We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation which the solutions can be obtained in fixed number forward backward alternating substitutions. The use novel combination implicit explicit time discretizations one-dimensional case Strang splitting method several dimensional case. are described equati...

Via the semi-inverse method, variational principles are established for the Drinfeld-Sokolov-Wilson (DSW) equation. Based on the obtained variational formulations, exact solitary solution and exact singular periodic wave solution can be obtained using the variational approach.