# Coherent structure of Alice-Bob modified Korteweg de-Vries Equation

## TL;DR

This paper investigates exact solutions of the integrable Alice-Bob modified Korteweg de-Vries system, revealing new localized structures and symmetry-breaking solutions using Darboux transformations.

## Contribution

It constructs the general Nth Darboux transformation for the AB-mKdV system and explicitly derives various shifted symmetry-breaking solutions.

## Key findings

- Derived explicit one-soliton, two-soliton, and rogue wave solutions.
- Discovered abundant new localized structures in AB-mKdV systems.
- Identified symmetry-breaking solutions not present in standard mKdV.

## Abstract

To describe two-place events, Alice-Bob systems have been established by means of the shifted parity and delayed time reversal in Ref. [1]. In this paper, we mainly study exact solutions of the integrable Alice-Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation are constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.