By van Wijk K.
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Extra info for Answers to a Selection Problems from Classical Electrodynamics John David Jackson
4. Non–hyperbolic regions We have already discussed the potential for our numerical scheme to become unstable because of Courant-type instability and also due to the imperfect imposition of the constraint φ · φ = 1. However, there is a much more pathological instability which comes about since the equations of motion are not manifestly hyperbolic and their precise nature, hyperbolic, parabolic or elliptic, depends on the particular configuration being evolved . The problem is that the specific numerical scheme we have designed will work only for the hyperbolic case and we know of no way of treating all configurations within a single numerical scheme.
A. Battye & P. M. Sutcliffe where I denotes the integral I= 1 4π 1 + |z|2 dR 1 + |R|2 dz 4 2i dzd¯ z . 6), therefore, one first determines the rational map which minimizes I, which may be thought of as an energy functional on the space of rational maps. 6) using a gradient flow method to solve the appropriate boundary value problem. Thus, within the rational map ansatz, the problem of finding the minimal energy Skyrmion reduces to the simpler problem of calculating the rational map which minimizes the function I.
Now, let R(z) be a degree B rational map between Riemann spheres, that is, R = p/q where p and q are polynomials in z such that max[deg(p), deg(q)] = B, and p and q have no common factors. 5) given a particular rational map R. It can be shown that this Skyrme field has charge B, and for 1 ≤ B ≤ 9 rational maps were presented in  which reproduced Skyrmions with the same symmetries as those computed in . Furthermore, they were shown to have energies which are only about one or two percent above the numerically calculated values.
Answers to a Selection Problems from Classical Electrodynamics John David Jackson by van Wijk K.