Fábio Natali

Positivity Properties of the Fourier Transform and the Stability of Periodic Travelling-Wave Solutions

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg–de Vries-type

The Fourth-Order Dispersive Nonlinear Schrodinger Equation: Orbital Stability of a Standing Wave ∗

u_t+u^pu_x-Mu_x=0

A note on the stability for Kawahara–KdV type equations

, with M being a general pseudodifferential operator and where

(Non)linear instability of periodic traveling waves: Klein-Gordon and KdV type equations

p\geq1

Stability properties of periodic traveling waves for the intermediate long wave equation

is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obtain the stability of a family of periodic travelling waves solutions for the Benjamin–Ono equation. The present technique gives a new way to obtain the existence and stability of cnoidal and dnoidal waves solutions associated with the Korteweg–de Vries and modified Korteweg–de Vries equations, respectively. The theory has prospects for the study of periodic travelling-wave solutions of other partial differential equations.

Orbital stability of periodic traveling wave solutions for the Kawahara equation

In this paper we consider the one-dimensional fourth-order dispersive cubic nonlinear Schrodinger equation with mixed dispersion. Orbital stability, in the energy space, of a particular standing-wave solution is proved in the context of Hamiltonian systems. The main result is established by constructing a suitable Lyapunov function.

Stability and instability of periodic travelling wave solutions for the critical Korteweg–de Vries and nonlinear Schrödinger equations

Abstract In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara–KdV equation u t + u u x + u x x x − γ 1 u x x x x x = 0 , and the modified Kawahara–KdV equation u t + 3 u 2 u x + u x x x − γ 2 u x x x x x = 0 , where γ i ∈ R is a positive number when i = 1 , 2 . The main approach used to determine the stability of solitary traveling waves will be the theory developed by Albert (1992) in [9] .

Stability and instability of periodic standing wave solutions for some Klein–Gordon equations

Abstract. We prove the existence and nonlinear instability of periodic traveling wave solutions for the critical one-dimensional Klein–Gordon equation. We also establish a linear instability criterium for a KdV type system. An application of this approach is made to obtain the linear/nonlinear instability of vector cnoidal wave profiles. Finally, via a theoretical and numerical approach we show the linear stability or instability of periodic positive and sign changing waves, respectively, for the critical Korteweg–de Vries equation.

Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization

In this paper, we determine orbital and linear stability of periodic waves with the mean zero property related to the Intermediate Long Wave equation. Our arguments follow the recent developments in \cite{andrade-pastor}, \cite{natali} and \cite{DK} to deduce the orbital/linear stability of periodic traveling waves.

Exponential stability for the

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)].