Dynamics of Partial Differential Equations

Volume 13 (2016)

Number 3

Periodic solutions for a class of one-dimensional Boussinesq systems

Pages: 241 – 261

DOI: https://dx.doi.org/10.4310/DPDE.2016.v13.n3.a3


José R. Quintero (Departamento de Matemáticas, Universidad del Valle, Cali, Colombia)

Alex M. Montes (Departamento de Matemáticas, Universidad del Cauca, Popayán, Colombia)


In this paper we show the local and global well-posedness for the periodic Cauchy problem associated with a special class of 1D Boussinesq systems that emerges in the study of the evolution of long water waves with small amplitude in the presence of surface tension. By a variational approach, we establish the existence of periodic travelling waves. We see that those periodic solutions are characterized as critical points of some functional, for which the existence of critical points follows as a consequence of the Arzela–Ascoli Theorem and the fact that the action functional associated is coercive and is (sequentially) weakly lower semi-continuous in an appropriate set.


Boussinesq systems, well-posedness, variational methods, periodic travelling waves

2010 Mathematics Subject Classification

34B10, 35A01, 35C07, 35Q35

Published 23 June 2016