The large-time development of the solution to an initial–boundary value problem for the Korteweg–de Vries equation on the negative quarter-plane

In this paper, we consider an initial–boundary value problem for the Korteweg–de Vries equation on the negative quarter-plane. The normalized Korteweg–de Vries equation considered is given byuτ+uux+uxxx=0,x<0,τ>0, where x and τ   represent dimensionless distance and time, respectively. In particular, we consider the case when the initial and boundary conditions are given by u(x,0)=uiu(x,0)=ui for x<0x<0 and u(0,τ)=ubu(0,τ)=ub, ∂∂xu(0,τ)=ubx for τ>0τ>0. Here the initial value ui<0ui<0 and we restrict attention to boundary values ubub and ubxubx in the ranges 0