Space Curves from Nonlinear Schrödinger Solutions: A Direct Approach

Abstract

The connection between vortex filament evolution in the local induction approximation and non-linear Schrödinger (NLS) equation by Hasimoto [H. Hasimoto, J. Fluid Mechanics 51, (1972) 477] has led to space curves corresponding to NLS solitons in the past. Utilizing this map, we propose a direct construction of parametric curve evolution from any NLS solution. It includes ordered (or nested) integrals of products of local matrices akin to the causal evolution of quantum theory, necessitating the implementation of the Magnus expansion. Such a straightforward mapping may be a simple tool to study the evolution of various systems of physical concern, although the actual computation can be a challenge for most NLS solutions.

Figures (4)

• Figure 1: Plot for the periodic solution of the NLSE.
• Figure 2: Frenet-Serret curve for the periodic NLS solution with time $t$ as a parameter.
• Figure 3: Plot for the soliton solution of the NLSE.
• Figure 4: Evolution of the Frenet-Serret curve corresponding to the NLS soliton with $t$ as a parameter.