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May  2005, 5(2): 489-512. doi: 10.3934/dcdsb.2005.5.489

The complex KdV equation with or without dissipation

Juan-Ming Yuan 1, and  Jiahong Wu 1,

1. 

Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, United States

Received  August 2003 Revised  July 2004 Published  February 2005

Solutions of the complex KdV equation and the complex KdV- Burgers equation are studied theoretically and numerically. Attention is focused on whether their solutions are regular for all time. This is a difficult issue partially because the conservation laws of the KdV equation no longer yield a priori bounds for its complex-valued solutions in the $L^2$-space. The problem is tackled here on several fronts including investigating how the regularity of the real part is related to that of the imaginary part, studying blow-up of series solutions, and assessing the impact of dissipation. Systematic numerical simulations are performed to complement the theoretical results.
Keywords: regularity., complex KdV equation, complex KdV-Burgers equation.
Mathematics Subject Classification: 35B35, 35B45, 35B65, 65M15, 65M50, 65M6.
Citation: Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489
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