Comparison of quarter-plane and two-point boundary value problems: The BBM-equation

The focus of the present study is the BBM equation which models unidirectional propagation of small amplitude long waves in shallow water and other dispersive media. Interest will be turned to the two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of the medium of propagation. The principal new result is an exact theory of convergence of the two-point boundary value problem to the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. The latter problem has been featured in modeling waves generated by a wavemaker in a flume and in describing the evolution of long crested, deep water waves propagating into the near shore zone of large bodies of water. In addition to their intrinsic interest, our results provide justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem.