2009/04/29 by Angulo, Jaime, M. Scialom, Scialom, Marcia +2
Mathematics · Physics and Astronomy · #35B35 #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.0904.4623
openalex publication_date 2009/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 H1/2per 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 H1per.