Is there a way to guess/find unitary transformations for Hamiltonians? Many papers that I have read which concern the dynamics of open systems and even otherwise typically start with a Hamiltonian of the system and then apply a unitary transformation depending upon the form of their Hamiltonian and the effect that they want to bring out. The effect that these unitary transformations explicitly bring out interesting effects which, of course, are already present (or can be guessed to be present) in the system Hamiltonian.
Consider the following Hamiltonian,
$$H=\Delta a^{\dagger}a + \omega b^{\dagger}b+g_o\bigg(b^{\dagger}b+\frac{1}{2}\bigg)(a^{\dagger}+a)$$
where,
$a$ and $b$ are usual photon and phonon bosonic operators respectively.
It is known that this kind of $H$ has a non-linearity in $b^{\dagger}b$ and to show this explicitly the author in this paper uses what is known as a polaron transformation:
$$H'=UHU^{\dagger}$$
with $U=\exp[\big(b^{\dagger}b+\frac{1}{2}\big)(a^{\dagger}a)]$
and gets the Hamiltonian $H \propto$ $(b^{\dagger}b)^2$
Similarly, other papers have different such unitary transformations (even non-linear ones, check  this paper if interested) which are used to write the hamiltonian in an effective way that clearly can indicate an interesting phenomenon (like the above non-linearity is an indicator of phonon blockade in the system ).
Finding out the 'right' unitary transformation becomes crucial for many studies and in cases where one has a system Hamiltonian which one suspects should show a certain phenomenon under optimal parameters. Say one suspects his system to show phonon blockade. He can try to coming up or building a unitary transformation that transforms the $H$ to become proportional to $(b^{\dagger}b)^2$. The question is how shall he find such a transformation assuming that such a transformation exists looking at the form of the Hamiltonian and the form he wants to bring it to?
All the papers that I have read do not show how they come up with the unitary transformation for their hamiltonian.
Are there any general approaches or studies for constructing unitary transformations for a given hamiltonian or is it just guesswork (which I doubt it cannot be)?
 A: Quadratic Hamiltonians
For quadratic Hamiltonians canonical transformation corresponds to the rotation of the basis in non-second quantized representation. Thus, the new operators can be represented as a linear transformation of the current operators, and the canonical transformation is easily derive dusing matrix algebra.
Bogoliubov transformation
The non-interacting procedure works for some interacting Hamiltonians as well. A notable case is Bogoliubov transformation applied to BCS Hamiltonian - in this specific case one considers linear combination of creationa nd annihilation operators, which, strictly speaking, is not simple basis rotation.
Perturbative procedures
One can obtain approximate transformation using perturbative methods, e.g., with the help of the Baker-Campbell_Houssdorff formula. The classical example here is the derivation of the Schrieffer-Wolff transformation, transforming Anderson model to Kondo model. (See the Wikipedia article or the original paper by Schriffer and Wolff for more details.)
Known transformations
There are many known transformations, obtained through educated guesses, experience, and trial and error. Such are: Holstein-Primakoff transformation, Jordan-WIgner transformation, Polaron transformation, bosonization transformation (as described by Tomonaga, Schotte&Schotte and Haldane), and others.
