The height of an $n$th-order fundamental rogue wave for the nonlinear Schr\"odinger equation

Abstract

The height of an $n$th-order fundamental rogue wave $q_{\rm rw}^{[n]}$ for the nonlinear Schr\"odinger equation, namely $(2n+1)c$, is proved directly by a series of row operations on matrices appeared in the $n$-fold Darboux transformation. Here the positive constant $c$ denotes the height of the asymptotical plane of the rogue wave.