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This work mainly focuses on the spatial decay properties of solutions to theZakharov-Kuznetsov equation. In earlier studies for the two- andthree-dimensional cases, it was established that if the initial condition $u_0$verifies $\langle \sigma\cdot x\rangle^{r}u_{0}\in L^{2}(\left\{\sigma\cdotx\geq \kappa\right\}),$ for some $r\in\mathbb{N}$, $\kappa \in\mathbb{R}$,being $\sigma$ be a suitable non-null vector in the Euclidean space, then thecorresponding solution $u(t)$ generated from this initial condition verifies$\langle \sigma\cdot x\rangle ^{r}u(t)\in L^2\left(\left\{\sigma\cdotx>\kappa-\nu t\right\}\right)$, for any $\nu >0$. In this regard, we firstextend such results to arbitrary dimensions, decay power $r>0$ not necessarilyan integer, and we give a detailed description of the gain of regularitypropagated by solutions in terms of the magnitude of the weight $r$. Thededuction of our results depends on a new class of pseudo-differentialoperators, which is useful to quantify decay and smoothness properties on afractional scale. Secondly, we show that if the initial data $u_{0}$ has adecay of exponential type on a particular half space, that is, $e^{b\,\sigma\cdot x}u_{0}\in L^{2}(\left\{\sigma\cdot x\geq \kappa\right\}),$ thenthe corresponding solution satisfies $e^{b\, \sigma\cdot x} u(t)\inH^{p}\left(\left\{\sigma\cdot x>\kappa-\nu t\right\}\right),$ for all$p\in\mathbb{N}$, and time $t\geq \delta,$ where $\delta>0$. To our knowledge,this is the first study of such property. As a further consequence, we alsoobtain well-posedness results in anisotropic weighted Sobolev spaces inarbitrary dimensions.  Finally, as a by-product of the techniques considered here, we show that ourresults are also valid for solutions of the Korteweg-de Vries equation.

On decay properties for solutions of the Zakharov-Kuznetsov equation