The effect of dissipation on solutions of the complex KdV equation (Q2567261)

It is well-known that there exists a smooth initial datum \(u_0\) such that the solution of the initial value problem for the complex Korteweg-de Vries (KdV) equation with a periodic boundary condition \[ u_t + 2 uu_x + \delta u_{xxx} = 0, u(x,t) = u(x+1,t), u(x,0) = u_0(x) \] with \(\delta > 0\) and \(u=u(x,t)\) being complex-valued blows up in a finite time. The authors conduct a theoretical and numerical study to investigate the effects of dissipation on the regularity of solutions of the complex KdV equation. Namely, they consider the dissipative complex KdV equation of the form \[ u_t + 2 uu_x + \delta u_{xxx} + \nu (-\Delta)^\alpha u = 0, u(x,t) = u(x+1,t), \] where \(\nu > 0\) and \((-\Delta)^\alpha\) is a Fourier multiplier operator. The authors prove that if \(\nu\) and the initial datum \(u_0\) satisfy for some constant \(C\) the inequality \(\nu \geq \| u_0\| _{L^2}\), then the \(L^2-\)norm of the solution of the above dissipative complex KdV equation with \(u(x,0) = u_0\) is bounded uniformly for all time. Furthermore, the solution decays exponentially in time and becomes real analytic for large time. Numerical simulations are also performed to provide detailed information on the behavior of solutions in different parameter ranges.