# American Institute of Mathematical Sciences

May  2007, 7(3): 465-495. doi: 10.3934/dcdsb.2007.7.465

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

 1 Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States 2 Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, 38152, United States 3 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, United States 4 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, United States

Received  October 2006 Revised  January 2007 Published  February 2007

This paper is concerned with the Korteweg-de Vries equation which models unidirectional propagation of small amplitude long waves in dispersive media. The two-point boundary value problem wherein the wave motion is specified at both ends of a finite stretch of length $L$ of the media of propagation is considered. It is shown that the solution of the two-point boundary value problem converges as $L\rightarrow +\infty$ to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch of the medium is disturbed at its finite end. In addition to its intrinsic interest, our result provides justification for the use of the two-point boundary value problem in numerical studies of the quarter plane problem for the KdV equation.
Citation: Jerry L. Bona, Hongqiu Chen, Shu-Ming Sun, Bing-Yu Zhang. Comparison of quarter-plane and two-point boundary value problems: The KdV-equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 465-495. doi: 10.3934/dcdsb.2007.7.465
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