A note on higher-charge configurations for the Faddeev-Hopf model

Abstract

We identify higher-charge configurations that satisfy Euler-Lagrange equations for the (strong coupling limit of) Faddeev-Hopf model, by means of adequate changes of the domain metric and a reduction technique based on α-Hopf construction. In the last case it is proved that the solutions are local minima for the reduced σ 2 -energy and we identify among them those who are global minima for the unreduced energy.2010 Mathematics Subject Classification. Primary 58E20, 53B50; Secondary 58E30, 81T20.

A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution

“…For this question the only known example with smooth associated map ϕ is the (anti-) Hopf field (k = ±1). Beside this classsic example, S-integrable (non-vanishing) Beltrami fields with non-constant proportionality factor and arbitrary linking number can be deduced from the Faddeev-Skyrme solution in [4,25], where the associated map ϕ has two circles of singular points (where dϕ fails to be continuous).…”

Steady Euler flows and the Faddeev-Skyrme model with mass term

“…Steady Euler flows with higher helicities on S 3 can be easily constructed by particularizing the class of solutions given in [22,Example 4.5]. On the other side, there are several attempts to find higher Hopf charge solutions for the σ 2 -variational problem: [8,30,31] for the pure σ 2 -energy case and [1,15] for the case with potential ( [15] provides numerical solutions). Nevertheless the existing higher charge field configurations we are aware of are not classical solutions; they may be weak solutions (in some sense to be defined) or energy minimizers with low regularity.…”