No abstract

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely,For 2 k 7, the IVP above is shown to be locally well posed for data in H s (R 2 ), s > 3/4. For k 8, local well-posedness is shown to hold for data in H s (R 2 ), s > s k , where s k = 1 − 3/(2k − 4). Furthermore, for k 3, if u 0 ∈ H 1 (R 2 ) and satisfies u 0 H 1 1, then the solution is shown to be global in H 1 (R 2 ). For k = 2, if u 0 ∈ H s (R 2 ), s > 53/63, and satisfies u 0 L 2 < √ 3 ϕ L 2 , where ϕ is the corresponding ground state solution, then the solution is shown to be global in H s (R 2 ).

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 [27]. 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.2000 Mathematics Subject Classification. 76B25, 35Q51, 35Q53.

We establish the well-posedness of two Cauchy problems for the two-dimensional Zakharov-Kuznetsov equation motivated by the study of transverse stability properties of the N -soliton N of the Korteweg-de Vries equation. They differ by the functional setting: Sobolev spaces H s × s > 3 2 for the first one, N + H 1 2 for the second one. In the latter case, the solution is shown to be global.

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.