On the evolution of a rogue wave along the orthogonal direction of the (t, x)-plane

The localization characters of the first-order rogue wave (RW) solution u of the Kundu-Eckhaus equation is studied in this paper. We discover a full process of the evolution for the contour line with height c2+d along the orthogonal direction of the (t, x)-plane for a first-order RW |u|2: A point at height 9c2 generates a convex curve for 3c2 ≤ d < 8c2, whereas it becomes a concave curve for 0 < d < 3c2, next it reduces to a hyperbola on asymptotic plane (i.e. equivalently d=0), and the two branches of the hyperbola become two separate convex curves when −c2