In this case, the system is still described by the EKG Eqs. (27 – 32), with the the additional simplification that the scalar field is strictly real, . In order to find equilibrium configurations, one expands both metric components and the scalar field as a truncated Fourier series[216, 5], the scalar field consists only of odd components while the metric terms consist only of even ones. Solutions are obtained by substituting the expansions of Eq. (65) into the spherically symmetric Eqs. (27 – 32). By matching terms of the same frequency, the system of equations reduce to a set of coupled ODEs. The boundary conditions are determined by requiring regularity at the origin and that the fields become asymptotically flat at large radius. These form an eigenvalue problem for the coefficients corresponding to a given central value . As pointed out in , the frequency is determined by the coefficient and is, therefore, called an output value. Although the equations are non-linear, the Fourier series converges rapidly, and so a small value of usually suffices.
A careful analysis of the high frequency components of this construction reveals difficulties in avoiding infinite total energy while maintaining the asymptotically flat boundary condition . Therefore, the truncated solutions constructed above are not exactly time periodic. Indeed, very accurate numerical work has shown that the oscillatons radiate scalar field on extremely long time scales while their frequency increases [84, 97]. This work finds a mass loss rate of just one part in per oscillation period, much too small for most numerical simulations to observe. The solutions are, therefore, only near-equilibrium solutions and can be extremely long-lived.
Although the geometry is oscillatory in nature, these oscillatons behave similar to BSs. In particular, they similarly transition from long-lived solutions to a dynamically unstable branch separated at the maximum mass . Figure 6 displays the total mass curve, which shows the mass as a function of central value. Compact solutions can be found in the Newtonian framework when the weak field limit is performed appropriately, reducing to the so-called Newtonian oscillations . The dynamics produced by perturbations are also qualitatively similar, including gravitational cooling, migration to more dilute stars, and collapse to black holes . More recently, these studies have been extended by considering the evolution in 3D of excited states  and by including a quartic self-interaction potential . In , a variational approach is used to construct oscillatons in a reduced system similar to that of the sine-Gordon breather solution.
Closely related, are oscillons that exist in flatspace and that were first mentioned as “pulsons” in 1977 . There is an extensive literature on such solutions, many of which appear in . A series of papers establishes that oscillons similarly radiate on very long time scales [79, 80, 81, 82]. An interesting numerical approach to evolving oscillons adopts coordinates that blueshift and damp outgoing radiation of the massive scalar field [115, 117]. A detailed look at the long term dynamics of these solutions suggests the existence of a fractal boundary in parameter space between oscillatons that lead to expansion of a true-vacuum bubble and those that disperse .
Living Rev. Relativity 15, (2012), 6
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