(Hope this question is o.k. for MO)

I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular Fourier multiplier); to solve it.

But mainly with underlying framework function spaces, which I have seen so far, are functions/distributions on $\mathbb R^{n}.$ (For instance,Sobolev spaces $H^{s}(\mathbb R^{n})$, Besove spaces $B^{p,q}(\mathbb R^{n})$, Modulation spaces $M^{p,q}(\mathbb R^{n})$, etc..)

My Question is: Bit vague:How the theory had gone(or have been going), if one consider the underlying framework function spaces on torus $\mathbb T$ (for instance $H^{s}(\mathbb T)$, $A(\mathbb T)$, etc...) ? Is there something common in handling the non linear dispersive equations on compact group $\mathbb T$ and on group $\mathbb R$ ?What are the reference (book/ recent monograph/ fundamental papers) for the study of non linear dispersive equations on torus $\mathbb T$ ?