This paper is concerned with the Cauchy problem of modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove sharp estimate which implies local in time well-posedness Sobolev space $H^s(\mathbb{R}^2)$ for $s \geq 1/4$. $d 3$, by employing $U^p$ and $V^p$ spaces, establish small data global scaling critical $H^{s_c}(\mathbb{R}^d)$ where $s_c = d/2-1$.

The generalized Riccati equation mapping is extended with the basic G′/G -expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the 2 1 -dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equation G′ η w uG η...

We study in this article the stochastic Zakharov-Kuznetsov equation driven by a multiplicative noise. We establish, in space dimensions two and three the global existence of martingale solutions, and in space dimension two the global pathwise uniqueness and the existence of pathwise solutions. New methods are employed to deal with a special type of boundary conditions and to verify the pathwise...