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

摘要

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 c(2) + d along the orthogonal direction of the (t, x)-plane for a first-order RW vertical bar u vertical bar(2): A point at height 9c(2) generates a convex curve for 3c(2) < d < 8c(2), whereas it becomes a concave curve for 0 < d < 3c(2), 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 c(2) < d < 0, and finally they reduce to two separate points at d = c(2). Using the contour line method, the length, width, and area of the RW at height c(2) + d(0 < d < 8c(2)), i.e. above the asymptotic plane, are defined. We study the evolutions of three above-mentioned localization characters on d through analytical and visual methods. The phase difference between the Kundu Eckhaus and the nonlinear Schrodinger equation is also given by an explicit formula.(C) 2016 Elsevier B.V. All rights reserved.