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We have verified that (i) the choice of various integrators, (ii) initial conditions of spatially more regular structures than random data, do not affect the reported results. We used the 6-th order Yoshida algorithm [18], a 2-nd order symplectic and adaptive RK integrators. We ensured a high degree of accuracy, e.g., the energy is conserved up to the 9-th digit for a runtime $T=10^6$.

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