By Peter Cornelis Schuur
E-book by way of Schuur, Peter Cornelis
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Extra info for Asymptotic Analysis of Soliton Problems: An Inverse Scattering Approach
Let C be the operation of taking the complex conjugate. The results in (fiber) block decomposition of L(n) , mentioned above is given in the following. 2  1. Let Hm := [L2rad ]4 and define U : X → H, where H = m∈Z Hm , so that on smooth compactly supported v it acts by the formula 2π (Uv)m (r) = Jm−1 ∫ iθ χ−1 m (θ)ρn (e )v(x)dθ. 0 where χm (θ) are characters of U(1), that is, all homomorphisms U(1) → U(1) (explicitly we have χm (θ) = eimθ ) and 2 2 2 2 ⊕ ei(m−n)θ Lrad ⊕ −iei(m−1)θ Lrad ⊕ iei(m+1)θ Lrad Jm : Hm → ei(m+n)θ Lrad acting in the obvious way.
1) admit several symmetries, that is, transformations that map solutions to solutions. 6) “9781118853986c02” — 2015/4/21 — 10:55 — page 22 — #4 22 MAGNETIC VORTICES, ABRIKOSOV LATTICES, AND AUTOMORPHIC FUNCTIONS One of the analytically interesting aspects of the Ginzburg–Landau theory is the fact that, because of the gauge transformations, the symmetry group is infinitedimensional. 1) (see Ref.  for the regularity results). t. |Ψ(x)| ≥ δ > 0 for x : |x| = R. ) For more on the degree on Sobolev spaces see Ref.
2. For type II superconductors, the ±1-vortices are stable, while the n-vortices with |n| ≥ 2 are unstable. This stability behavior was long conjectured (see Ref. , ) leading to a “vortex interaction” picture wherein intervortex interactions are always attractive in the type-I case but become repulsive for like-signed vortices in the type-II case. This result agrees √ earlier, that the surface tension is pos√ with the fact, mentioned negative for κ > 1/ 2, so the itive for κ < 1/ 2 and √ √ vortices try to minimize their “surface” for κ < 1/ 2 and maximize it for κ > 1/ 2.
Asymptotic Analysis of Soliton Problems: An Inverse Scattering Approach by Peter Cornelis Schuur