PhD thesis in English

5. BEC excitation by modulation of scattering lengththis is only an ad-hoc approximation. We stress that this issue concerning the trueresonant behavior can not be settled either by relying on a numerical calculationdue to inherent numerical artifacts related to finite numerical precision and finitecomputational time. To resolve it, one should rely on an analytical considerationalong the lines of Ref. [118] or use some analytical tool applicable at resonances,such as the resonant Bogoliubov-Mitropolsky method [124].10 3 1.94 1.96 1.98 2 2.02 2.04 2.06 2.0810 210 110 010 -110 -210 -3GP numerics 1GP numerics 2GP numerics 3Gaussian app.ΩFrequencyωFigure 5.12: Part of the Fourier spectrum of the time-dependent condensate widthfor P = 0.4, Q = 0.2, Ω = 2. For numerical solution of GP equation we use severaldiscretization schemes: GP numerics 1 (time step ε = 10 −3 , spacing h = 4 × 10 −2 ),GP numerics 2 (ε = 5 × 10 −4 , h = 2 × 10 −2 ), GP numerics 3 (ε = 5 × 10 −5 ,h = 5 × 10 −3 ). Details of the used numerical algorithms are given in AppendixA. For comparison we also show the corresponding spectrum obtained from theGaussian approximation (dotted-dashed line) and analytical result (5.21) for theposition of breathing mode (solid vertical line).In addition to comparison of our analytical results with numerical solutions basedon the Gaussian variational approximation, we present a comparison with the fullnumerical solution of the GP equation. In order to be able to perform Fourieranalysis with sufficient resolution, it is necessary to obtain an accurate solutionfor long evolution times. We do this by using the split-step method in combinationwith the semi-implicit Crank-Nicolson method [87]. As we refine the GP numerics byusing finer space and time discretization parameters, our numerical results becomestable as shown in Fig. 5.12. From the same figure, we observe quantitatively goodagreement between GP numerics and Gaussian approximation, reflected in close125