# Generation of higher-order rogue waves from multi-breathers by double degeneracy in an optical fibre

## TL;DR

This paper introduces a method to generate higher-order rogue waves in optical fibers by creating special breather solutions of the nonlinear Schrödinger equation and demonstrates their experimental observability.

## Contribution

The paper develops the concept of breather-positons as a new class of solutions and proposes a practical approach to observe higher-order rogue waves in optical fibers.

## Key findings

- Analytical plots confirm the double degeneration process.
- Numerical simulations match analytical solutions with low noise.
- Proposed measurement method supports experimental realization.

## Abstract

In this paper, we construct a special kind of breather solution of the nonlinear Schr\"{o}dinger (NLS) equation, the so-called breather-positon ({\it b-positon} for short), which can be obtained by taking the limit $\lambda_{j}$ $\rightarrow$ $\lambda_{1}$ of the Lax pair eigenvalues in the order-$n$ periodic solution which is generated by the $n$-fold Darboux transformation from a special "seed" solution--plane wave. Further, an order-$n$ {\it b-positon} gives an order-$n$ rogue wave under a limit $\lambda_1\rightarrow \lambda_0$. Here $\lambda_0$ is a special eigenvalue in a breather of the NLS equation such that its period goes to infinity. Several analytical plots of order-2 breather confirm visually this double degeneration. The last limit in this double degeneration can be realized approximately in an optical fiber governed by the NLS equation, in which an injected initial ideal pulse is created by a frequency comb system and a programable optical filter (wave shaper) according to the profile of an analytical form of the {\it b-positon} at a certain position $z_0$. We also suggest a new way to observe higher-order rogue waves generation in an optical fiber, namely, measure the patterns at the central region of the higher-order {\it b-positon} generated by above ideal initial pulses when $\lambda_1$ is very close to the $\lambda_0$. The excellent agreement between the numerical solutions generated from initial ideal inputs with a low signal noise ratio and analytical solutions of order-2 {\it b-positon}, supports strongly this way in a realistic optical fiber system. Our results also show the validity of the generating mechanism of a higher-order rogue waves from a multi-breathers through the double degeneration.