The large-time development of the solution to an initial–boundary value problem for the Korteweg–de Vries equation. I. Steady state solutions

In this paper, we consider an initial–boundary value problem for the Korteweg–de Vries equation on the positive quarter-plane. The normalized Korteweg–de Vries equation considered is given byuτ+uux+uxxx=0,00, 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 00τ>0 respectively. Here the initial value ui⩽0ui⩽0 and we restrict attention to values of the boundary value, ubub, for which ub⩽−2uiub⩽−2ui when ui<0ui<0 and ub<0ub<0 when ui=0ui=0. We consider the three cases (ui<0ui<0, ui