The Interaction Energy of Well-Separated Skyrme Solitons

Abstract

We prove that the asymptotic field of a Skyrme soliton of any degree has a nontrivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space. We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons' multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons.

Global solutions of the evolutionary Faddeev model with small initial data

“…The aim of this paper is to prove the global well-posedness of the Cauchy problem of the Fadeev model (1.3) in R 1+2 under the assumption that the initial data is small in some generalized Sobolev space. These results provide a starting point for further studies of evolutions of intereacting particle like approximate solutions, see [26] and [24]. Theorem 1.1.…”

Nonexistence of Skyrmion–Skyrmion and Skyrmion–anti-Skyrmion static equilibria

“…By Taylor's theorem with remainder, U must converge to the identity element faster than any polynomial in a neighborhood of P . Then by the strong unique continuation property of the static Skyrme model (see Theorems 2.4 and 2.5 in [29]) U must be constant.…”

On the existence of minima in the Skyrme model

“…The upshot of this chain of arguments, described in detail in [7], is that every Skyrme soliton has a leading Lie-algebra valued multipole field…”

“…The computation of this interaction energy for the multipole fields f M and g N is not easy because the combined field of the two multipoles only has cylindrical symmetry about the x 3 -axis and not the full rotational symmetry of each of the multipoles. As explained in [7], Fourier transformation in the x 1 x 2 -plane turns out to be an efficient tool for evaluating the integral in (2.12). Assuming without loss of generality that M ≤ N, the answer is…”

“…In the cases where the infima I l and I k−l are attained by Skyrme solitons, the inequality (1.13) is equivalent to the existence of attractive forces between those Skyrme solitons. In [7] the existence of attractive forces is studied in detail for Skyrme solitons obeying certain regularity assumptions. In the second part of this talk I review the arguments and results of that paper and in the third part I discuss their relation to Esteban's result.…”