Three-component WKI equation and curve motion flow in Euclidean and Minkowski space

Motion of curves in the four-dimensional Euclidean and Minkowski space are discussed. It is shown that the three-component WKI equation and its hyperbolic type arise from certain curve motion flows. They are obtained by using the relation between curvatures of the curves and their graph. Group-invariant solutions to the three-component WKI equation and its hyperbolic type are also derived.