The effect of dissipation on solutions of the complex KdV equation

It is known that some periodic solutions of the complex KdV equation with smooth initial data blow up in finite time. In this paper, we investigate the effect of dissipation on the regularity of solutions of the complex KdV equation. It is shown here that if the initial datum is comparable to the dissipation coefficient in the L2-norm, then the corresponding solution does not develop any finite-time singularity. The solution actually decays exponentially in time and becomes real analytic as time elapses. Numerical simulations are also performed to provide detailed information on the behavior of solutions in different parameter ranges.