The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. Cauchy problem shown to be globally well-posed for small initial data critical spaces and it proved that solutions scatter free as $t \to \pm \infty$. proof based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $(d-1)$-dimensional Schr\"odinger equation, ii) transversal bilinear rest...

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...

Abstract We propose a new modification of homotopy perturbation method (HPM) called the δ -homotopy transform ( -HPTM). This consists Laplace method, HPM, and control parameter . convergence in this helps adjusting controlling region series solution overcome some limitations HPM HPTM. The -HPTM q-homotopy analysis (q-HATM) are considered to study generalized time-fractional perturbed $(3+1)$ <m...

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...