Conservation laws, exact solutions and stability analysis for time-fractional extended quantum Zakharov–Kuznetsov equation

In this paper, we analyze Riemann–Liouville (R-L) time-fractional \((2+1)\) dimensional extended quantum Zakharov–Kuznetsov (EQZK) equation by using the Lie symmetry method which arises in hydrodynamic that describes the nonlinear propagation of the quantum ion-acoustic waves. By using its symmetry, we convert the equation under consideration to a fractional order non-linear ordinary differential equation (ODE). In this reduced ODE, we use a special type of derivative which is known as Erdélyi–Kober (EK) derivative. This enables us to obtain explicit solutions with convergence analysis of the considered problem. By using Ibragimov’s conservation laws theorem, we compute the conservation laws of the problem under investigation. Moreover, by employing the two potent methods explicit power series and (\(\frac{1}{G^\prime }\))-expansion technique, we get the explicit solutions to the problem under discussion. This analysis leads to the derivation of various key findings, including the identification of symmetries, the establishment of similarity reductions involving the EK fractional differential operator, the determination of exact solutions, and the formulation of conservation laws for the considered equation. We have confidence that these remarkable findings can provide valuable insights and contribute to the exploration of additional evolutionary mechanisms associated with the studied equation.