abbreviate formula_9. in the soliton region formula_10 the solutions splits into a finite linear combination solitons. in the region formula_11 the solution is asymptotically given by a modulated sine function whose amplitude decays like formula_12. in the region formula_13 the solution is asymptotically given by a sum of two modulated sine function as in the previous case. in the region formula_14 the solution decays rapidly.