Dynamics of internal waves in a stratified ocean modeled by the extended Gardner equation with time-dependent coefficients

•Extended Gardner equation with time-dependent coefficients.•Painlev'e analysis is also performed to check integrability.•Lie classical method is applied to investigate the symmetries.•Some physically important exact solutions are derived.

In this paper, we have considered the variable-coefficients Gardner (vc-Gardner) equation. We have check the Painlevé properties of vc-Gardner by using the Painlevé analysis and the symmetries for the equations are obtained using Lie group analysis. The exact solutions obtained from the symmetries and Painlevé analysis are presented.