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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 137; no. 1; pp. 203 - 223
• Main Authors: Wu, Jiahong, Yuan, Juan-Ming
• Format: Journal Article
• Language: English
• Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.02.2007; Cambridge University Press
• Subjects: Mathematics; Theory
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/S0308210505000946

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.