Superposition of gravitational waves and the dark energy Is it theoretically possible to create a superposition of gravitational waves that form a locally static negative curvature, something like the dark energy?
 A: Superposition of gravitational waves if averaged over the regions larger than the typical wavelength would produce an effective stress-energy tensor that would serve as a matter source in the Einstein field equations. But this stress-energy tensor will have the same properties as stress-energy of any other type of incoherent radiation: the pressures would be positive, in the isotropic case the equation of state would be $p=\frac13 \rho$ (in relativistic units). Dark energy  (if it exists) would have negative pressures with the equation of state $p=-\rho$ (in the simplest case, other models of dark energy are also considered). So, gravitational waves could not be contributing to the accelerating expansion of the universe, the main observational evidence of dark energy.
On the other hand it is possible to confine a large number of gravitational waves within a small region of space trapping them by their own gravity, thus forming an approximately static object, a geon. Here is a description taken from this paper of one such geon solution, constructed by Brill & Hartle:

Brill and Hartle [1], BH, developed a very useful method for finding approximate solutions to Einstein’s equations that correspond to high frequency gravitational waves propagating in a background geometry, which is created by the average stress-energy of the waves themselves. In their paper they applied this method to the case of a static spherically symmetric background geometry and found that gravitational waves can remain confined in a region for a time much longer than the region’s light-crossing time. This so-called gravitational geon is generated by a large number of high frequency, small amplitude gravitational waves. The time average of the curvature due to these waves creates the background geometry of the geon, and this background geometry traps the waves for a long time in a region of space called the “active” region. The BH solution is important because it serves as an example in which the gravitational field both creates and responds to its own effective stress-energy. It is also an example of a nontrivial (approximate) solution to the vacuum Einstein equations that has no curvature singularities.

As we can see, such geons are not truly static, with enough time gravitational waves would escape and the geon would dissolve.
