I have a vague but interesting question. The field equation of general relativity is non-linear. Also, general relativity is simply classical field theory. The framework of chaos is something which is fairly well defined in classical physics. My question is: Do we have something like chaotic solutions of general relativity? If not, why so?
If yes, is there some interesting research being done in this topic?
Yes, we definitely have chaos in GR.
One of the earlier references on the topic is Barrow's Chaotic behaviour in general relativity (e-print).
A particularly relevant result is that Relativistic Chaos is Coordinate Invariant (arXiv, e-print), by A. Motter. And there's also a book called Deterministic Chaos in General Relativity by Hobill et al. one can check and, of course, many papers, such as this, this, and this.
As for the last question (post v. 3), it's opinion-based, but fwiw, I give my take:
is there some interesting research being done in this topic?
I'd say yes, and point to quantum gravity.
For instance$^1$,
Hawking proposed in 2014 (Information Preservation and Weather Forecasting for Black Holes) that chaos could be the resolution for the information paradox in black holes.
Shenker and Stanford suggested in the same year (Black holes and the butterfly effect) a connection between chaos and the geometry of black holes.
Maldacena et al. argued in 2015 that there's A bound on chaos for a broad class of quantum systems (the bound diverges in the classical limit), which can be linked to gravity via gauge/gravity duality.
Polchinski, also in 2015, building on the work of Shenker and Stanford (2014) proposed a connection of "the thermal properties of black holes to chaos" (Chaos in the black hole S-matrix).
Dittrich et al. argued in 2017 that an "alternative quantization method" is made necessary by the chaoticity/non-integrability of generic systems in general relativity (Can chaos be observed in quantum gravity?).
$^1$ These papers are not in my area, I hope people correct any mistakes in my understanding.
The framework of chaos is something which is fairly well defined in classical physics.
I am not exactly sure what you are aiming at here, but I fail to see that chaos theory emphasises classical physical systems. First of all, chaos is a mathematical phenomenon that can be seen in all sorts of dynamical systems (i.e., systems of differential equations) and thus chaos theory is somewhat ignorant of the physical background of a system. Second, most chaotic systems are not even physical in the sense that they do not have something like energy conservation, e.g., they are models of fuelled chemical reactions, population dynamics, or the weather.
Do we have something like chaotic solutions of general relativity?
As general relativity approximates classical mechanics on everyday scales, you can start with something as simple as the double pendulum. There is no reason to assume that it turns non-chaotic if general relativity is taken into account. In fact, if this were the case, we would have to doubt general relativity as an accurate description of reality. After all, we can also experimentally confirm the chaoticity of the double pendulum.
