A direct derivation of the dark soliton excitation energy

TL;DR

This paper provides a rigorous and intuitive derivation of the dark soliton excitation energy, establishing a solid theoretical foundation for understanding dark soliton dynamics in nonlinear wave systems.

Contribution

It offers an alternative derivation of the renormalized dark soliton energy from fundamental field energy, clarifying its theoretical basis.

Findings

Derivation confirms the known energy form from fundamental principles

Uses a limiting procedure conserving particle number

Strengthens the theoretical understanding of dark soliton energy

Abstract

Dark solitons are common topological excitations in a wide array of nonlinear waves. The dark soliton excitation energy, crucial for exploring dark soliton dynamics, is necessarily calculated in a renormalized form due to its existence on a finite background. Despite its tremendous importance and success, the renormalized energy form was firstly only suggested with no detailed derivation, and was then "derived" in the grand canonical ensemble. In this work, we revisit this fundamental problem and provide an alternative and intuitive derivation of the energy form from the fundamental field energy by utilizing a limiting procedure that conserves number of particles. Our derivation yields the same result, putting therefore the dark soliton energy form on a solid basis.