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Local well-posedness and local (in space) regularity results for the complex Korteweg–de Vries equation

Published online by Cambridge University Press:  09 February 2007

Jiahong Wu  and
Juan-Ming Yuan
Jiahong Wu
Affiliation:
Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, USA ([email protected])
Juan-Ming Yuan
Affiliation:
Department of Applied Mathematics, Providence University, 200 Chung-Chi Road, Shalu 433, Taichung Hsien, Taiwan, Republic of China ([email protected])
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Abstract

This paper is concerned with solutions of the complex Korteweg–de Vries (KdV) equation. We achieve two goals. First, we prove local well-posedness results for the complex KdV equation on a line, in a periodic domain and in a finite domain. These results are in line with the local well-posedness theory for the real KdV equation. Second, we establish a rigorous connection between the local (in space) regularity of the real part and that of the imaginary part of any solution to the complex KdV equation. This result partly validates the numerical observation that the real and imaginary parts of a singular solution of the complex KdV equation blow up at the same point and at the same time.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 1 , February 2007 , pp. 203 - 223
Copyright
2007 Royal Society of Edinburgh

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