| \imath}$,             |
|                       | }$                    | $X_3=0$               |
+-----------------------+-----------------------+-----------------------+
| Fuzzy hyperboloid     | $k$-representation of | $X_i = \frac{1}{k} K_ |
|                       | $\mathfrak{su}(1,1)$  | i$                    |
|                       | ($N=+\infty$)\        |                       |
|                       | $[K_1,K_2]=-\imath K_ |                       |
|                       | 3$,\                  |                       |
|                       | $[K_3,K_1]=\imath K_2 |                       |
|                       | $,\                   |                       |
|                       | $[K_2,K_3]=\imath K_1 |                       |
|                       | $                     |                       |
+-----------------------+-----------------------+-----------------------+

: \[NCmanifold\] Noncommutative manifolds used as initial conditions for the D2-brane (see also [@Sykora]). For the numerical simulations, the choosen parameters of the models are $\alpha=1.5$, $\delta=0.4$ and $k=12.5$.

Numerical simulations for the integration of eq. \[equX\] are realized by using a leap-frog integrator as in [@Asplund]. For the non-compact manifolds, a cutoff in the basis is realized which needs a numerical correction to avoid numerical artefacts (see \[AppA\]). The integration of the Schrödinger equation \[equSchro\] is realised by a split operator method for the evolution operator: $U(t_n,0) \simeq e^{-\imath H^{eff}(t_n) \Delta t}...e^{-\imath H^{eff}(0) \Delta t}$ for a partition $0<t_1<...<t_n$ of step $\Delta t$ equal to the one used in the leap-frog integrator.

Dynamics of the quantum fluctuations
====================================

D-brane equation with fluctuations
----------------------------------

The equation $\ddot{\mathbf X}_i - [\mathbf X_j,[\mathbf X_i,\mathbf X^j]]=0$ 