Nonlinear instability and solitons in a self-gravitating fluid

Abstract

We study a spherical, self-gravitating fluid model, which finds applications in cosmic structure formation. We argue that since the system features nonlinearity and gravity-induced dispersion, the emergence of solitons becomes possible. We thus employ a multiscale expansion method to study, in the weakly nonlinear regime, the evolution of small-amplitude perturbations around the equilibrium state. This way, we derive a spherical nonlinear Schr{ö}dinger (NLS) equation that governs the envelope of the perturbations. The effective NLS description allows us to predict a "nonlinear instability" (occurring in the nonlinear regime of the system), namely, the modulational instability which, in turn, may give rise to spherical soliton states. The latter feature a very slow (polynomial) curvature-induced decay in time. The soliton profiles may be used to describe the shape of dark matter halos at the rims of the galaxies.

Figures (5)

• Figure 1: The Jeans instability growth rate ${\rm Im}\,\omega(k)$ as a function of the wavenumber $k>0$. According to the linear theory, the instability band is defined by the interval $k<k_{\rm J}=1$, highlighted with the blue segment. The red segment depicts the extension of the instability band once nonlinearity is taken into account ---see discussion in Sec. \ref{['nMI']}.
• Figure 2: The dependence of the dispersion [solid (blue) line] and nonlinearity [dashed (red) line] coefficients, $\omega"$ and $\gamma$, on the wavenumber $k$, in the regime $k>k_J$. The dispersion coefficient is always negative, while the nonlinearity coefficient becomes zero at $k=\alpha k_J$, with $\alpha =(1/4)\sqrt{21+\sqrt{57}}$, and it is negative (positive) for $k<\alpha k_J$ ($k>\alpha k_J$).
• Figure 3: The instability growth rate, ${\rm Im}(\Omega)$ as a function of $K$ [see Eq. (\ref{['ngr']})], for $k=(1+\alpha)k_J/2 \approx 1.17$ [i.e., in the middle of the nonlinearity-induced extension of the instability band ---see Eq. (\ref{['extens']})], and for three different times: $T_2=5\times 10^{-3}$ [upper, dashed-dotted (blue) curve], $T_2=8\times 10^{-3}$ [middle, dashed (red) curve], and $T_2=10^{-2}$ [lower, solid (orange) curve]. Notice that both the maximum value of ${\rm Im}(\Omega)$ and the range of the nonlinear instability band decrease with $T_2$.
• Figure 4: The function $F(w)$ given in Eq. (\ref{['Fofw']}). It is observed that this function is initially increasing, then features a maximum ${\rm max} F(w_c) \approx 0.55$ at $w=w_c \approx 12$, and then it asymptotically decreases to zero.
• Figure 5: The ratio $|\omega"(k)/\gamma(k)|$ as a function of the wavenumber $k$ in the interval $k_J<k<\alpha k_J$, i.e., in the nonlinear instability band. As seen, this ratio is ${\cal O}(1)$ for wavenumbers around the middle of this interval.