space Curves: Lax Pairs, Soliton Solution and Equivalent Spin Chain

The close connection between the Hasimoto type theory of vortex filament motion, soliton equations and continuum spin chains is reviewed. Using space curve formalism, we show that the completely integrable homogeneous and inhomogeneous nonlinear Schrödinger (NLS) type equations, such as mixed derivative NLS, extended NLS, higher order NLS, inhomogeneous NLS, circularly and radially symmetric NLS, and generalized inhomogeneous radially symmetric NLS equations, can be mapped on to certain types of moving helical space curves. We also briefly discuss the Lax pairs, one soliton solution and equivalent spin chain of the above integrable NLS type equations.