Modified (1+1)-Dimensional Displacement Shallow Water Wave System

Recently, a (1+1)-dimensional displacement shallow water wave system (1DDSWWS) was constructed by applying variational principle of the analytic mechanics under the Lagrange coordinates. However, fluid viscidity is not considered in the 1DDSWWS, which is the same as the famous Korteweg-de Vries (KdV) equation. We modify the 1DDSWWS and add the term related to fluid viscosity to the model by means of dimension analysis. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the modified 1DDSWWS (M1DDSWWS) will degenerate to 1DDSWWS. The KdV-Burgers equation and the Abel equation can be derived from the M1DDSWWS. The calculation on symmetry shows that the system is invariant under the Galilean transformations and the spacetime translations. Two types of exact solutions and some evolution graphs of the M1DDSWWS are proposed.