Investigation of the Nicole model

Abstract

We study soliton solutions of the Nicole model - a non-linear four-dimensional field theory consisting of the CP^1 Lagrangian density to the non-integer power 3/2 - using an ansatz within toroidal coordinates, which is indicated by the conformal symmetry of the static equations of motion. We calculate the soliton energies numerically and find that they grow linearly with the topological charge (Hopf index). Further we prove this behaviour to hold exactly for the ansatz. On the other hand, for the full three-dimensional system without symmetry reduction we prove a sub-linear upper bound, analogously to the case of the Faddeev-Niemi model. It follows that symmetric solitons cannot be true minimizers of the energy for sufficiently large Hopf index, again in analogy to the Faddeev-Niemi model.Comment: Latex, 35 pages, 1 figur

Integrability and Diffeomorphisms on Target Space

“…In this appendix we provide the details of the calculation of the conserved currents (10) for the class of Lagrangians (11). The Lagrangians L = L(a, b, c, ξ, d, e) have the following canonical four-momenta…”

“…For models with infinitely many conservation laws (the so-called AFZ model and related models), static and time-dependent solutions have been constructed, e.g., in [2,3,4,5], and in [6,7], respectively. Solutions in integrable subsectors of models which are, themselves, not integrable, have been constructed, e.g., in [8,9,10,11] (the Nicole model and versions thereof) and in [12,13] (diverse models on base space S 3 ). All these models share the property that their target space has dimension two.…”

Soliton stability in some knot soliton models

“…A Hopf map is a map φ : S 3 → S 2 or, via the stereographic projection from S 3 to one-point compactified IR 3 0 , a map φ : IR 3 0 → S 2 , and is characterized by the integer Hopf index (for details we refer to the Appendix A of Ref. [23]). Therefore, the above fields n and u can be identified with Hopf maps provided that the following two conditions hold.…”

“…Consequently, the field equations are no longer integrable, and the solutions are no longer available in closed, analytic form (except for the simplest case with Hopf index one 2 ). For a detailed investigation of soliton solutions within the ansatz in toroidal coordinates we refer to [23]. It is the main purpose of this paper to study stability issues of these two latter models with explicit solutions.…”

Conservation laws in Skyrme-type models

“…For models with infinitely many conservation laws (the so-called AFZ model and related models), static and time-dependent solutions have been constructed, e.g., in [2], [3], [4], [5], and in [6], [7], respectively. Solutions in integrable subsectors of models which are, themselves, not integrable, have been constructed, e.g., in [8], [9]- [11] (the Nicole model and versions thereof) and in [12], [13] (diverse models on base space S 3 ).…”