Moving curves of the sine-Gordon equation: New links

We provide a general scheme for mapping integrable nonlinear partial differential equations of real functions to moving space curves using an approach different from the one proposed by Lamb. We apply our method to the sine-Gordon equation and obtain links to five new classes of space curves, in addition to the two found by Lamb. For each class, we display the rich variety of moving curves associated with the one-soliton, the breather, the two-soliton and the soliton–antisoliton solutions, and suggest possible applications. Our results also provide new insights with regard to the two-soliton (soliton–antisoliton) scattering process.