A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane (Q2759065)

The Korteweg-de Vries equation was used, originally, to model long crested, small amplitude, long waves propagating on the surface of water. Now it is often used to model the uni-directional propagation of waves in a variety of other situations. The analysis of this partial differential equation has been principally centred on the associated initial value problem. However, in practical cases there is frequently a need for an analysis of associated initial-boundary value problems. The authors take as a naturally occurring example that which arises when modelling either the effects in a channel of a wave generator mounted at one end or when modelling near shore zone motions generated by waves propagating from deep water.NEWLINENEWLINENEWLINEThe present study improves results of earlier investigations of nonlinear, dispersive wave equations by making extensive use of modern mathematical methods. The classes to which the initial data and the boundary data must belong in order to ensure local well-posedness are determined precisely. Furthermore, the authors establish the classes of initial and boundary data that ensure global well-posedness. Finally, conditions are obtained which indicate when the required solution might be approximated, arbitrarily well, by solving a finite number of linear problems.