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

TL;DR

This paper establishes the well-posedness, stability, and existence of periodic travelling wave solutions for the regularized Benjamin-Ono equation, and extends the analysis to related equations like the Benjamin-Bona-Mahony equation.

Contribution

It develops a comprehensive stability and existence theory for periodic solutions of the regularized Benjamin-Ono equation, including nonlinear stability results and extensions to related models.

Findings

Existence of smooth periodic travelling wave solutions.

Nonlinear stability of these solutions in the energy space.

Extension of stability results to the Benjamin-Bona-Mahony equation.

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 $H^1_{per}$.