# Moving planes for domain walls in a coupled system

## TL;DR

This paper proves the monotonicity, one-dimensionality, and uniqueness of domain wall solutions in a coupled Bose-Einstein condensate system with Rabi coupling, and shows non-existence of solutions for large coupling.

## Contribution

It introduces new estimates enabling the moving plane method to analyze non-cooperative systems with no maximum principle, establishing key properties of domain wall solutions.

## Key findings

- Monotonicity and one-dimensionality of solutions
- Uniqueness of solutions up to translation
- Non-existence of solutions for large Rabi coupling

## Abstract

The system leading to phase segregation in two-component Bose-Einstein condensates can be generalized to hyperfine spin states with a Rabi term coupling. This leads to domain wall solutions having a monotone structure for a non-cooperative system. We use the moving plane method to prove mono-tonicity and one-dimensionality of the phase transition solutions. This relies on totally new estimates for a type of system for which no Maximum Principle a priori holds. We also derive that one dimensional solutions are unique up to translations. When the Rabi coefficient is large, we prove that no non-constant solutions can exist.