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

Abstract

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 $|u{|}^2$: A point at height $9c^2$ generates a convex curve for $3c^2\le d<8c^2$, 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.

17 pages, 13 figures. We have newly added the disucssion on the phase difference between the KE and the NLS. This version has been accepted by CNSNS(Aug.2016)

References in corpus (3)

• Capillary rogue waves
• Few-cycle optical rogue waves:complex modified Korteweg-de Vries equation
• Higher-Order Nonlinear Schrodinger equation with derivative non-Kerr nonlinear terms: A model for sub-10fs pulse propagation