# Multi-breathers and high order rogue waves for the nonlinear Schr\"odinger equation on the elliptic function background

## TL;DR

This paper constructs and analyzes multi-breather and rogue wave solutions of the focusing nonlinear Schrödinger equation on elliptic function backgrounds, revealing their complex dynamics and stability properties.

## Contribution

It introduces explicit multi-breather and rogue wave solutions on elliptic backgrounds using Darboux transformation and theta functions, advancing understanding of their behaviors and instabilities.

## Key findings

- Multi-breather solutions expressed via theta functions.
- Asymptotic analysis of breather dynamics at large times.
- Exact rogue wave solutions derived from breather limits.

## Abstract

We construct the multi-breather solutions of the focusing nonlinear Schr\"odinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.