I am solving the equation of motion of an isolated vortex oscillation in a superconductor. I am assuming that the driving force is generated by an RF current that oscillates at 1.3 GHz.

The main point of this calculation is to see what the effect of the instability of the flow of flow in the RF unit is.

The code that I wrote is reported below.

```
f = 1.3 10 ^ 9; (* Hz *)
[Omega] = 2. Pi f;
Trf = 1./f;
Tc = 9.25; (* K *)
T = 1.5; (* K *)
[Mu]0 = 4. Pi 10 ^ -7; (* H / m *)
[Phi]0 = 2.07 10 ^ -15; (* Wb *)
n = 5.56 10 ^ 28; (* m ^ -3 *)
e = 1.6 10 ^ -19; (*DO*)
m = 9.1 10 ^ -31; (* kg *)
[Tau] = 0.5 10 ^ -9; (* s *)
vf = 1.37 10 ^ 6; (*Mrs*)
Bc2 = 410. 10 ^ -3; (* T *)
l = 100 10 ^ -9; (*subway*)
[Lambda] = 49.5 10 ^ -9; (*subway*)
[Xi] = 28 10 ^ -9; (*subway*)
[Kappa] = [Lambda]/ [Xi];
[Gamma] = ([Phi]0) / ([Mu]0 [Lambda] );
v0 = Sqrt[(l vf Sqrt[14 Zeta[3] (1 - T / Tc)]) / (3 Pi [Tau])];
[Rho] = (m vf) / (n e ^ 2 l);
[Eta] = ([Phi]0 Bc2) / [Rho];
g = Log[[[[[Kappa]]+ 0.5 + Exp.[-04-08Log[[-04-08Log[[-04-08Log[[-04-08Log[[Kappa]]-0.1 (Log[[[[[Kappa]]) ^ 2];
[Epsilon] = ([Phi]0 ^ 2 g) / (4. Pi [Mu]0 [Lambda]^ 2);
Tmax = 3. Trf;
Zmax = 1. 10 ^ -6; (*subway*)
B = 100. 10 ^ -3; (* T *)
sol = NDSolve[{
(v0 ^ 2 [Eta]) / (v0 ^ 2 + D[x[t, z], t]^ 2) D[x[t, z], t]==[Epsilon] re[x[t, z], z, z]+ [Gamma] B Cos[[[[[Omega] t]Exp[-(z/[-(z/[-(z/[-(z/[Lambda])],
X[0., z] == 0.,
X[t, Zmax] == 0.,
(RE[x[t, z], z]/. z -> 0) == 0.
}, x, {t, 0., Tmax}, {z, 0., Zmax},
AccuracyGoal -> 13,
PrecisionGoal -> 2,
MaxSteps -> Infinity];
you[t_, z_] = Evaluate[x[t, z] /. Sun][[1]];
du[t_, z_] = D[u[t, z], t];
phaseSpace =
Parallel table[{Re[u[t, 0.]]10 ^ 6, Re[du[t, 0.]]10 ^ -3}, {t, Trf,
Tmax, Tmax / 500.}];
ListPlot[phaseSpace, Joined -> True, PlotRange -> All]
```

I can get an almost correct result from this calculation only if I fix the domain of the solution to 1 micron (Zmax = 1. 10 ^ -6). However, this is not correct because the vortex should be able to oscillate more than that within the volume, therefore, literally, I am cutting my solution using too small a domain.

If I fix Zmax to a larger number (for example, 5 um), I need to correct MaxStepSize to a small number to avoid errors and the calculation would take more than 2 days on my PC (actually I stopped it after 2 days because the PC froze !). Does anyone know how to solve this problem?

Also, I was wondering if there is any way to solve the equation when the velocity of the vortex is greater than v0 (the start of the velocity for the instability of the flow of flow). If I fix B to say 200 mT, then the velocity of the vortex would be greater than v0 and the calculation will not yield a correct solution, just a series of peaks. Is there a method I can use to solve this problem?

Please let me know!

Thanks in advance.

Mattia