THE BENJAMIN-ONO AND BBM EQUATIONS AFTER REGULARIZATION

Abstract

The periodic solutions of the Benjamin-Bona-Mahony (BBM) equation and the regularized Benjamin-Ono (rBO) equation demonstrate a characteristic of nonlinear stability under wavelength-sharing perturbations is the subject of this work. This work clarify that these perturbations are stable through analytical analysis, which advances our knowledge of the stability characteristics of these nonlinear wave equations. Further, in periodic and non-periodic (line) situations, this work enhances the global well-posedness conjecture associated with the rBO equation. The difficulties raised in these settings by the Cauchy problem are emphasized a great deal. In particular, we show that the widely used iteration strategy The application of the Duhamel formula to the Cauchy problem associated with the rBO equation proves to be ineffective in resolving such issues with negative Sobolev indices. This result points to a significant flaw in the usual iterative methods for solving these equations, requiring the development of new techniques or frameworks to handle the Cauchy issue when such indices are present. Additionally, the research advances our knowledge of the mathematical frameworks that underlie nonlinear wave equations, particularly with regard to the effects of regularization on stability and well-posedness. The findings have ramifications for theoretical studies as well as real-world applications that include wave phenomena stability and solvability. In summary, this study offers fresh perspectives on the behavior of the rBO and BBM equations and presents a more thorough framework for examining their stability and resolving related Cauchy issues.