arXiv subjects
Publications and source records attributed to Giovana Alves.
In this paper, we establish the monotonicity of the period map in terms of the energy levels for certain periodic solutions of the equation $-φ''+φ-φ^{k}=0$, where $k>1$ is a real number. We present a new approach to demonstrate this property, utilizing spectral information of the corresponding linearized operator around the periodic solution and tools related to Floquet theory.
In this paper, we prove existence and orbital stability results of periodic standing waves for the cubic-quintic nonlinear Schrödinger equation. We use the implicit function theorem to construct a smooth curve of explicit periodic waves with \textit{dnoidal} profile and such construction can be used to prove that the associated period map is strictly increasing in terms of the energy levels. The monotonicity is also useful to obtain the behaviour of the non-positive spectrum for the associated linearized operator around the wave. Concerning the stability, we prove that the dnoidal waves are orbitally stable in the energy space restricted to the even functions.
The present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves. Our method can be seen as an extension to spatially periodic waves of the theory of solitary waves recently developed in \cite{st}. Firstly, our main result do not depend on the parametrization of the periodic wave itself. Secondly, motived by the well known orbital stability criterion for solitary waves, we show that the same criterion holds for periodic waves. In addition, we show that the positiveness of the principal entries of the Hessian matrix related to the "energy surface function" are also sufficient to obtain the stability. Consequently, we can establish the orbital stability of periodic waves for several nonlinear dispersive models. We believe our method can be applied in a wide class of evolution equations; in particular it can be extended to regularized dispersive wave equations.