We present a detailed numerical study of solutions to the Zakharov–Kuznetsov equation in three spatial dimensions. The is three-dimensional generalization Korteweg–de Vries equation, though, not completely integrable. This L2-subcritical, and thus, exist globally, for example, H1 energy space. first stability solitons with various perturbations sizes symmetry, show asymptotic formation radiatio...

This paper is concerned with the Cauchy problem of $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal up to endpoint. utilize nonlinear version classical Loomis-Whitney inequality develop an almost orthogonal decomposition set resonant frequencies. As a corollary, we obt...

In this paper, we propose a class of adaptive multiresolution (also called the sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including Korteweg--de Vries (KdV) equation and its two-dimensional generalization, Zakharov--Kuznetsov (ZK) equation. The UWDG formulation, which relies on repeated integration by parts, was proposed Kd...

We consider a model that describes electromigration in nanoconductors known as surface (SEM) equation. Our purpose here is to establish local well-posedness for the associated initial value problem Sobolev spaces from two different points of view. In first one, we study pure Cauchy and $H^s(\mathbb{R}^2)$, $s>1/2$. second on background Korteweg-de Vries solitary traveling wave less regular spac...

In this paper we study the initial-value problem associated with dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation,ut+Dxa+1∂xu+uxyy+uux=0,a∈(0,1). More specifically, persistence property of solution in weighted anisotropic Sobolev spacesH(1+a)s,2s(R2)∩L2((x2r1+y2r2)dxdy), for appropriate s, r1 and r2. By establishing unique continuation properties also show that our results are sh...