Emergent singular solutions of non-local density-magnetization equations in one dimension

TL;DR

This paper studies how singular solutions, called clumpons, emerge in a non-local density-magnetization model, revealing their formation from smooth data and classifying their interactions through analysis of a simplified two-clumpon system.

Contribution

It introduces a coupled density-magnetization model with non-local effects, analyzes stability, and classifies clumpon interactions, advancing understanding of singular solutions in magnetic systems.

Findings

Singular solutions emerge from smooth initial conditions.

Clumpons behave as interacting particles with classifiable dynamics.

Non-local effects significantly influence stability and solution behavior.

Abstract

We investigate the emergence of singular solutions in a non-local model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the non-local effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of non-locality on the system's stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.