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.