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This paper has various goals: first, we develop a local and globalwell-posedness theory for the regularized Benjamin-Ono equation in the periodicsetting, second, we show that the Cauchy problem for this equation (in bothperiodic and non-periodic case) cannot be solved by an iteration scheme basedon the Duhamel formula for negative Sobolev indices, third, a proof of theexistence of a smooth curve of periodic travelling wave solutions, for theregularized Benjamin-Ono equation, with fixed minimal period 2L, is given. Itis also shown that these solutions are nonlinearly stable in the energy space$H^{1/2}_{per}$ by perturbations of the same wavelength. Finally, an extensionof the theory developed for the regularized Benjamin-Ono equation is given andas an example it is proved that the cnoidal wave solutions associated to theBenjamin-Bona-Mahony equation are nonlinearly stable in $H^1_{per}$.

Nonlinear Stability of Periodic Travelling Wave Solutions for the Regularized Benjamin-Ono and BBM Equations