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

## TL;DR

This paper proves that the height of the nth-order fundamental rogue wave in the nonlinear Schrödinger equation is exactly (2n+1) times the asymptotic plane height, using matrix row operations in Darboux transformations.

## Contribution

It provides a direct proof of the rogue wave height formula for the nonlinear Schrödinger equation using matrix operations, clarifying the structure of higher-order rogue waves.

## Key findings

- The height of nth-order rogue wave is (2n+1)c.
- The proof employs row operations on matrices from Darboux transformations.
- The result confirms the linear relation between rogue wave order and height.

## 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.