Approximate analytic solutions to the Skyrme equation

Abstract

The asymptotic properties and the stability of the solutions of the Skyrme nonlinear differential equation for the profile function F are discussed. Using rational approximants for σ=cos F some different ways of finding approximate analytic solutions to the equation are discussed. For boundary conditions appropriate to soliton number 1 we construct a sequence of approximate analytic solutions that rapidly approaches the true solution according to several quality tests.

Approximate solutions for the Skyrmion

“…while the results for higher terms may be found elsewhere [9]. In terms of the dimensionless variable r = gf π r, expansion (14) becomes…”

“…The PA of order [M, N] to the hedgehog solution is found by comparing its Taylor expansion at the origin with series (14). From all the possible combinations (M, N) allowed by the input series expansion ( 14), we are interested in tho se that may provide a suitable representation to the hedgehog solution.…”

“…The PA representations built from the exact series solution near the origin do not guarantee the exact behaviour at intermediate and large values of r. However, one could use the asymptotic solution at infinity given in (19) in order to find better representations of the hedgehog [14]. Yet, the right behaviour at infinity may be incorporated using the 2-point Padé Approximant procedure.…”

Approximate solutions for the single soliton in a Skyrmion-type model with a dilaton scalar field

“…Previous works [11,12] that addressed solving the equation of motion of the original Skyrme Model have shown the utility, for practical calculations, of implementing an approach that incorporates both asymptotic behaviors of the chiral angle F (r), near the origin and at large r, in one single representation.…”