225 research outputs found
Uniqueness results for Zakharov-Kuznetsov equation
In this paper we study uniqueness properties of solutions to the
Zakharov-Kuznetsov equation of plasma physic. Given two sufficiently regular
solutions we prove that, if decays fast enough at two
distinct times, then Comment: 33 page
On the support of solutions to the Zakharov-Kuznetsov equation
In this article we prove that sufficiently smooth solutions of the
Zakharov-Kuznetsov equation that have compact support for two different times
are identically zero.Comment: Version of Dec 17/2010 contains simpler proof of Theorem 1.3 and new
references are adde
A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results
We consider the generalized two-dimensional Zakharov-Kuznetsov equation
, where is an integer
number. For we prove local well-posedness in the -based Sobolev
spaces , where is greater than the critical scaling
index . For we also establish a sharp criteria to obtain
global solutions. A nonlinear scattering result in is
also established assuming the initial data is small and belongs to a suitable
Lebesgue space
Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation
This paper addresses well-posedness issues for the initial value problem
(IVP) associated with the generalized Zakharov-Kuznetsov equation, namely,
\{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x \Delta
u+u^ku_x = 0,}\qquad (x,y) \in \mathbb{R}^2, \,\,\,\, t>0, {\displaystyle
u(x,y,0)=u_0(x,y)}. \{array} \right. \{equation*} For , the IVP
above is shown to be locally well-posed for data in ,
. For , local well-posedness is shown to hold for data in
, , where . Furthermore, for
, if and satisfies , then
the solution is shown to be global in . For , if
, , and satisfies , where is the corresponding ground state solution, then
the solution is shown to be global in
Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons
We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK)
equation, a physically relevant high dimensional generalization of the
Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed
KdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptotically
stable in the energy space in the physical region. We also prove that the sum
of well-arranged solitons is stable in the same space. Orbital stability of ZK
solitons is well-known since the work of de Bouard. Our proofs follow the ideas
by Martel and Martel and Merle, applied for generalized KdV equations in one
dimension. In particular, we extend to the high dimensional case several
monotonicity properties for suitable half-portions of mass and energy; we also
prove a new Liouville type property that characterizes ZK solitons, and a key
Virial identity for the linear and nonlinear part of the ZK dynamics, obtained
independently of the mixed KdV-NLS dynamics. This last Virial identity relies
on a simple sign condition, which is numerically tested for the two and three
dimensional cases, with no additional spectral assumptions required. Possible
extensions to higher dimensions and different nonlinearities could be obtained
after a suitable local well-posedness theory in the energy space, and the
verification of a corresponding sign condition.Comment: 61 pages, 10 figures, accepted version including referee comment
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