Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed

•System of coupled Drinfel’d–Sokolov–Wilson equations describes waves travelling horizontally on the seabed.•The killing form is used to get the invariants.•One dimensional optimal sub algebras are classified and analytical solutions are attained via one parameter Lie symmetry analysis.•The solutions are asymptotic, bell-shaped, bright and dark soliton, bright soliton, parabolic, bright and kink, kink, and periodic in nature.