LAUSR

Abstract The Korteweg-de Vries (KdV)-type equations appear during the studies on the cosmic plasmas, planetary atmospheres and oceans. In this paper, via the Lie symmetry method, we investigate a three-coupled KdV system. Lie point symmetry generators and Lie symmetry groups are given. Based on the Lie point symmetry generators, the system is reduced to some ordinary differential systems. Applying the power-series, polynomial, Jacobi elliptic function and (G′/G) expansion methods on those reduced systems, we derive some group-invariant solutions including the explicit power-series, soliton, snoidal-wave and cnoidal-wave solutions. We graphically analyze the solitons and snoidal waves. Besides, we demonstrate the nonlinear self-adjointness of the system, based on which we present the conservation laws according to the Lie point symmetries.