Kuznetsov independence of variables X and Y means that, for any pair of bounded functions f(X) and g(Y ), E [f(X)g(Y )] = E [f(X)] E [g(Y )], where E [·] denotes interval-valued expectation and denotes interval multiplication. We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that st...

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 $s_c = d/2-1$.

In this study, we have applied the modified exp -expansion function method (MEFM) to obtain exact travelling wave solutions for (3+1) dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and (3+1)-dimensional Jimbo-Miwa (JM) equation. Dark soliton solution, dark-bright hyperbolic solution trigonometric of KZK JM been found by using method. After that, scratched 2D 3D graphs all obtained i...