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

Recently,a(1+1)-dimensional displacement shallow water wave system(IDDSWWS) was constructed by applying variational principle of the analytic mechanics under the Lagrange coordinates.However,fluid viscidity is not considered in the IDDSWWS,which is the same as the famous Korteweg-de Vries(KdV) equation.We modify the IDDSWWS 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 IDDSWWS(MIDDSWWS)will degenerate to IDDSWWS.The KdV-Burgers equation and the Abel equation can be derived from the MIDDSWWS.The calculation on symmetry shows that the system is invariant under the GaUlean transformations and the spacetime translations.Two types of exact solutions and some evolution graphs of the MIDDSWWS are proposed.