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

TL;DR

This paper performs a symmetry analysis of the 2+1 generalized quantum Zakharov-Kuznetsov equation, classifies its Lie symmetries, and derives similarity solutions, enhancing understanding of its mathematical structure and solution space.

Contribution

It provides a comprehensive group classification for the generalized quantum Zakharov-Kuznetsov equation and constructs similarity solutions using Lie invariants.

Findings

Classified Lie symmetries for various forms of the equation.

Derived explicit similarity solutions.

Identified conditions on functions for additional symmetries.

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.