The Laue pattern and the Rydberg formula in classical soliton models

TL;DR

This paper analyzes soliton oscillations in potential wells using the Klein-Gordon equation, revealing Laue patterns and a Rydberg-like frequency spectrum, thus connecting classical soliton dynamics with quantum-like spectral properties.

Contribution

It introduces a method to describe soliton oscillations in potentials, linking classical models to quantum spectral features and predicting Rydberg formula-based absorption frequencies.

Findings

Laue pattern emerges in soliton scattering in cyclic potentials.

A discrete frequency spectrum of solitons is derived in bounded potentials.

Solitons in Coulomb potential absorb waves at Rydberg frequencies.

Abstract

In recent researches of the dynamics of solitons, it is gradually revealed that oscillation modes play a crucial role when we analyze the dynamics of solitons. Some dynamical properties of solitons on external potentials are studied with both numerical methods and analytical methods. In this paper, we propose a method to deal with such oscillation modes of solitons in potential wells. We show that oscillations of a soliton is described by the Klein-Gordon equation with an external potential. Although this analysis does not seems to give quantitative scattering amplitude of a soliton itself, it explains qualitative pictures of scattering. As a result of our analysis, when a soliton is scattered in a cyclic potential, the Laue pattern emerges. Furthermore, since our analysis is based on the Klein-Gordon equation, a discrete frequency spectrum of a soliton is obtained when it is bounded by some potentials. What is especially important is that this analysis predicts a frequency spectrum of a soliton in the Coulomb potential and then we find that this system absorbs external waves with specific frequencies described by the Rydberg formula.