A110818 - OEIS

%I #8 Aug 04 2023 15:47:42

%S 1,2,19,374,12559,645992,47367124,4701142286,607384076311,

%T 99104140036610,19933965307701547,4846421980399770152,

%U 1401149529610562030404,475128611089824908724944,186768400411319414544569368,84248002219370115308687582078

%N Coefficient (times -1) of the 1/r^(2n) term in the radial far-field expansion of the squared amplitude of a unit topological point charge (-1 or +1 vortex) in the two-dimensional Ginzburg-Landau equation.

%C Ginzburg-Landau vortex solutions are fundamental in the study of superconductors and superfluids.

%H María Aguareles Carrero, <a href="https://hdl.handle.net/2117/93169">Interaction of spiral waves in the general complex Ginzburg-Landau equation</a>, Universitat Politècnica de Catalunya, Doctoral thesis, 2007. See Eqs. (1.11)-(1.13).

%e a(3) = 19 because A(r)^2 = 1 - 1/r^2 - 2/r^4 - 19/r^6 - ...

%t n = 17;

%t v = 1; (* change to 2 to get A111100 *)

%t sol = AsymptoticDSolveValue[{4 z^3 f''[z] + 4 z^2 f'[z] - f[z] v^2 z + (1 - f[z]^2) f[z] == 0, f[0] == 1}, f[z], {z, 0, n}];

%t Rest@CoefficientList[1 - sol^2 + O[z]^n, z] (* _Andrey Zabolotskiy_, Aug 04 2023 *)

%Y Cf. A111100.

%K nonn

%O 1,2

%A _Greg Huber_, Sep 15 2005

%E Terms a(13) and beyond from _Andrey Zabolotskiy_, Aug 04 2023