No abstract

In this paper, we investigate the orbital stability of periodic traveling waves for the Kawahara equation. We prove that the periodic traveling wave, under certain conditions, minimizes a convenient functional by using an adaptation of the method developed by Grillakis et al. [J. Funct. Anal. 74, 160–197 (1987)]. The required spectral properties to ensure the orbital stability are obtained by knowing the positiveness of the Fourier transform of the associated periodic wave established by Angulo and Natali [SIAM J. Math. Anal. 40, 1123–1151 (2008)].

In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work [11], in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in [28] to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in [20] and [33] to deduce the orbital stability of the periodic traveling waves in the energy space.2000 Mathematics Subject Classification. 76B25, 35Q51, 35Q53.

In this paper, we present the first result concerning the orbital stability of periodic traveling waves for the modified Kawahara equation. Our method is based on the Fourier expansion of the periodic wave in order to know the behaviour of the nonpositive spectrum of the associated linearized operator around the periodic wave combined with a recent development which significantly simplifies the obtaining of orbital stability results.2000 Mathematics Subject Classification. 76B25, 35Q51, 35Q53.

We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.