The question for linear stability of spatially periodic waves for the Boussinesq equation (the cases p = 2, 3) and the Klein-Gordon-Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability respectively), when the perturbations are taken with the same period T . In particular, our results allow us to completely re...

We develop a general theory to treat the linear stability of standing waves of second order in time evolutionary PDEs. We apply these results to several concrete examples: the KleinGordon equation, for which we consider both ground states and certain excited states, the KleinGordon-Zakharov system and the beam equation. We also discuss possible applications to some non-standard ground and excit...

Transverse instabilities correspond to a class of perturbations traveling in a direction transverse to the direction of the basic solitary wave. Solitary waves traveling in one space direction generally come in one-parameter families. We embed them in a two-parameter family and deduce a new geometric condition for transverse instability of solitary waves. This condition is universal in the sens...

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In 1968 V.E. Zakharov derived the Nonlinear Schrödinger equation for the 2D water wave problem in the absence of surface tension, i.e., for the evolution of gravity driven surface water waves, in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. In this paper we give a rigorous proof that the wave packets in the two-dimensional water...

This paper address a new vision for the generalized Mittag-Leffler stability of the fractional differential equations. We mainly focus on a new method, consisting of decomposing a given fractional differential equation into a cascade of many sub-fractional differential equations. And we propose a procedure for analyzing the generalized Mittag-Leffler stability for the given fractional different...

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research the Institute Computer Science, Czestochowa University Technology