Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior

We consider the subcritical generalized Korteweg–de Vries equationut+(uxx+u4)x=0,t,x∈R. Let Rj(t,x)=Qcj(x−xj−cjt)Rj(t,x)=Qcj(x−xj−cjt) (j=1,…,Nj=1,…,N) be N   soliton solutions to this equation. Denote U(t)U(t) the KdV linear group, and let V be in an adequate weighted Sobolev space.We construct a solution u(t)u(t) to the generalized Korteweg–de Vries equation such thatlimt→∞‖u(t)−U(t)V−∑j=1NRj(t)‖H1=0.