Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids

We perform symmetry classification of a variable-coefficient combined KdV–mKdV equation. That is, the equation combining the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations, or also known as the Gardner equation. The direct method of group classification is utilized to specify the forms of these time-dependent coefficients.

► Symmetry classification of the variable-coefficient Gardner equation arising in stratified fluids.
► Time-dependent coefficients are specified using the direct method.
► The obtained symmetry Lie algebra can be utilized to derive physical solutions in fluid stratification.