We study the stability of spatially periodic solutions to the Kawahara equation, a fifth order, nonlinear partial differential equation. The equation models the propagation of nonlinear water-waves in the long-wavelength regime, for Weber numbers close to 1/3 where the approximate description through the Korteweg - de Vries (KdV) equation breaks down. Beyond threshold, Weber number larger than 1/3, this equation possesses solitary waves just as the KdV approximation. Before threshold, true solitary waves typically do not exist. In this case, the origin is surrounded by a family of periodic solutions and only generalized solitary waves exist which are asymptotic to one of these periodic solutions at infinity. We show that these periodic solutions are spectrally stable at small amplitude.
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Acknowledgment. The authors gratefully acknowledge financial support by DAAD/Procope, Nr. D/0031082 and F/03132UD. Arnd Scheel was partially supported by the National Science Foundation through grant DMS-0203301.
- Dispersive equations
- Periodic solutions