Emergent singular solutions of nonlocal density-magnetization equations in one dimension

We investigate the emergence of singular solutions in a nonlocal 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 nonlocal effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of nonlocality 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.