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July  2005, 13(4): 921-940. doi: 10.3934/dcds.2005.13.921

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

Jerry Bona 1, , Hongqiu Chen 2, , Shu Ming Sun 3, and  B.-Y. Zhang 4,

1. 

Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago , 851 S. Morgan Street MC 249, Chicago, Illinois 60607-7045
2. 

Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, and Department of Mathematics, Statistics & Computer Science, University of Illinois at Chicago, Chicago, Illinois, United States
3. 

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, United States
4. 

Department of Mathematics, University of Cincinnati, Cincinnati, Ohio, United States

Received  February 2005 Revised  May 2005 Published  August 2005

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.
Keywords: two-point boundary value problems, BBM equation, comparison principles., quarter-plane problems, Nonlinear dispersive wave equations.
Mathematics Subject Classification: Primary: 35Q53, 35Q80, 76B03, 76B15; Secondary: 45G10, 65M99.
Citation: Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.-Y. Zhang. Comparison of quarter-plane and two-point boundary value problems: the BBM-equation. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 921-940. doi: 10.3934/dcds.2005.13.921
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