American Institute of Mathematical Sciences

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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

Jerry L. Bona 1, , Hongqiu Chen 2, , Shu-Ming Sun 3, and  Bing-Yu Zhang 4,

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.
Keywords: Two-point boundary value problem., Quarter-plane problem, KdV equation.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 76B0.
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 and Continuous Dynamical Systems - B, 2007, 7 (3) : 465-495. doi: 10.3934/dcdsb.2007.7.465
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