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

Abstract

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