The complex KdV equation with or without dissipation

Juan-Ming Yuan; Jiahong Wu

doi:10.3934/dcdsb.2005.5.489

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2005, Volume 5, Issue 2: 489-512. Doi: 10.3934/dcdsb.2005.5.489

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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

Received: July 31, 2003

Revised: June 30, 2004

Published: April 2008

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Abstract

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.

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Mathematics Subject Classification: 35B35, 35B45, 35B65, 65M15, 65M50, 65M60.

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