Symmetry analysis for the $2+1$ generalized quantum Zakharov-Kuznetsov equation

Abstract

We solve the group classification problem for the $2+1$ generalized quantum Zakharov-Kuznetsov equation. Particularly we consider the generalized equation $u_{t}+f\left( u\right) u_{z}+u_{zzz}+u_{xxz}=0$, and the time-dependent Zakharov-Kuznetsov equation $u_{t}+\delta \left( t\right) uu_{z}+\lambda \left( t\right) u_{zzz}+\varepsilon \left( t\right) u_{xxz}=0$% . Function $f\left( u\right) $ and $\delta \left( t\right) ,~\lambda \left( t\right) $,~$\varepsilon \left( t\right) $ are determine in order the equations to admit additional Lie symmetries.\ Finally, we apply the Lie invariants to find similarity solutions for the generalized quantum Zakharov-Kuznetsov equation.