PhD thesis in English

5. BEC excitation by modulation of scattering lengthnext step, we use these solutions and solve for x 2 (t) and y 2 (t) and so on. At eachlevel n ≥ 1 we impose the initial conditions u ρn (0) = 0, ˙u ρn (0) = 0, u zn (0) = 0,and ˙u zn (0) = 0. At the first level of perturbation theory, equations for x and yare decoupled, i.e. x 1 (t) and y 1 (t) are normal modes: x 1 (t) describes quadrupoleoscillations, while y 1 (t) describes breathing oscillations. However, at the secondorder of perturbation theory y 1 (t) enters the equation for x 2 (t) and also x 1 (t) appearsin equation for y 2 (t), i.e. we have a nonlinear mode coupling.We have performed the explicit calculation to the second order by using thesoftware package MATHEMATICA [54]. We have obtained an excellent agreementof second-order analytical results and numerical results, as can be seen in Fig. 5.13for a moderate value of a modulation amplitude Q. The first secular terms appearat the level n = 3. The expressions are cumbersome, but the relevant behavior isobtained from the following terms in the equation for x 3 (t):ẍ 3 (t) + ωQ0x 2 3 (t) + C Q cos ω Q0 t + . . . = 0, (5.31)which leads tox 3 (t) = − C Q2ω Q0t sin ω Q0 t + . . . (5.32)The last term can be absorbed into the first-order solutionu ρ (t) = A Q cos ω Q0 t − C QQ 2t sin ω Q0 t + . . .2ω Q0≈ A Q cos [(ω Q0 + ∆ω Q0 )t], (5.33)and can be interpreted as a frequency shift of the quadrupole mode, quadratic in Q:ω Q = ω Q0 + ∆ω Q0 = ω Q0 + Q 2 + . . . (5.34)2ω Q0 A QThe coefficients A Q and C Q are calculated using the MATHEMATICA code availableat our site [60], and their explicit form is too long to be presented here. Along thesame lines we have also calculated the frequency shift of the breathing mode.C Q5.5.2 Results and discussionThe main results of our calculation in this section are shown in Figs. 5.15 and5.16. In Fig. 5.15 we plot the analytically obtained frequency of the quadrupole129